Risk Aversion Asymptotics for Power Utility
Maximization
Marcel Nutz
ETH Zurich, Department of Mathematics, 8092 Zurich, Switzerland
[email protected]
This Version: March 16, 2010.
Abstract
We consider the economic problem of optimal consumption and investment with power utility. We study the optimal strategy as the
relative risk aversion tends to innity or to one. The convergence of
the optimal consumption is obtained for general semimartingale models while the convergence of the optimal trading strategy is obtained
for continuous models. The limits are related to exponential and logarithmic utility. To derive these results, we combine approaches from
optimal control, convex analysis and backward stochastic dierential
equations (BSDEs).
Keywords power utility, risk aversion asymptotics, opportunity process, BSDE.
AMS 2000 Subject Classications Primary 91B28; secondary 93E20, 60G44.
JEL Classication G11, C61.
Acknowledgements.
Financial support by Swiss National Science Founda-
tion Grant PDFM2-120424/1 is gratefully acknowledged. The author thanks
Freddy Delbaen and Semyon Malamud for discussions and Martin Schweizer
for comments on the draft.
1
Introduction
This paper considers the maximization of expected utility, a classical problem of mathematical nance. The agent obtains utility from the wealth he
possesses at some given time horizon
𝑇 ∈ (0, ∞) and, in an alternative case,
𝑇 . More specically, we study
also from intermediate consumption before
preferences given by power utility random elds for an agent who can invest
in a nancial market which is modeled by a general semimartingale. We defer
the precise formulation to the next section to allow for a brief presentation
𝑈 (𝑝) (𝑥) = 𝑝1 𝑥𝑝 , where
there exists for each 𝑝
of the contents and focus on the power utility function
𝑝 ∈ (−∞, 0) ∪ (0, 1).
Under standard assumptions,
an optimal trading and consumption strategy that maximizes the expected
1
𝑈 (𝑝) . Our main interest concerns the behavior of
these strategies in the limits 𝑝 → −∞ and 𝑝 → 0.
(𝑝) tends to innity for 𝑝 → −∞. Hence
The relative risk aversion of 𝑈
utility corresponding to
economic intuition suggests that the agent should become reluctant to take
risks and, in the limit, not invest in the risky assets. Our rst main result
conrms this intuition. More precisely, we prove in a general semimartingale
model that the optimal consumption, expressed as a proportion of current
wealth, converges pointwise to a deterministic function. This function corresponds to the consumption which would be optimal in the case where
trading is not allowed. In the continuous semimartingale case, we show that
the optimal trading strategy tends to zero in a local
𝐿2 -sense
and that the
corresponding wealth process converges in the semimartingale topology.
Our second result pertains to the same limit
𝑝 → −∞
but concerns the
problem without intermediate consumption. In the continuous case, we show
that the optimal trading strategy scaled by
1−𝑝 converges to a strategy which
is optimal for exponential utility. We provide economic intuition for this fact
via a sequence of auxiliary power utility functions with shifted domains.
The limit
𝑝→0
is related to the logarithmic utility function. Our third
main result is the convergence of the corresponding optimal consumption for
the general semimartingale case, and the convergence of the trading strategy
and the wealth process in the continuous case.
All these results are readily observed for special models where the optimal
strategies can be calculated explicitly.
While the corresponding economic
intuition extends to general models, it is
a priori unclear how to go about
get our hands on the optimal
proving the results. Indeed, the problem is to
controls, which is a notorious question in stochastic optimal control.
Our main tool is the so-called opportunity process, a reduced form of
the value process in the sense of dynamic programming. We prove its convergence using control-theoretic arguments and convex analysis. On the one
hand, this yields the convergence of the value function. On the other hand,
we deduce the convergence of the optimal consumption, which is directly related to the opportunity process. The optimal trading strategy is also linked
to this process, by the so-called Bellman equation.
We study the asymp-
totics of this backward stochastic dierential equation (BSDE) to obtain the
convergence of the strategy. This involves nonstandard arguments to deal
with nonuniform quadratic growth in the driver and solutions that are not
locally bounded.
To derive the results in the stated generality, it is important to
combine
ideas from optimal control, convex analysis and BSDE theory rather than to
rely on only one of these ingredients; and one may see the problem at hand
as a
model problem of control
in a semimartingale setting.
The paper is organized as follows.
In the next section, we specify the
optimization problem in detail. Section 3 summarizes the main results on
the risk aversion asymptotics of the optimal strategies and indicates connec-
2
tions to the literature. Section 4 introduces the main tools, the opportunity
process and the Bellman equation, and explains the general approach for the
proofs.
In Section 5 we study the dependence of the opportunity process
𝑝 and establish some related estimates. Sections 6 deals with the limit
𝑝 → −∞; we prove the main results stated in Section 3 and, in addition, the
on
convergence of the opportunity process and the solution to the dual problem
(in the sense of convex duality). Similarly, Section 7 contains the proof of
the main theorem for
2
𝑝→0
and additional renements.
Preliminaries
𝑥, 𝑦 ∈ ℝ are reals, 𝑥 ∧ 𝑦 = min{𝑥, 𝑦}
𝑑
and 𝑥 ∨ 𝑦 = max{𝑥, 𝑦}. We use 1/0 := ∞ where necessary. If 𝑧 ∈ ℝ
𝑖
⊤
is a 𝑑-dimensional vector, 𝑧 is its 𝑖th coordinate, 𝑧
its transpose, and
∣𝑧∣ = (𝑧 ⊤ 𝑧)1/2 the Euclidean norm. If 𝑋 is an ℝ𝑑 -valued semimartingale
𝑑
and 𝜋 is an ℝ -valued predictable integrand, the vector stochastic integral,
∫
denoted by
𝜋 𝑑𝑋 or 𝜋 ∙ 𝑋 , is a scalar semimartingale with initial value
The following notation is used.
If
zero. Relations between measurable functions hold almost everywhere unless
otherwise mentioned. Dellacherie and Meyer [8] and Jacod and Shiryaev [17]
are references for unexplained notions from stochastic calculus.
2.1
The Optimization Problem
We consider a xed time horizon 𝑇 ∈ (0, ∞) and a ltered probability space
(Ω, ℱ, 𝔽 = (ℱ𝑡 )𝑡∈[0,𝑇 ] , 𝑃 ) satisfying the usual assumptions of right-continuity
𝑑
and completeness, as well as ℱ0 = {∅, Ω} 𝑃 -a.s. Let 𝑅 be an ℝ -valued càdlàg
semimartingale with 𝑅0 = 0. Its components are interpreted as the returns
1
𝑑
of 𝑑 risky assets and the stochastic exponential 𝑆 = (ℰ(𝑅 ), . . . , ℰ(𝑅 ))
represents their prices. Let M be the set of equivalent 𝜎 -martingale measures
for 𝑆 . We assume
M ∕= ∅,
(2.1)
so that arbitrage is excluded in the sense of the NFLVR condition (see Delbaen and Schachermayer [7]).
Our agent also has a bank account at his
disposal. As usual in mathematical nance, the interest rate is assumed to
be zero.
The agent is endowed with a deterministic initial capital
ing strategy
𝑥0 > 0. A trad𝜋 , where 𝜋 𝑖 is
𝑑
is a predictable 𝑅-integrable ℝ -valued process
interpreted as the fraction of the current wealth (or the portfolio proportion)
invested in the
𝑐≥0
such that
𝑖∫th
𝑇
0
risky asset. A
consumption rate
𝑐𝑡 𝑑𝑡 < ∞ 𝑃 -a.s.
is an optional process
We want to consider two cases simulta-
neously: Either consumption occurs only at the terminal time
𝑇
(utility from
terminal wealth only); or there is intermediate and a bulk consumption at
3
the time horizon. To unify the notation, we dene the measure
{
0
𝜇(𝑑𝑡) :=
𝑑𝑡
𝜇
on
[0, 𝑇 ],
in the case without intermediate consumption,
in the case with intermediate consumption.
𝜇∘ := 𝜇 + 𝛿{𝑇 } , where 𝛿{𝑇 } is the unit Dirac measure at 𝑇 .
process 𝑋(𝜋, 𝑐) of a pair (𝜋, 𝑐) is dened by the linear equation
Moreover, let
The
wealth
𝑡
∫
∫
𝑋𝑠− (𝜋, 𝑐)𝜋𝑠 𝑑𝑅𝑠 −
𝑋𝑡 (𝜋, 𝑐) = 𝑥0 +
0
The set of
admissible
𝑐𝑠 𝜇(𝑑𝑠),
0 ≤ 𝑡 ≤ 𝑇.
0
trading and consumption pairs is
{
𝒜(𝑥0 ) = (𝜋, 𝑐) : 𝑋(𝜋, 𝑐) > 0
The convention
𝑡
𝑐𝑇 = 𝑋𝑇 (𝜋, 𝑐)
and
}
𝑐𝑇 = 𝑋𝑇 (𝜋, 𝑐) .
is merely for notational convenience and
𝑇 . We x the
𝒜 for 𝒜(𝑥0 ). Moreover, 𝑐 ∈ 𝒜 indicates
(𝜋, 𝑐) ∈ 𝒜; an analogous convention is used for
means that all the remaining wealth is consumed at time
initial capital
𝑥0
that there exists
and usually write
𝜋
such that
similar expressions.
It will be convenient to parametrize the consumption strategies as fractions of the wealth. Let
(𝜋, 𝑐) ∈ 𝒜 and let 𝑋 = 𝑋(𝜋, 𝑐) be the corresponding
wealth process. Then
𝜅 :=
propensity to consume
is called the
𝑐
𝑋
(𝜋, 𝑐). In general, a
∫𝑇
𝜅 ≥ 0 such that 0 𝜅𝑠 𝑑𝑠 < ∞
parametrizations by 𝑐 and by 𝜅 are equivalent (see
and we abuse the notation by identifying 𝑐 and 𝜅
corresponding to
propensity to consume is an optional process
𝑃 -a.s.
and
𝜅𝑇 = 1.
The
Nutz [27, Remark 2.1])
when
𝜋
is given. Note that the wealth process can be expressed as
(
)
𝑋(𝜋, 𝜅) = 𝑥0 ℰ 𝜋 ∙ 𝑅 − 𝜅 ∙ 𝜇 .
(2.2)
The preferences of the agent are modeled by a random utility function with constant relative risk aversion. More precisely, let
adapted positive process and x
𝑝 ∈ (−∞, 0) ∪ (0, 1).
𝐷
be a càdlàg
We dene the utility
random eld
(𝑝)
𝑈𝑡 (𝑥) := 𝑈𝑡 (𝑥) := 𝐷𝑡 𝑝1 𝑥𝑝 ,
𝑥 ∈ (0, ∞), 𝑡 ∈ [0, 𝑇 ],
where we assume that there are constants
𝑘1 ≤ 𝐷𝑡 ≤ 𝑘2 ,
The process
𝐷
0 < 𝑘1 ≤ 𝑘2 < ∞
0 ≤ 𝑡 ≤ 𝑇.
𝑝
4
in
such that
(2.4)
𝑝; interpretations are discussed
𝑈 (𝑝) will sometimes be suppressed
is taken to be independent of
in [27, Remark 2.2]. The parameter
(2.3)
in the notation and made explicit when we want to recall the dependence.
The same applies to other quantities in this paper.
The constant
expected
utility
[∫
𝑇
0
𝐸
1−𝑝 > 0
is called the
relative risk aversion
of 𝑈 . The
𝑐
∈
𝒜
is given by
∫𝑇
𝐸[ 0 𝑈𝑡 (𝑐𝑡 ) 𝑑𝑡 + 𝑈𝑇 (𝑐𝑇 )].
corresponding to a consumption rate
𝑈𝑡 (𝑐𝑡 ) 𝜇∘ (𝑑𝑡)
]
𝐸[𝑈𝑇 (𝑐𝑇 )]
, which is either
or
We will always assume that the optimization problem is nondegenerate, i.e.,
𝑢𝑝 (𝑥0 ) := sup 𝐸
𝑐∈𝒜(𝑥0 )
𝑇
[∫
0
]
(𝑝)
𝑈𝑡 (𝑐𝑡 ) 𝜇∘ (𝑑𝑡) < ∞.
(2.5)
𝑝, but not on 𝑥0 . Note that
𝑢𝑝 (𝑥0 ) < ∞ for any 𝑝 < 𝑝0 ; and for 𝑝 < 0 the condi(𝑝) < 0. A strategy (𝜋, 𝑐) ∈ 𝒜(𝑥 ) is optimal if
tion (2.5) is void since then 𝑈
0
[∫𝑇
]
∘
𝐸 0 𝑈𝑡 (𝑐𝑡 ) 𝜇 (𝑑𝑡) = 𝑢(𝑥0 ). Note that 𝑈𝑡 is irrelevant for 𝑡 < 𝑇 when there
This condition depends on the choice of
𝑢𝑝0 (𝑥0 ) < ∞
implies
is no intermediate consumption. We recall the following existence result.
For each 𝑝, if 𝑢𝑝 (𝑥0 ) < ∞,
there exists an optimal strategy (ˆ𝜋, 𝑐ˆ) ∈ 𝒜. The corresponding wealth process
ˆ = 𝑋(ˆ
𝑋
𝜋 , 𝑐ˆ) is unique. The consumption rate 𝑐ˆ can be chosen to be càdlàg
and is unique 𝑃 ⊗ 𝜇∘ -a.e.
Proposition 2.1 (Karatzas and šitkovi¢ [20]).
ˆ = 𝑋(ˆ
𝑐ˆ denotes this càdlàg version, 𝑋
𝜋 , 𝑐ˆ) is the
ˆ is the optimal propensity to consume.
and 𝜅
ˆ = 𝑐ˆ/𝑋
In the sequel,
wealth process
2.2
Decompositions and Spaces of Processes
In some of the statements, we will assume that the price process
alently
that
optimal
𝑅
𝑅) is continuous.
satises the
𝑆
(or equiv-
In this case, it follows from (2.1) and Schweizer [31]
structure condition, i.e.,
∫
𝑅=𝑀+
𝑀
Let 𝜉
where
is a continuous local martingale with
(2.6)
𝑀0 = 0
and
𝜆 ∈ 𝐿2𝑙𝑜𝑐 (𝑀 ).
be a scalar special semimartingale, i.e., there exists a (unique)
canonical decomposition
martingale,
continuous,
to
𝑑⟨𝑀 ⟩𝜆,
𝜉 = 𝜉0 + 𝑀 𝜉 + 𝐴𝜉 ,
where
𝐴𝜉 is predictable of nite variation,
𝑀 𝜉 has a Kunita-Watanabe (KW)
and
𝜉0 ∈ ℝ, 𝑀 𝜉 is a local
𝑀0𝜉 = 𝐴𝜉0 = 0. As 𝑀 is
decomposition with respect
𝑀,
𝜉 = 𝜉0 + 𝑍 𝜉 ∙ 𝑀 + 𝑁 𝜉 + 𝐴𝜉 ,
where
[𝑀 𝑖 , 𝑁 𝜉 ] = 0
Stricker [1,
cas
for
1 ≤ 𝑖 ≤ 𝑑
and
𝑍 𝜉 ∈ 𝐿2𝑙𝑜𝑐 (𝑀 );
(2.7)
see Ansel and
3]. Analogous notation will be used for other special semi-
martingales and, with a slight abuse of terminology, we will refer to (2.7) as
the KW decomposition of
𝜉.
5
𝒮
𝑃 -semimartingales and 𝑟 ∈ [1, ∞). If
𝑋 = 𝑋0 + 𝑀 𝑋 + 𝐴𝑋 , we dene
∫ 𝑇
1/2 ∥𝑋∥ℋ𝑟 := ∣𝑋0 ∣ + 0 ∣𝑑𝐴𝑋 ∣𝐿𝑟 + [𝑀 𝑋 ]𝑇 𝐿𝑟 .
[
]
2
In particular, we will often use that ∥𝑁 ∥ 2 = 𝐸 [𝑁 ]𝑇 for a local martingale
ℋ
𝑁 with 𝑁0 = 0. If 𝑋 is a non-special semimartingale, ∥𝑋∥ℋ𝑟 := ∞. We
𝑟
can now dene ℋ := {𝑋 ∈ 𝒮 : ∥𝑋∥ℋ𝑟 < ∞}. The same space is some𝑟
times denoted by 𝒮 in the literature; moreover, there are many equivalent
𝑟
𝑟
denitions for ℋ (see [8, VII.98]). The localized spaces ℋ𝑙𝑜𝑐 are dened in
𝑛
𝑟
the usual way. In particular, if 𝑋, 𝑋
∈ 𝒮 we say that 𝑋 𝑛 → 𝑋 in ℋ𝑙𝑜𝑐
if there exists a localizing sequence of stopping times (𝜏𝑚 )𝑚≥1 such that
lim𝑛 ∥(𝑋 𝑛 − 𝑋)𝜏𝑚 ∥ℋ𝑟 = 0 for all 𝑚. The localizing sequence may depend
𝑛
on the sequence (𝑋 ), causing this convergence to be non-metrizable. On
𝒮 , the Émery distance is dened by
[
]
𝑑(𝑋, 𝑌 ) := ∣𝑋0 − 𝑌0 ∣ + sup 𝐸 sup 1 ∧ ∣𝐻 ∙ (𝑋 − 𝑌 )𝑡 ∣ ,
Let
𝑋∈𝒮
be the space of all càdlàg
has the canonical decomposition
∣𝐻∣≤1
𝑡∈[0,𝑇 ]
where the supremum is taken over all predictable processes bounded by one
in absolute value.
topology
𝒮
This complete metric induces on
(cf. Émery [9]).
An optional process
𝑋
satises a certain property
the
semimartingale
prelocally
if there ex-
𝜏𝑚 such that 𝑋 𝜏𝑚 − := 𝑋1[0,𝜏𝑚 ) +
each 𝑚. When 𝑋 is continuous, pre-
ists a localizing sequence of stopping times
𝑋𝜏𝑚 − 1[𝜏𝑚 ,𝑇 ]
satises this property for
local simply means local.
Let 𝑋, 𝑋 𝑛 ∈ 𝒮 and 𝑟 ∈ [1, ∞). Then 𝑋 𝑛 → 𝑋 in
the semimartingale topology if and only if every subsequence of (𝑋 𝑛 ) has a
subsequence which converges to 𝑋 prelocally in ℋ𝑟 .
Proposition 2.2 ([9]).
We denote by
𝐵𝑀 𝑂
∥𝑁 ∥2𝐵𝑀 𝑂
where
𝜏
norm).
𝑁 with 𝑁0 = 0
]
[
:= sup 𝐸 [𝑁 ]𝑇 − [𝑁 ]𝜏 − ℱ𝜏 ∞ < ∞,
the space of martingales
satisfying
𝐿
𝜏
𝐵𝑀 𝑂2 ℋ𝜔 be the
𝑋0 = 0 and
ranges over all stopping times (more precisely, this is the
There exists a similar notion for semimartingales:
ℋ1 consisting of all special semimartingales 𝑋 with
]
[(
)1/2 ∫ 𝑇
:= sup 𝐸 [𝑀 𝑋 ]𝑇 − [𝑀 𝑋 ]𝜏 −
+ 𝜏 − ∣𝑑𝐴𝑋 ∣ ℱ𝜏 subspace of
∥𝑋∥2ℋ𝜔
let
Finally, let
𝐿∞
𝜏
ℛ𝑟
< ∞.
be the space of scalar adapted processes which are right-
continuous and such that
∥𝑋∥ℛ𝑟 := sup ∣𝑋𝑡 ∣
0≤𝑡≤𝑇
𝐿𝑟
< ∞.
With a mild abuse of notation, we will use the same norm also for leftcontinuous processes.
6
3
Main Results
In this section we present the main results about the limits of the optimal
strategies. To state an assumption in the results, we rst have to introduce
opportunity process 𝐿(𝑝);
the
this is a reduced form of the value process in
the language of dynamic programming. Fix
𝑝
such that
𝑢𝑝 (𝑥0 ) < ∞.
Using
the scaling properties of our utility function, we can show that there exists
a unique càdlàg semimartingale
𝐿𝑡 (𝑝)
for all
1
𝑝
(
𝐿(𝑝)
[
)𝑝
𝑋𝑡 (𝜋, 𝑐) = ess sup 𝐸
(𝜋, 𝑐) ∈ 𝒜,
where
𝑇
∫
𝑐˜∈𝒜(𝜋,𝑐,𝑡)
such that
𝑡
]
𝑈𝑠 (˜
𝑐𝑠 ) 𝜇∘ (𝑑𝑠)ℱ𝑡 ,
0≤𝑡≤𝑇
{
𝒜(𝜋, 𝑐, 𝑡) := (˜
𝜋 , 𝑐˜) ∈ 𝒜 : (˜
𝜋 , 𝑐˜) = (𝜋, 𝑐)
on
(3.1)
[0, 𝑡]
}
.
While we refer to [27, Proposition 3.1] for the proof, we shall have more to
say about
𝐿(𝑝)
later since it will be an important tool in our analysis.
We can now proceed to state the main results. The proofs are postponed
to Sections 6 and 7. Those sections also contain statements about the convergence of the opportunity processes and the solutions to the dual problems,
as well as some renements of the results below.
3.1
The Limit
𝑝 → −∞
The relative risk aversion
1−𝑝
of
𝑈 (𝑝)
increases to innity as
𝑝 → −∞.
Therefore we expect that in the limit, the agent does not invest at all. In
that situation the optimal propensity to consume is
𝜅𝑡 = (1 + 𝑇 − 𝑡)−1
since
this corresponds to a constant consumption rate. Our rst result shows that
this coincides with the limit of the
𝑈 (𝑝) -optimal
propensities to consume.
The following convergences hold as 𝑝 → −∞.
(i) Let 𝑡 ∈ [0, 𝑇 ]. In the case with intermediate consumption,
Theorem 3.1.
𝜅
ˆ 𝑡 (𝑝) →
1
1+𝑇 −𝑡
𝑃 -a.s.
If 𝔽 is continuous, the convergence is uniform in 𝑡, 𝑃 -a.s.; and holds
also in ℛ𝑟𝑙𝑜𝑐 for all 𝑟 ∈ [1, ∞).
(ii) If 𝑆 is continuous and 𝐿(𝑝) is continuous for all 𝑝 < 0, then
𝜋
ˆ (𝑝) → 0
ˆ
and 𝑋(𝑝)
→ 𝑥0 exp −
(
∫⋅
𝜇(𝑑𝑠) )
0 1+𝑇 −𝑠
in 𝐿2𝑙𝑜𝑐 (𝑀 )
in the semimartingale topology.
The continuity assumptions in (ii) are always satised if the ltration
is generated by a Brownian motion; see also Remark 4.2.
7
𝔽
Literature.
We are not aware of a similar result in the continuous-time litera-
ture, with the exception that when the strategies can be calculated explicitly,
the convergences mentioned in this section are often straightforward to obtain. E.g., Grasselli [16] carries out such a construction in a complete market
model. There are also related systematic results. Carassus and Rásonyi [5]
and Grandits and Summer [15] study convergence to the superreplication
problem for increasing (absolute) risk aversion of general utility functions
in discrete models. Note that superreplicating the contingent claim
𝐵 ≡0
corresponds to not trading at all. For the maximization of exponential utility
− exp(−𝛼𝑥)
without claim, the optimal strategy is proportional to the
inverse of the absolute risk aversion
in the limit
𝛼 → ∞.
𝛼
and hence trivially converges to zero
The case with claim is also studied. See, e.g., Mania
and Schweizer [24] for a continuous model, and Becherer [2] for a related
result. The references given here and later in this section do not consider
intermediate consumption.
We continue with our second main result, which concerns only the case
without intermediate consumption.
We rst introduce in detail the expo-
𝐵 ∈ 𝐿∞ (ℱ𝑇 )
nential hedging problem already mentioned above. Let
contingent claim.
utility (here with
𝛼 = 1)
of the terminal wealth including the claim,
[
(
)]
max 𝐸 − exp 𝐵 − 𝑥0 − (𝜗 ∙ 𝑅)𝑇 ,
(3.2)
𝜗∈Θ
where
𝜗
is the trading strategy parametrized by the
𝑖
𝜗 :=
vested in the assets (setting
To describe the set
Θ,
𝜗
∙
𝑆 =𝜗
∙
𝑅
[ 𝑑𝑄
𝑑𝑃
log
( 𝑑𝑄 )]
𝑑𝑃
in-
and
number of shares of the assets).
entropy of 𝑄 ∈ M relative to 𝑃
we dene the
𝐻(𝑄∣𝑃 ) := 𝐸
Now
monetary amounts
𝑖 yields
1{𝑆 𝑖 ∕=0} 𝜗𝑖 /𝑆−
−
corresponds to the more customary
and let
be a
Then the aim is to maximize the expected exponential
by
[
( 𝑑𝑄 )]
= 𝐸 𝑄 log
𝑑𝑃
}
M 𝑒𝑛𝑡 = 𝑄 ∈ M : 𝐻(𝑄∣𝑃 ) < ∞ .
We assume in the following that
M 𝑒𝑛𝑡 ∕= ∅.
(3.3)
{
}
𝑄-supermartingale for all 𝑄 ∈ M 𝑒𝑛𝑡 is
of admissible strategies for (3.2). If 𝑆 is locally bounded, there
ˆ ∈ Θ for (3.2) by Kabanov and Stricker [19,
optimal strategy 𝜗
{
Θ := 𝜗 ∈ 𝐿(𝑅) : 𝜗
the class
exists an
∙
𝑅
is a
Theorem 2.1]. (See Biagini and Fritelli [3, 4] for the unbounded case.)
As there is no intermediate consumption, the process
to a random variable
𝐷𝑇 ∈
𝐿∞ (ℱ
𝐷
in (2.3) reduces
𝑇 ). If we choose
𝐵 := log(𝐷𝑇 ),
we have the following result.
8
(3.4)
Theorem 3.2. Let 𝑆 be continuous and assume that 𝐿(𝑝) is continuous for
all 𝑝 < 0. Under (3.3) and (3.4),
in 𝐿2𝑙𝑜𝑐 (𝑀 ).
(1 − 𝑝) 𝜋
ˆ (𝑝) → 𝜗ˆ
Here 𝜋ˆ (𝑝) is in the fractions of wealth parametrization, while 𝜗ˆ denotes the
monetary amounts invested for the exponential utility.
As this convergence may seem surprising at rst glance, we give the
following heuristics.
𝐵 = log(𝐷𝑇 ) = 0 for simplicity. The preferences
=
ℝ+ are not directly comparable to the ones
exponential utility, which are dened on ℝ. We consider the
Remark 3.3. Assume
(𝑝) (𝑥)
induced by 𝑈
given by the
1 𝑝
𝑝 𝑥 on
shifted power utility functions
(
)
˜ (𝑝) (𝑥) := 𝑈 (𝑝) 𝑥 + 1 − 𝑝 ,
𝑈
Then
˜ (𝑝)
𝑈
𝑥 ∈ (𝑝 − 1, ∞).
again has relative risk aversion
denition increases to
ℝ
as
˜ (𝑝) (𝑥) =
(1 − 𝑝)1−𝑝 𝑈
𝑝 → −∞.
( 𝑥
1−𝑝
𝑝
1−𝑝
1−𝑝 > 0
and its domain of
Moreover,
+1
)𝑝
→ −𝑒−𝑥 ,
𝑝 → −∞,
(3.5)
and the multiplicative constant does not aect the preferences.
Let the agent with utility function
∗
capital 𝑥0
˜ (𝑝) be
𝑈
𝑥∗0 < 0,
endowed with some initial
∈ ℝ independent of 𝑝. (If
we consider only values of
∗
𝑝 such that 𝑝 − 1 < 𝑥0 .) The change of variables 𝑥 = 𝑥
˜ + 1 − 𝑝 yields
˜ (𝑝) (˜
ˆ
𝑈 (𝑝) (𝑥) = 𝑈
𝑥). Hence the corresponding optimal wealth processes 𝑋(𝑝)
˜
˜
ˆ
and 𝑋(𝑝) are related by 𝑋(𝑝) = 𝑋(𝑝) − 1 + 𝑝 if we choose the initial capital
𝑥0 := 𝑥∗0 + 1 − 𝑝 > 0 for the agent with 𝑈 (𝑝) . We conclude
(
)
˜
ˆ
ˆ 𝜋 (𝑝) 𝑑𝑅 = 𝑋(𝑝)
˜
𝑑𝑋(𝑝)
= 𝑑𝑋(𝑝)
= 𝑋(𝑝)ˆ
+1−𝑝 𝜋
ˆ (𝑝) 𝑑𝑅,
i.e., the optimal monetary investment
˜
𝜗(𝑝)
for
˜ (𝑝)
𝑈
is given by
(
)
˜
˜
𝜗(𝑝)
= 𝑋(𝑝)
ˆ (𝑝).
+1−𝑝 𝜋
In view of (3.5), it is reasonable that
˜
𝜗(𝑝)
monetary investment for the exponential utility.
fractions of wealth) does not depend on
conditions of Theorem 3.1.
𝜗ˆ,
should converge to
𝑥0
the optimal
We recall that
𝜋
ˆ (𝑝)
(in
and converges to zero under the
Thus, loosely speaking,
˜ 𝜋 (𝑝) ≈ 0
𝑋(𝑝)ˆ
for
−𝑝
large, and hence
More precisely, one can
˜
𝜗(𝑝)
≈ (1 − 𝑝)ˆ
𝜋 (𝑝).
(
)
˜ 𝜋 (𝑝)
show that lim𝑝→−∞ 𝑋(𝑝)ˆ
∙
𝑅 = 0
semimartingale topology, using arguments as in Appendix A.
9
in the
Literature.
To the best of our knowledge, the statement of Theorem 3.2
is new in the systematic literature.
𝐵 = 0.
the dual side for the case
However, there are known results on
The problem dual to exponential utility
maximization is the minimization of
𝑄𝐸
∈
𝐻(𝑄∣𝑃 )
over to
M 𝑒𝑛𝑡
and the optimal
M 𝑒𝑛𝑡 is called minimal entropy martingale measure.
tional assumptions on the model, the solution
𝑌ˆ (𝑝)
Under addi-
of the dual problem for
power utility (4.3) introduced below is a martingale and then the measure
𝑄𝑞
𝑑𝑄𝑞 /𝑑𝑃 = 𝑌ˆ𝑇 (𝑝)/𝑌ˆ0 (𝑝) is called 𝑞 -optimal martingale measure,
where 𝑞 < 1 is conjugate to 𝑝. This measure can be dened also for 𝑞 > 1,
𝑞
in which case it is not connected to power utility. The convergence of 𝑄 to
𝑄𝐸 for 𝑞 → 1+ was proved by Grandits and Rheinländer [14] for continuous
dened by
semimartingale models satisfying a reverse Hölder inequality. Under the additional assumption that
𝑞→1
𝔽
is continuous, the convergence of
and more generally the continuity of
𝑄𝑞 for
𝑞 7→
𝑞≥0
𝑄𝑞
to
𝑄𝐸
for
were obtained
by Mania and Tevzadze [25] (see also Santacroce [29]) using BSDE convergence together with
𝐵𝑀 𝑂
arguments.
The latter are possible due to the
reverse Hölder inequality; an assumption which is not present in our results.
3.2
As
The Limit
𝑝
𝑝→0
tends to zero, the relative risk aversion of the power utility tends to
log(𝑥).
which corresponds to the utility function
𝑢log (𝑥0 ) := sup 𝐸
−∞
]
log(𝑐𝑡 ) 𝜇∘ (𝑑𝑡) ;
0
𝑐∈𝒜(𝑥0 )
here integrals are set to
𝑇
[∫
1,
Hence we consider
if they are not well dened in
ℝ.
A
log-utility
agent exhibits a very special (myopic) behavior, which allows for an explicit
solution of the utility maximization problem (cf. Goll and Kallsen [11, 12]).
If in particular
𝑆
is continuous, the
𝜋𝑡 = 𝜆𝑡 ,
by [11, Theorem 3.1], where
𝜆
log-optimal
𝜅𝑡 =
strategy is
1
1+𝑇 −𝑡
is dened by (2.6). Our result below shows
𝐷 ≡ 1 converges to the logoptimal one as 𝑝 → 0. In general, the randomness of 𝐷 is an additional source
that the optimal strategy for power utility with
of risk and will cause an excess hedging demand.
Consider the bounded
semimartingale
𝜂𝑡 := 𝐸
[∫
𝑡
𝑇
]
𝐷𝑠 𝜇 (𝑑𝑠)ℱ𝑡 .
∘
𝜂 = 𝜂0
𝑀 + 𝑁 𝜂 + 𝐴𝜂 denotes the Kunita-Watanabe
decomposition of 𝜂 with respect to 𝑀 and the standard case 𝐷 ≡ 1 corre∘
𝜂
sponds to 𝜂𝑡 = 𝜇 [𝑡, 𝑇 ] and 𝑍 = 0.
If
𝑆
is continuous,
+ 𝑍𝜂
∙
10
Assume 𝑢𝑝0 (𝑥0 ) < ∞ for some 𝑝0 ∈ (0, 1). As 𝑝 → 0,
(i) in the case with intermediate consumption,
Theorem 3.4.
𝜅
ˆ 𝑡 (𝑝) →
𝐷𝑡
𝜂𝑡
uniformly in 𝑡, 𝑃 -a.s.
(ii) if 𝑆 is continuous,
𝜋
ˆ (𝑝) → 𝜆 +
𝑍𝜂
𝜂−
in 𝐿2𝑙𝑜𝑐 (𝑀 )
and the corresponding wealth processes converge in the semimartingale
topology.
Remark 3.5. If we consider the limit
that
𝑢𝑝0 (𝑥0 ) < ∞
for some
𝑝 0 > 0.
𝑝 → 0−,
we need not
a priori
assume
Without that condition, the assertions
of Theorem 3.4 remain valid if (i) is replaced by the weaker statement that
lim𝑝→0− 𝜅
ˆ 𝑡 (𝑝) → 𝐷𝑡 /𝜂𝑡 𝑃 -a.s.
𝑡.
for all
If
𝔽
is continuous, (i) remains valid
without changes. In particular, these convergences hold even if
Literature.
In the following discussion we assume
𝐷 ≡1
𝑢log (𝑥0 ) = ∞.
for simplicity. It
log-optimal strategy can be obtained
𝑝 = 0. Initiated by Jouini and Napp [18],
is part of the folklore that the
from
𝜋
ˆ (𝑝)
a re-
by formally setting
cent branch of the literature studies the stability of the utility maximization
problem under perturbations of the utility function (with respect to pointwise convergence) and other ingredients of the problem. To the best of our
knowledge, intermediate consumption was not considered so far and the results for continuous time concern continuous semimartingale models.
log(𝑥) = lim𝑝→0 (𝑈 (𝑝) (𝑥) − 𝑝−1 )
We note that
and here the additive con-
stant does not inuence the optimal strategy, i.e., we have pointwise conver-
𝑈 (𝑝) . Now Larsen [23, Theorem 2.2]
ˆ𝑇 for 𝑈 (𝑝) converges in probabilwealth 𝑋
gence of utility functions equivalent to
implies that the optimal terminal
ity to the
log-optimal one and that the value functions at time zero converge
pointwise (in the continuous case without consumption). We use the specic
form of our utility functions and obtain a stronger result. Finally, we can
mention that on the dual side and for
the continuity of
For general
of
𝐷
𝑞 -optimal
𝐷
and
𝑝,
𝑝 → 0−,
measures as mentioned after Remark 3.3.
it seems dicult to determine the precise inuence
on the optimal trading strategy
𝜋
ˆ (𝑝).
We can read Theorem 3.4(ii) as
a partial result on the excess hedging demand
𝜋
ˆ (𝑝, 1)
the convergence is related to
𝜋
ˆ (𝑝) − 𝜋
ˆ (𝑝, 1)
𝐷 ≡ 1.
due to
𝐷;
here
denotes the optimal strategy for the case
Corollary 3.6.
Suppose that the conditions of Theorem 3.4(ii) hold. Then
in 𝐿2𝑙𝑜𝑐 (𝑀 ); i.e., the asymptotic excess hedging
by 𝑍 𝜂 /𝜂− .
𝜋
ˆ (𝑝) − 𝜋
ˆ (𝑝, 1) → 𝑍 𝜂 /𝜂−
demand due to 𝐷 is given
11
The stability theory mentioned above considers also perturbations of the
probability measure
𝑃
(see Kardaras and šitkovi¢ [21]) and our corollary
can be related as follows. In the special case when
under
𝑃
𝑈 (𝑝)
is a martingale,
corresponds to the standard power utility function optimized under
the measure
demand due
4
𝐷
𝑑𝑃˜ = (𝐷𝑇 /𝐷0 ) 𝑑𝑃 (see [27, Remark
to 𝐷 then represents the inuence of
2.2]). The excess hedging
the subjective beliefs
𝑃˜.
Tools and Ideas for the Proofs
In this section we introduce our main tools and then present the basic ideas
how to apply them for the proofs of the theorems.
4.1
Opportunity Processes
We x
𝑝
and assume
𝑢𝑝 (𝑥0 ) < ∞
throughout this section. We rst discuss
the properties of the (primal) opportunity process
in (3.1).
𝐿 = 𝐿(𝑝) as introduced
𝐿𝑇 = 𝐷𝑇 and that
Moreover, 𝐿 has the fol-
Directly from that equation we have that
𝑢𝑝 (𝑥0 ) = 𝐿0 𝑝1 𝑥𝑝0
is the value function from (2.5).
lowing properties by [27, Lemma 3.5] in view of (2.4).
The opportunity process satises 𝐿, 𝐿− > 0.
(i) If 𝑝 ∈ (0, 1), 𝐿 is a supermartingale satisfying
Lemma 4.1.
(
)−𝑝 [
𝐿𝑡 ≥ 𝜇∘ [𝑡, 𝑇 ]
𝐸
∫
𝑡
𝑇
]
𝐷𝑠 𝜇∘ (𝑑𝑠)ℱ𝑡 ≥ 𝑘1 .
(ii) If 𝑝 < 0, 𝐿 is a bounded semimartingale satisfying
(
∘
0 < 𝐿𝑡 ≤ 𝜇 [𝑡, 𝑇 ]
)−𝑝
𝐸
𝑇
[∫
𝑡
]
(
)1−𝑝
𝐷𝑠 𝜇∘ (𝑑𝑠)ℱ𝑡 ≤ 𝑘2 𝜇∘ [𝑡, 𝑇 ]
.
If in addition there is no intermediate consumption, then 𝐿 is a submartingale.
In particular,
𝐿
𝛽 :=
is always a special semimartingale. We denote by
1
> 0,
1−𝑝
𝑞 :=
𝑝
∈ (−∞, 0) ∪ (0, 1)
𝑝−1
the relative risk tolerance and the exponent conjugate to
These constants are of course redundant given
𝑝,
𝑝,
(4.1)
respectively.
but turn out to simplify
the notation.
In the case with intermediate consumption, the opportunity process and
the optimal consumption are related by
𝑐ˆ𝑡 =
( 𝐷 )𝛽
𝑡
𝐿𝑡
ˆ𝑡
𝑋
and hence
12
𝜅
ˆ𝑡 =
( 𝐷 )𝛽
𝑡
𝐿𝑡
(4.2)
according to [27, Theorem 5.1]. Next, we introduce the convex-dual analogue
of
𝐿;
cf. [27, Ÿ4] for the following notions and results. The
inf 𝐸
𝑌 ∈Y
[∫
𝑇
dual problem
]
𝑈𝑡∗ (𝑌𝑡 ) 𝜇∘ (𝑑𝑡) ,
is
(4.3)
0
{
}
𝑈𝑡∗ (𝑦) = sup𝑥>0 𝑈𝑡 (𝑥) − 𝑥𝑦 = − 1𝑞 𝑦 𝑞 𝐷𝑡𝛽 is the conjugate of 𝑈𝑡 .
Only three properties of the domain Y = Y (𝑝) are relevant for us. First,
each element 𝑌 ∈ Y is a positive càdlàg supermartingale. Second, the set Y
𝑝−1
depends on 𝑝 only by a normalization: with the constant 𝑦0 (𝑝) := 𝐿0 (𝑝)𝑥0
,
′
−1 Y (𝑝) does not depend on 𝑝. As the elements of Y will
the set Y := 𝑦0 (𝑝)
where
occur only in terms of certain fractions, the constant plays no role. Third,
𝑃 -density
the
The
process of any
𝑄∈M
is contained in
Y
(modulo scaling).
dual opportunity process 𝐿∗ is the analogue of 𝐿 for the dual problem
and can be dened by
⎧
]
[∫

⎨ess sup𝑌 ∈Y 𝐸 𝑇 𝐷𝑠𝛽 (𝑌𝑠 /𝑌𝑡 )𝑞 𝜇∘ (𝑑𝑠)ℱ𝑡
𝑡
𝐿∗𝑡 :=
]
[∫

⎩ess inf 𝑌 ∈Y 𝐸 𝑇 𝐷𝑠𝛽 (𝑌𝑠 /𝑌𝑡 )𝑞 𝜇∘ (𝑑𝑠)ℱ𝑡
𝑡
Here the extremum is attained at the minimizer
denote by
𝑌ˆ = 𝑌ˆ (𝑝).
if
𝑝<0,
(4.4)
if
𝑌 ∈Y
𝑝 ∈ (0, 1).
for (4.3), which we
Finally, we shall use that the primal and the dual
opportunity process are related by the power
𝐿∗ = 𝐿𝛽 .
4.2
(4.5)
Bellman BSDE
We continue with a xed
𝑝
such that
𝑢𝑝 (𝑥0 ) < ∞.
We recall the Bellman
equation, which in the present paper will be used only for continuous
𝑍𝐿
𝑆.
In this case, recall (2.6) and let 𝐿 = 𝐿0 +
𝑀+
+ 𝐴𝐿 be the KW
𝐿
𝐿
decomposition of 𝐿 with respect to 𝑀 . Then the triplet (𝐿, 𝑍 , 𝑁 ) satises
∙
𝑁𝐿
the Bellman BSDE
𝑑𝐿𝑡 =
(
(
𝑞
𝑍 𝐿 )⊤
𝑍𝐿 )
𝐿𝑡− 𝜆𝑡 + 𝑡
𝑑⟨𝑀 ⟩𝑡 𝜆𝑡 + 𝑡
− 𝑝𝑈𝑡∗ (𝐿𝑡− ) 𝜇(𝑑𝑡)
2
𝐿𝑡−
𝐿𝑡−
+ 𝑍𝑡𝐿 𝑑𝑀𝑡 + 𝑑𝑁𝑡𝐿 ;
(4.6)
𝐿𝑇 = 𝐷𝑇 .
Put dierently, the nite variation part of
𝐴𝐿
𝑡
𝑞
=
2
∫
𝑡
𝐿𝑠−
0
(
𝐿
satises
(
𝑍𝐿 )
𝑍 𝐿 )⊤
𝜆𝑠 + 𝑠
𝑑⟨𝑀 ⟩𝑠 𝜆𝑠 + 𝑠
−𝑝
𝐿𝑠−
𝐿𝑠−
∫
𝑡
𝑈𝑠∗ (𝐿𝑠− ) 𝜇(𝑑𝑠).
0
(4.7)
13
𝑈∗
Here
is dened as in (4.3). Moreover, the optimal trading strategy
𝜋
ˆ
can
be described by
(
𝑍𝐿 )
𝜋
ˆ 𝑡 = 𝛽 𝜆𝑡 + 𝑡 .
𝐿𝑡−
(4.8)
See Nutz [26, Corollary 3.12] for these results.
Finally, still under the as-
sumption of continuity, the solution to the dual problem (4.3) is given by
the local martingale
(
1
𝑌ˆ = 𝑦0 ℰ − 𝜆 ∙ 𝑀 +
𝐿−
with the constant
𝑦0 = 𝑢′𝑝 (𝑥0 ) = 𝐿0 𝑥0𝑝−1
Remark 4.2. Continuity of
𝑆
∙
)
𝑁𝐿 ,
(4.9)
(cf. [26, Remark 5.18]).
does not imply that
𝐿 is continuous; the local
𝐿 may still have jumps (see also [26, Remark 3.13(i)]). If the
martingale 𝑁
ltration
𝔽
follows that
is continuous (i.e., all
𝐿 and 𝑆
𝔽-martingales
are continuous), it clearly
are continuous. The most important example with this
property is the Brownian ltration.
4.3
The Strategy for the Proofs
We can now summarize the basic scheme that is common for the proofs of
the three theorems.
The rst step is to prove the
process
𝐿
pointwise convergence
or of the dual opportunity process
𝐿∗ ;
of the opportunity
the choice of the process
depends on the theorem. The convergence of the optimal propensity to consume
of
𝐿
𝜅
ˆ
then follows in view of the feedback formula (4.2). The denitions
and
𝐿∗
via the value processes lend themselves to control-theoretic ar-
guments and of course Jensen's inequality will be the basic tool to derive
𝐿∗ = 𝐿𝛽 from (4.5), it is essentially equiv𝐿 or 𝐿∗ , as long as 𝑝 is xed. However, the
estimates. In view of the relation
alent whether one works with
dual problem has the advantage of being dened over a set of supermartingales, which are easier to handle than consumption and wealth processes.
This is particularly useful when passing to the limit.
The second step is the convergence of the trading strategy
𝜋
ˆ.
Note that
𝐿 from the KW decomposition
its formula (4.8) contains the integrand 𝑍
of
𝐿
with respect to
𝑀.
Therefore, the convergence of
convergence of the martingale part
𝑀 𝐿 (resp.
𝜋
ˆ
is related to the
∗
𝑀 𝐿 ). In general, the pointwise
convergence of a semimartingale is not enough to conclude the convergence
of its martingale part; this requires some control over the semimartingale
decomposition. In our case, this control is given by the Bellman BSDE (4.6),
which can be seen as a description for the dependence of the nite variation
part
𝐴𝐿
on the martingale part
𝑀 𝐿.
As we use the BSDE to show the
𝐿
convergence of 𝑀 , we benet from techniques from the theory of quadratic
14
BSDEs. However, we cannot apply standard results from that theory since
our assumptions are not strong enough.
In general, our approach is to extract as much information as possible
by basic control arguments and convex analysis
before
tackling the BSDE,
rather than to rely exclusively on (typically delicate) BSDE arguments. For
instance, we use the BSDE only after establishing the pointwise convergence
of its left hand side, i.e., the opportunity process. This essentially eliminates
the need for an
a priori
estimate or a comparison principle and constitutes
a key reason for the generality of our results.
Our procedure shares basic
features of the viscosity approach to Markovian control problems, where one
also works directly with the value function before tackling the HamiltonJacobi-Bellman equation.
5
Auxiliary Results
We start by collecting inequalities for the dependence of the opportunity
processes on
𝑝.
The precise formulations are motivated by the applications
in the proofs of the previous theorems, but the comparison results are also
of independent interest.
5.1
Comparison Results
𝑢𝑝0 (𝑥0 ) < ∞ for a given exponent 𝑝0 . For
𝑝
convenience, we restate the quantities 𝛽 = 1/(1−𝑝) > 0 and 𝑞 =
𝑝−1 dened
in (4.1). It is useful to note that 𝑞 ∈ (−∞, 0) for 𝑝 ∈ (0, 1) and vice versa.
When there is a second exponent 𝑝0 under consideration, 𝛽0 and 𝑞0 have the
obvious denition. We also recall from (2.4) the bounds 𝑘1 and 𝑘2 for 𝐷 .
We assume the entire section that
Proposition 5.1.
𝐿∗𝑡 (𝑝)
≤ 𝐸
𝐿𝑡 (𝑝) ≤
(
Let 0 < 𝑝 < 𝑝0 < 1. For each 𝑡 ∈ [0, 𝑇 ],
[∫
𝑡
∘
𝑇
𝐷𝑠𝛽
]1−𝑞/𝑞0 (
)𝑞/𝑞0
𝜇 (𝑑𝑠)ℱ𝑡
𝑘1𝛽−𝛽0 𝐿∗𝑡 (𝑝0 )
,
∘
)1−𝑝/𝑝0
𝑘2 𝜇 [𝑡, 𝑇 ]
𝐿𝑡 (𝑝0 )𝑝/𝑝0 .
(5.1)
(5.2)
If 𝑝 < 𝑝0 < 0, the converse inequalities hold, if in (5.1) 𝑘1 is replaced by 𝑘2 .
If 𝑝 < 0 < 𝑝0 < 1, the converse inequalities hold, if in (5.2) 𝑘2 is replaced by
𝑘1 .
Proof.
We x
𝑡
and begin with (5.1). To unify the proofs, we rst argue a
Jensen's inequality: if
𝐸
[∫
𝑡
𝑋 = (𝑋𝑠 )𝑠∈[𝑡,𝑇 ] > 0
is optional and
𝛼 ∈ (0, 1),
then
𝑇
]𝛼
]
]1−𝛼 [ ∫ 𝑇
[∫ 𝑇
𝐷𝑠𝛽 𝑋𝑠 𝜇∘ (𝑑𝑠)ℱ𝑡 .
𝐷𝑠𝛽 𝑋𝑠𝛼 𝜇∘ (𝑑𝑠)ℱ𝑡 ≤ 𝐸
𝐷𝑠𝛽 𝜇∘ (𝑑𝑠)ℱ𝑡
𝐸
𝑡
𝑡
(5.3)
15
To see this, introduce the probability space
[
𝜈(𝐼 × 𝐺) := 𝐸 𝜉 −1
∫
(
[𝑡, 𝑇 ]×Ω, ℬ([𝑡, 𝑇 ])⊗ℱ, 𝜈
]
1𝐺 𝐷𝑠𝛽 𝜇∘ (𝑑𝑠) ,
)
, where
𝐺 ∈ ℱ, 𝐼 ∈ ℬ([𝑡, 𝑇 ]),
𝐼
𝜉 := 𝐸[
with the normalizing factor
∫𝑇
𝑡
𝐷𝑠𝛽 𝜇∘ (𝑑𝑠)∣ℱ𝑡 ].
On this space,
𝑋
is a
random variable and we have the conditional Jensen's inequality
]𝛼
]
[ [ 𝐸 𝜈 𝑋 𝛼 [𝑡, 𝑇 ] × ℱ𝑡 ≤ 𝐸 𝜈 𝑋 [𝑡, 𝑇 ] × ℱ𝑡
𝜎 -eld [𝑡, 𝑇 ] × ℱ𝑡 := {[𝑡, 𝑇 ] × 𝐴 : 𝐴 ∈ ℱ𝑡 }. But this inequality
0
0
coincides with (5.3) if we identify 𝐿 ([𝑡, 𝑇 ] × Ω, [𝑡, 𝑇 ] × ℱ𝑡 ) and 𝐿 (Ω, ℱ𝑡 ) by
for the
using that an element of the rst space is necessarily constant in its time
variable.
0 < 𝑝 ≤ 𝑝0 < 1 and let 𝑌ˆ := 𝑌ˆ (𝑝0 ) be the solution of the dual
problem for 𝑝0 . Using (4.4) and then (5.3) with 𝛼 := 𝑞/𝑞0 ∈ (0, 1) and
(
)𝛼
𝑋𝑠𝛼 := (𝑌ˆ𝑠 /𝑌ˆ𝑡 )𝑞0 = (𝑌ˆ𝑠 /𝑌ˆ𝑡 )𝑞 ,
]
[∫ 𝑇
(
)𝑞
𝐿∗𝑡 (𝑝) ≤ 𝐸
𝐷𝑠𝛽 𝑌ˆ𝑠 /𝑌ˆ𝑡 𝜇∘ (𝑑𝑠)ℱ𝑡
𝑡
]1−𝑞/𝑞0 [ ∫ 𝑇
]𝑞/𝑞0
[∫ 𝑇
𝛽 ∘
≤𝐸
𝐷𝑠 𝜇 (𝑑𝑠)ℱ𝑡
𝐸
𝐷𝑠𝛽 (𝑌ˆ𝑠 /𝑌ˆ𝑡 )𝑞0 𝜇∘ (𝑑𝑠)ℱ𝑡
.
Let
𝑡
𝛽
Now 𝐷𝑠
≤
𝑡
𝑘1𝛽−𝛽0 𝐷𝑠𝛽0 since
𝛽 − 𝛽 0 < 0,
which completes the proof of the
rst claim in view of (4.4). In the cases with
replaced by a supremum and
𝛼 = 𝑞/𝑞0
𝑝 < 0, the inmum in (4.4) is
> 1 or < 0, reversing the
is either
direction of Jensen's inequality.
We turn to (5.2).
Let
0 < 𝑝 ≤ 𝑝0 < 1
and
ˆ = 𝑋(𝑝)
ˆ , 𝑐ˆ = 𝑐ˆ(𝑝).
𝑋
Using (3.1) and (the usual) Jensen's inequality twice,
𝑇
[∫
]
𝐷𝑠 𝑐ˆ𝑝𝑠 0 𝜇∘ (𝑑𝑠)ℱ𝑡
𝑡
]𝑝0 /𝑝
[∫ 𝑇
∘
1−𝑝0 /𝑝
≥ 𝜇 [𝑡, 𝑇 ]
𝐸
𝐷𝑠𝑝/𝑝0 𝑐ˆ𝑝𝑠 𝜇∘ (𝑑𝑠)ℱ𝑡
ˆ 𝑝0 ≥ 𝐸
𝐿𝑡 (𝑝0 )𝑋
𝑡
𝑡
)1−𝑝0 /𝑝 (
)
ˆ 𝑝 𝑝0 /𝑝
≥ 𝑘2 𝜇 [𝑡, 𝑇 ]
𝐿𝑡 (𝑝)𝑋
𝑡
(
∘
and the claim follows. The other cases are similar.
A useful consequence is that
the possibly critical exponent
Corollary 5.2.
𝐿(𝑝)
gains moments as
𝑝
moves away from
𝑝0 .
(i) Let 0 < 𝑝 < 𝑝0 < 1. Then
𝐿(𝑝) ≤ 𝐶𝐿(𝑝0 )
(5.4)
with a constant 𝐶 independent of 𝑝0 and 𝑝. In the case without intermediate consumption we can take 𝐶 = 1.
16
(ii) Let 𝑟 ≥ 1 and 0 < 𝑝 ≤ 𝑝0 /𝑟. Then
[
]
𝐸 (𝐿𝜏 (𝑝))𝑟 ≤ 𝐶𝑟
for all stopping times 𝜏 , with a constant 𝐶𝑟 independent of 𝑝0 , 𝑝, 𝜏 . In
particular, 𝐿(𝑝) is of class (D) for all 𝑝 ∈ (0, 𝑝0 ).
Proof.
𝐿 = 𝐿(𝑝0 ). By Lemma 4.1, 𝐿/𝑘1 ≥ 1, hence 𝐿𝑝/𝑝0 =
𝑝/𝑝
≤ 𝑘1 0 (𝐿/𝑘1 ) as 𝑝/𝑝0 ∈ (0, 1). Proposition 5.1 yields the
( ∘
)1−𝑝/𝑝0
result with 𝐶 = 𝜇 [0, 𝑇 ]𝑘2 /𝑘1
; note that 𝐶 ≤ 1 ∨ (1 + 𝑇 )𝑘2 /𝑘1 . In
the absence of intermediate consumption we may assume 𝑘1 = 𝑘2 = 1 by the
subsequent Remark 5.3 and then 𝐶 = 1.
(ii) Let 𝑟 ≥ 1, 0 < 𝑝 ≤ 𝑝0 /𝑟 , and 𝐿 = 𝐿(𝑝0 ). Proposition 5.1 shows
(i)
Denote
𝑝/𝑝
𝑘1 0 (𝐿/𝑘1 )𝑝/𝑝0
(
)𝑟(1−𝑝/𝑝0 ) 𝑟𝑝/𝑝0 (
)𝑟 𝑟𝑝/𝑝
𝐿𝑡 (𝑝)𝑟 ≤ 𝑘2 𝜇∘ [𝑡, 𝑇 ]
𝐿𝑡
≤ (1 ∨ 𝑘2 )(1 + 𝑇 ) 𝐿𝑡 0 .
𝑟𝑝/𝑝0 ∈ (0, 1), thus 𝐿𝑟𝑝/𝑝0
𝑟𝑝/𝑝
𝑟𝑝/𝑝
𝐸[𝐿𝜏 0 ] ≤ 𝐿0 0 ≤ 1 ∨ 𝑘2 .
Note
is a supermartingale by Lemma 4.1 and
Remark 5.3. In the case without intermediate consumption we may assume
𝐷 ≡ 1
Indeed, 𝐷 reduces to the random
𝐷𝑇 and can be absorbed into the measure 𝑃 as follows. Under the
˜ with 𝑃 -density process 𝜉𝑡 = 𝐸[𝐷𝑇 ∣ℱ𝑡 ]/𝐸[𝐷𝑇 ], the opportunity
measure 𝑃
˜ (𝑥) = 1 𝑥𝑝 is 𝐿
˜ = 𝐿/𝜉 by [27, Remark 3.2].
process for the utility function 𝑈
𝑝
˜
˜ 0 ) and then
If Corollary 5.2(i) is proved for 𝐷 ≡ 1, we conclude 𝐿(𝑝)
≤ 𝐿(𝑝
the inequality for 𝐿 follows.
in the proof of Corollary 5.2(i).
variable
Inequality (5.4) is stated for reference as it has a simple form; however,
note that it was deduced using the very poor estimate
the pure investment case, we have
𝐶=1
𝑎𝑏 ≥ 𝑎 for 𝑎, 𝑏 ≥ 1.
In
and so (5.4) is a direct comparison
result. Intermediate consumption destroys this monotonicity property: (5.4)
fails for
𝐶 = 1
𝐷 ≡1
𝑝 = 0.1
in that case, e.g., if
a standard Brownian motion, and
by explicit calculation.
and
and
𝑅𝑡 = 𝑡 + 𝑊𝑡 , where 𝑊 is
𝑝0 = 0.2, as can be seen
This is not surprising from a BSDE perspective,
because the driver of (4.6) is not monotone with respect to
of the
𝑑𝜇-term.
𝑝 in the presence
In the pure investment case, the driver is monotone and so
the comparison result can be expected, even for the entire parameter range.
This is conrmed by the next result; note that the inequality is
to (5.2) for the considered parameters.
Proposition 5.4.
Let 𝑝 < 𝑝0 < 0, then
𝐿𝑡 (𝑝) ≤
)𝑝 −𝑝
𝑘2 ( ∘
𝜇 [𝑡, 𝑇 ] 0 𝐿𝑡 (𝑝0 ).
𝑘1
In the case without intermediate consumption, 𝐿(𝑝) ≤ 𝐿(𝑝0 ).
17
converse
The proof is based on the following auxiliary statement.
Let 𝑌 > 0 be a supermartingale. For xed 0 ≤ 𝑡 ≤ 𝑠 ≤ 𝑇 ,
Lemma 5.5.
1
( [
]) 1−𝑞
𝑞
𝑞 7→ 𝜙(𝑞) := 𝐸 (𝑌𝑠 /𝑌𝑡 ) ℱ𝑡
𝜙 : (0, 1) → ℝ+ ,
is a monotone decreasing function
we have
(
[ 𝑃 -a.s. If 𝑌 is a martingale,
])
𝜙(1) := lim𝑞→1− 𝜙(𝑞) = exp − 𝐸 (𝑌𝑠 /𝑌𝑡 ) log(𝑌𝑠 /𝑌𝑡 ) ℱ𝑡 𝑃 -a.s., where the
conditional expectation has values in ℝ ∪ {+∞}.
Lemma 5.5 can be obtained using Jensen's inequality and a suitable
change of measure; we refer to [27, Lemma 4.10] for details.
Proof of Proposition 5.4.
Let 0 < 𝑞0 < 𝑞 < 1 be the dual exponents and
1−𝑞
𝑌ˆ := 𝑌ˆ (𝑝). By Lemma 5.5 and Jensen's inequality for 1−𝑞
∈ (0, 1),
0
∫ 𝑇(
∫ 𝑇
1−𝑞
] ∘
]) 1−𝑞
[
[
𝑞
0
ˆ
ˆ
𝐸 (𝑌ˆ𝑠 /𝑌ˆ𝑡 )𝑞0 ℱ𝑡
𝜇∘ (𝑑𝑠)
𝐸 (𝑌𝑠 /𝑌𝑡 ) ℱ𝑡 𝜇 (𝑑𝑠) ≤
denote
𝑡
𝑡
( 1−𝑞 )( ∫
1− 1−𝑞
0
≤ 𝜇∘ [𝑡, 𝑇 ]
𝑇
]
[
𝐸 (𝑌ˆ𝑠 /𝑌ˆ𝑡 )𝑞0 ℱ𝑡 𝜇∘ (𝑑𝑠)
) 1−𝑞
1−𝑞0
.
𝑡
Using (2.4) and (4.4) twice, we conclude that
𝐿∗𝑡 (𝑝)
∫
𝑇
≤
𝑘2𝛽
)( ∫
≤
1−𝑞
1−𝑞
1− 1−𝑞
𝛽 −𝛽0 1−𝑞0 ∘
0
𝜇 [𝑡, 𝑇 ]
𝑘2 𝑘1
)
≤
1−𝑞
1−𝑞
−𝛽0 1−𝑞
1− 1−𝑞
0 ∘
0
𝑘2𝛽 𝑘1
𝜇 [𝑡, 𝑇 ]
]
[
𝐸 (𝑌ˆ𝑠 /𝑌ˆ𝑡 )𝑞 ℱ𝑡 𝜇∘ (𝑑𝑠)
𝑡
(
(
Now (4.5) and
𝛽 = 1−𝑞
𝑇
) 1−𝑞
1−𝑞0
] ∘
[ 𝛽0
𝑞
𝐸 𝐷𝑠 (𝑌ˆ𝑠 /𝑌ˆ𝑡 ) 0 ℱ𝑡 𝜇 (𝑑𝑠)
𝑡
1−𝑞
𝐿∗𝑡 (𝑝0 ) 1−𝑞0 .
yield the rst result. In the case without inter-
mediate consumption, we may assume
𝐷≡1
and hence
𝑘1 = 𝑘2 = 1,
as in
Remark 5.3.
Remark 5.6. Our argument for Proposition 5.4 extends to
𝑝 = −∞
(cf.
Lemma 6.7 below). The proposition generalizes [25, Proposition 2.2], where
the result is proved for the case without intermediate consumption and under
the additional condition that
𝑞0 -optimal
𝑌ˆ (𝑝0 ) is a martingale (or equivalently, that the
equivalent martingale measure exists).
Propositions 5.1 and 5.4 combine to the following continuity property of
𝑝 7→ 𝐿(𝑝)
at interior points of
(−∞, 0).
We will not pursue this further as
we are interested mainly in the boundary points of this interval.
Corollary 5.7.
1−𝑝/𝑝0
𝐶𝑡
Assume 𝐷 ≡ 1 and let 𝐶𝑡 := 𝜇∘ [𝑡, 𝑇 ]. If 𝑝 ≤ 𝑝0 < 0,
1−𝑝0 /𝑝+𝑝0 −𝑝
𝐿(𝑝0 )𝑝/𝑝0 ≤ 𝐿(𝑝) ≤ 𝐶𝑡𝑝0 −𝑝 𝐿(𝑝0 ) ≤ 𝐶𝑡
𝐿(𝑝)𝑝0 /𝑝 .
In particular, 𝑝 7→ 𝐿𝑡 (𝑝) is continuous on (−∞, 0) uniformly in 𝑡, 𝑃 -a.s.
18
𝜅
ˆ (𝑝) is not monotone with
𝐷 ≡ 1 and 𝑅𝑡 =
motion, and 𝑝 ∈ {−1/2, −1, −2}.
Remark 5.8. The optimal propensity to consume
respect to
𝑡 + 𝑊𝑡 ,
𝑝
in general. For instance, monotonicity fails for
𝑊
where
is a standard Brownian
One can note that
𝑝
determines both the risk aversion and the elasticity of
intertemporal substitution (see, e.g., Gollier [13, Ÿ15]). As with any timeadditive utility specication, it is not possible in our setting to study the
dependence on each of these quantities in an isolated way.
𝐵𝑀 𝑂
5.2
Estimate
In this section we give
𝐵𝑀 𝑂
estimates for the martingale part of
𝐿.
The
following lemma is well known; we state the proof since the argument will
be used also later on.
Let 𝑋 be a submartingale satisfying 0 ≤ 𝑋 ≤ 𝛼 for some
constant 𝛼 > 0. Then for all stopping times 0 ≤ 𝜎 ≤ 𝜏 ≤ 𝑇 ,
Lemma 5.9.
]
]
[
[
𝐸 [𝑋]𝜏 − [𝑋]𝜎 ℱ𝜎 ≤ 𝐸 𝑋𝜏2 − 𝑋𝜎2 ℱ𝜎 .
Proof.
𝑋𝑡2
=
𝑋 ∫= 𝑋0 + 𝑀 𝑋 + 𝐴𝑋 be the Doob-Meyer
decomposition.
∫𝜏
𝑡
𝑋
𝑋
)
+
[𝑋]
and
2
+ 2 0 𝑋𝑠− (𝑑𝑀𝑠𝑋 + 𝑑𝐴𝑋
𝑠− 𝑑𝐴𝑠 ≥ 0,
𝑡
𝑠
𝜎
∫ 𝜏
2
2
[𝑋]𝜏 − [𝑋]𝜎 ≤ 𝑋𝜏 − 𝑋𝜎 − 2
𝑋𝑠− 𝑑𝑀𝑠𝑋 .
Let
𝑋02
As
𝜎
𝑋− ∙ 𝑀 𝑋 is a
𝑋
𝑋
1
martingale. Indeed, 𝑋 is bounded and sup𝑡 ∣𝑀𝑡 ∣ ≤ 2𝛼 + 𝐴𝑇 ∈ 𝐿 , so the
𝑋 1/2 ∈ 𝐿1 , hence [𝑋 ∙ 𝑀 𝑋 ]1/2 ∈ 𝐿1 ,
BDG inequalities [8, VII.92] show [𝑀 ]𝑇
−
𝑇
𝑋
1
which by the BDG inequalities implies that sup𝑡 ∣𝑋− ∙ 𝑀𝑡 ∣ ∈ 𝐿 .
The claim follows by taking conditional expectations because
𝐿(𝑝)
We wish to apply Lemma 5.9 to
in the case
𝑝 < 0.
However,
the submartingale property fails in general for the case with intermediate
consumption (cf. Lemma 4.1). We introduce instead a closely related process
having this property.
Lemma 5.10.
tion. Then
Let 𝑝 < 0 and consider the case with intermediate consump∫ 𝑡
( 1 + 𝑇 − 𝑡 )𝑝
1
𝐵𝑡 :=
𝐿𝑡 +
𝐷𝑠 𝑑𝑠
1+𝑇
(1 + 𝑇 )𝑝 0
is a submartingale satisfying 0 < 𝐵𝑡 ≤ 𝑘2 (1 + 𝑇 )1−𝑝 .
Proof.
Choose
(𝜋, 𝑐) ≡ (0, 𝑥0 /(1 + 𝑇 ))
in [27, Proposition 3.4] to see that
is a submartingale. The bound follows from Lemma 4.1.
We are now in the position to exploit Lemma 5.9.
19
𝐵
(i) Let 𝑝1 < 0. There exists a constant 𝐶 = 𝐶(𝑝1 ) such
that ∥𝑀 𝐿(𝑝) ∥𝐵𝑀 𝑂 ≤ 𝐶 for all 𝑝 ∈ (𝑝1 , 0). In the case without intermediate consumption one can take 𝑝1 = −∞.
(ii) Assume 𝑢𝑝0 (𝑥0 ) < ∞ for some 𝑝0 ∈ (0, 1) and let 𝜎 be a stopping
time such that 𝐿(𝑝0 )𝜎 ≤ 𝛼 for a constant 𝛼 > 0. Then there exists
𝐶 ′ = 𝐶 ′ (𝛼) such that ∥(𝑀 𝐿(𝑝) )𝜎 ∥𝐵𝑀 𝑂 ≤ 𝐶 ′ for all 𝑝 ∈ (0, 𝑝0 ].
Lemma 5.11.
Proof.
(i) Let
𝑝1 < 𝑝 < 0 and let 𝜏 be a stopping time.
]
[
𝐸 [𝐿(𝑝)]𝑇 − [𝐿(𝑝)]𝜏 ℱ𝜏 ≤ 𝐶.
In the case without intermediate consumption,
martingale with
𝐿 ≤ 𝑘2
We rst show that
𝐿 = 𝐿(𝑝)
(5.5)
is a positive sub-
(Lemma 4.1), so Lemma 5.9 implies (5.5) with
−𝑡 𝑝
𝐶 = 𝑘22 . In the other case, dene 𝐵 as in Lemma 5.10 and 𝑓 (𝑡) := ( 1+𝑇
1+𝑇 ) .
∫ 𝑡 −2
Then [𝐿]𝑡 − [𝐿]0 =
(𝑠) 𝑑[𝐵]𝑠 and 𝑓 −2 (𝑠) ≤ 1 as 𝑓 is increasing with
0 𝑓
∫𝑇
𝑓 (0) = 1. Thus [𝐿]𝑇 − [𝐿]𝜏 = 𝜏 𝑓 −2 (𝑠) 𝑑[𝐵]𝑠 ≤ [𝐵]𝑇 − [𝐵]𝜏 . Now (5.5)
1−𝑝 and Lemma 5.9 imply
follows since 𝐵 ≤ 𝑘2 (1 + 𝑇 )
]
[
𝐸 [𝐵]𝑇 − [𝐵]𝜏 ℱ𝜏 ≤ 𝑘22 (1 + 𝑇 )2−2𝑝 ≤ 𝑘22 (1 + 𝑇 )2−2𝑝1 =: 𝐶(𝑝1 ).
[𝐿] = 𝐿20 + [𝑀 𝐿 ] + [𝐴𝐿 ] + 2[𝑀 𝐿 , 𝐴𝐿 ]. Since 𝐴𝐿 is predictable,
𝑁 := 2[𝑀 𝐿 , 𝐴𝐿 ] is a local martingale with some localizing sequence (𝜎𝑛 ).
𝐿
𝐿
Moreover, [𝑀 ]𝑡 − [𝑀 ]𝑠 = [𝐿]𝑡 − [𝐿]𝑠 − ([𝐴]𝑡 − [𝐴]𝑠 ) − (𝑁𝑡 − 𝑁𝑠 ) and (5.5)
We have
imply
[
]
𝐸 [𝑀 𝐿 ]𝑇 ∧𝜎𝑛 − [𝑀 𝐿 ]𝜏 ∧𝜎𝑛 ℱ𝜏 ∧𝜎𝑛 ≤ 𝐶.
Choosing
𝜏 = 0
and
𝑛 → ∞
we see that
[𝑀 ]𝑇 ∈ 𝐿1 (𝑃 )
and thus Hunt's
Lemma [8, V.45] shows the a.s.-convergence in this inequality; i.e., we have
]
[
𝐸 [𝑀 𝐿 ]𝑇 − [𝑀 𝐿 ]𝜏 ℱ𝜏 ≤ 𝐶 . If 𝐿 is bounded by 𝛼, the jumps
bounded by 2𝛼 (cf. [17, I.4.24]), therefore
]
[
sup 𝐸 [𝑀 𝐿 ]𝑇 − [𝑀 𝐿 ]𝜏 − ℱ𝜏 ≤ 𝐶 + 4𝛼2 .
of
𝑀𝐿
are
𝜏
By Lemma 4.1 we can take
𝛼 = 𝑘2 (1 + 𝑇 )1−𝑝1 ,
and
𝛼 = 𝑘2
when there is no
intermediate consumption.
0 < 𝑝 ≤ 𝑝0 < 1. The assumption and Corollary 5.2(i) show
≤ 𝐶𝛼 for a constant 𝐶𝛼 independent of 𝑝 and 𝑝0 . We ap𝜎
ply Lemma 5.9 to the nonnegative process 𝑋(𝑝) := 𝐶𝛼 − 𝐿(𝑝) , which is
]
[
𝜎
𝜎
a submartingale by Lemma 4.1, and obtain 𝐸 [𝐿(𝑝) ]𝑇 − [𝐿(𝑝) ]𝜏 ℱ𝜏
=
[
]
2
𝐸 [𝑋(𝑝)]𝑇 − [𝑋(𝑝)]𝜏 ℱ𝜏 ≤ 𝐶𝛼 . Now the rest of the proof is as in (i).
(ii)
Let
𝜎
that 𝐿(𝑝)
Let 𝑆 be continuous and assume that either 𝑝 ∈ (0, 1)
and 𝐿 is bounded or that 𝑝 < 0 and 𝐿 is bounded away from zero. Then
𝜆 ∙ 𝑀 ∈ 𝐵𝑀 𝑂, where 𝜆 and 𝑀 are dened by (2.6).
Corollary 5.12.
20
Proof.
In both cases, the assumed bound and Lemma 4.1 imply that
is bounded away from zero and innity.
in (4.6), we obtain a constant
[∫
𝐸
𝑡
𝑇
𝐶>0
such that
]
(
(
𝑍 𝐿 )
𝑍 𝐿 )⊤
𝑑⟨𝑀 ⟩ 𝜆 +
𝐿− 𝜆 +
ℱ𝑡 ≤ 𝐶,
𝐿−
𝐿− 𝑀 𝐿 ∈ 𝐵𝑀 𝑂
Moreover, we have
and the Cauchy-Schwarz inequality, it follows that
We remark that uniform bounds for
𝐿
𝐸[
∫𝑇
index
𝑞
𝑡
𝜆⊤ 𝑑⟨𝑀 ⟩ 𝜆∣ℱ
𝐿
𝑡] ≤
𝐶 ′ > 0.
for a constant
(as in the condition of Corol-
lary 5.12) are equivalent to a reverse Hölder inequality
Y;
0 ≤ 𝑡 ≤ 𝑇.
by Lemma 5.11. Using the bounds for
𝐶 ′ (1 + ∥𝑍 𝐿 ∙ 𝑀 ∥𝐵𝑀 𝑂 ) ≤ 𝐶 ′ (1 + ∥𝑀 𝐿 ∥𝐵𝑀 𝑂 )
ment of the dual domain
𝐿
Taking conditional expectations
R𝑞 (𝑃 )
for some ele-
see [27, Proposition 4.5] for details. Here the
𝑞 < 1. Therefore, our corollary complements well known
R𝑞 (𝑃 ) with 𝑞 > 1 implies 𝜆 ∙ 𝑀 ∈ 𝐵𝑀 𝑂 (in a suitable
satises
results stating that
setting); see, e.g., Delbaen et al. [6, Theorems A,B].
6
The Limit
𝑝 → −∞
The rst goal of this section is to prove Theorem 3.1. Recall that the consumption strategy is related to the opportunity processes via (4.2) and (4.5).
From these relations and the intuition mentioned before Theorem 3.1, we
𝐿∗𝑡 = 𝐿𝛽𝑡 converges to 𝜇∘ [𝑡, 𝑇 ] as
𝛽 = 1/(1 − 𝑝) → 0, this implies that
expect that the dual opportunity process
𝑝 → −∞. Noting that the exponent
𝐿𝑡 (𝑝) → ∞ for all 𝑡 < 𝑇 , in the case with intermediate consumption. There∗
fore, we shall work here with 𝐿 rather than 𝐿. In the pure investment case,
the situation is dierent as then 𝐿 ≤ 𝑘2 (Lemma 4.1). There, the limit of 𝐿
yields additional information; this is examined in Section 6.1 below.
Proposition 6.1.
For each 𝑡 ∈ [0, 𝑇 ],
lim 𝐿∗𝑡 (𝑝) = 𝜇∘ [𝑡, 𝑇 ]
𝑝→−∞
𝑃 -a.s.
and in 𝐿𝑟 (𝑃 ), 𝑟 ∈ [1, ∞),
with a uniform bound. If 𝔽 is continuous, the convergences are uniform in 𝑡.
Remark 6.2. We will use later that the same convergences hold if
𝑡
is
replaced by a stopping time, which is an immediate consequence in view
of the uniform bound. Of course, we mean by uniform bound that there
exists a constant
𝐶 > 0,
independent of
𝑝
and
𝑡,
such that
0 ≤ 𝐿∗𝑡 (𝑝) ≤ 𝐶 .
Analogous terminology will be used in the sequel.
Proof.
We consider
0 > 𝑝 → −∞
and note that
𝑞 → 1−
and
𝛽 → 0+.
From
Lemma 4.1 we have
0 ≤ 𝐿∗𝑡 (𝑝) = 𝐿𝛽𝑡 (𝑝) ≤ 𝑘2𝛽 𝜇∘ [𝑡, 𝑇 ] → 𝜇∘ [𝑡, 𝑇 ],
21
(6.1)
𝑡. To obtain a lower bound, we consider the density process 𝑌
𝑄 ∈ M , which exists by assumption (2.1). From (4.4) we have
uniformly in
of some
𝐿∗𝑡 (𝑝)
≥
𝑘1𝛽
∫
𝑇
]
[
𝐸 (𝑌𝑠 /𝑌𝑡 )𝑞 ℱ𝑡 𝜇∘ (𝑑𝑠).
𝑡
𝑠 ≥ 𝑡, clearly (𝑌𝑠 /𝑌𝑡 )𝑞 → 𝑌𝑠 /𝑌𝑡 𝑃 -a.s. as 𝑞 → 1, and noting
𝑞
1
bound 0 ≤ (𝑌𝑠 /𝑌𝑡 ) ≤ 1 + 𝑌𝑠 /𝑌𝑡 ∈ 𝐿 (𝑃 ) we conclude by dominated
For xed
the
convergence that
Since
]
]
[
[
𝐸 (𝑌𝑠 /𝑌𝑡 )𝑞 ℱ𝑡 → 𝐸 𝑌𝑠 /𝑌𝑡 ℱ𝑡 ≡ 1 𝑃 -a.s., for all 𝑠 ≥ 𝑡.
]
[
𝑌 𝑞 is a supermartingale, 0 ≤ 𝐸 (𝑌𝑠 /𝑌𝑡 )𝑞 ℱ𝑡 ≤ 1. Hence, for
each
𝑡,
dominated convergence shows
∫
𝑇
]
[
𝐸 (𝑌𝑠 /𝑌𝑡 )𝑞 ℱ𝑡 𝜇∘ (𝑑𝑠) → 𝜇∘ [𝑡, 𝑇 ] 𝑃 -a.s.
𝑡
This ends the proof of the rst claim. The convergence in
𝐿𝑟 (𝑃 )
follows by
the bound (6.1).
Assume that
xed
𝔽 is continuous; then
(𝑠, 𝜔) ∈ [0, 𝑇 ] × Ω we consider (a
the martingale
𝑌
is continuous. For
version of )
]1/𝑞
[
𝑓𝑞 (𝑡) := 𝐸 (𝑌𝑠 /𝑌𝑡 )𝑞 ℱ𝑡
(𝜔),
𝑡 ∈ [0, 𝑠].
𝑡 and increasing in 𝑞 by Jensen's inequality,
𝑓𝑞 → 1 uniformly[ in 𝑡 on the
]compact
[0, 𝑠], by Dini's lemma. The same holds for 𝑓𝑞 (𝑡)𝑞 = 𝐸 (𝑌𝑠 /𝑌𝑡 )𝑞 ℱ𝑡 (𝜔).
′
Fix 𝜔 ∈ Ω and let 𝜀, 𝜀 > 0. By Egorov's theorem there exist a measurable
∘
set 𝐼 = 𝐼(𝜔) ⊆ [0, 𝑇 ] and 𝛿 = 𝛿(𝜔) ∈ (0, 1) such that 𝜇 ([0, 𝑇 ] ∖ 𝐼) < 𝜀 and
]
[
𝑞
′
sup𝑡∈[0,𝑠] ∣𝐸 (𝑌𝑠 /𝑌𝑡 ) ℱ𝑡 − 1∣ < 𝜀 for all 𝑞 > 1 − 𝛿 and all 𝑠 ∈ 𝐼 . For
𝑞 > 1 − 𝛿 and 𝑡 ∈ [0, 𝑇 ] we have
These functions are continuous in
and converge to
∫
𝑇
1
for each
𝑡.
Hence
[
]
𝐸 (𝑌𝑠 /𝑌𝑡 )𝑞 ℱ𝑡 − 1 𝜇∘ (𝑑𝑠)
𝑡
∫
≤
[
]
𝐸 (𝑌𝑠 /𝑌𝑡 )𝑞 ℱ𝑡 − 1 𝜇∘ (𝑑𝑠) +
∫
[
]
𝐸 (𝑌𝑠 /𝑌𝑡 )𝑞 ℱ𝑡 − 1 𝜇∘ (𝑑𝑠)
[𝑡,𝑇 ]∖𝐼
𝐼
′
≤ 𝜀 (1 + 𝑇 ) + 𝜀.
We have shown that
sup𝑡∈[0,𝑇 ] ∣𝐿∗𝑡 (𝑝)−𝜇∘ [𝑡, 𝑇 ]∣ → 0 𝑃 -a.s., and also in 𝐿𝑟 (𝑃 )
by dominated convergence and the uniform bound resulting from (6.1) in
view of
𝑘2𝛽 𝜇∘ [𝑡, 𝑇 ] ≤ (1 ∨ 𝑘2 )(1 + 𝑇 ).
Under additional continuity assumptions, we will prove that the martingale part of
For each
𝐿∗
converges to zero in
2 .
ℋ𝑙𝑜𝑐
We rst need some preparations.
𝑝, it follows from Lemma 4.1 that 𝐿∗
22
has a canonical decomposition
∗
∗
𝐿∗ = 𝐿∗0 + 𝑀 𝐿 + 𝐴𝐿
. When
sition with respect to
𝑀
𝐿
𝑆 is continuous, we denote the KW decompo∗
∗
∗
𝐿∗ = 𝐿∗0 + 𝑍 𝐿 ∙ 𝑀 + 𝑁 𝐿 + 𝐴𝐿 . If in addition
∗
𝛽
from 𝐿 = 𝐿 and (4.7) by Itô's formula that
by
is continuous, we obtain
∗
∗
∗
𝑀 𝐿 = 𝛽𝐿𝛽−1 ∙ 𝑀 𝐿 ; 𝑍 𝐿 /𝐿∗ = 𝛽𝑍 𝐿 /𝐿; 𝑁 𝐿 = 𝛽𝐿𝛽−1 ∙ 𝑁 𝐿 ; (6.2)
∫
∫
∫
(
( ∗ )−1
∗ )⊤
∗
∗
𝛽𝜆𝐿∗ + 2𝑍 𝐿
𝑑⟨𝑀 ⟩ 𝜆 + 𝑝2
𝐿
𝑑⟨𝑁 𝐿 ⟩ − 𝐷𝛽 𝑑𝜇.
𝐴𝐿 = 2𝑞
Here
𝑑𝜇
is a shorthand for
Lemma 6.3.
(
𝜇(𝑑𝑠).
Let 𝑝0 < 0. There exists a localizing sequence (𝜎𝑛 ) such that
𝐿∗ (𝑝)
)𝜎𝑛
−
> 1/𝑛
simultaneously for all 𝑝 ∈ (−∞, 𝑝0 ];
and moreover, if 𝑆 and 𝐿(𝑝) are continuous, (𝑀 𝐿
∗ (𝑝)
)𝜎𝑛 ∈ 𝐵𝑀 𝑂
for 𝑝 ≤ 𝑝0 .
Proof.
Fix 𝑝0 < 0 (and corresponding 𝑞0 ) and a sequence 𝜀𝑛 ↓ 0 in (0, 1). Set
𝜎𝑛 = inf{𝑡 ≥ 0 : 𝐿∗𝑡 (𝑝0 ) ≤ 𝜀𝑛 } ∧ 𝑇 . Then 𝜎𝑛 → 𝑇 stationarily because each
∗
path of 𝐿 (𝑝0 ) is bounded away from zero (Lemma 4.1). Proposition 5.1 im( ∗
)𝑞/𝑞0
∗
plies that there is a constant 𝛼 = 𝛼(𝑝0 ) > 0 such that 𝐿𝑡 (𝑝) ≥ 𝛼 𝐿𝑡 (𝑝0 )
for all
𝑝 ≤ 𝑝0 .
It follows that
1/𝑞0
𝐿∗(𝜎𝑛 ∧𝑡)− (𝑝) ≥ 𝛼𝜀𝑛
for all
𝑝 ≤ 𝑝0
and we
have proved the rst claim.
𝑝 ∈ (−∞, 𝑝0 ], let 𝑆 and 𝐿 = 𝐿(𝑝) be continuous and recall that
= 𝛽𝐿𝛽−1 ∙ 𝑀 𝐿 from (6.2). Noting that 𝛽 − 1 < 0, we have just shown
𝛽−1 is bounded on [0, 𝜎 ]. Since 𝑀 𝐿 ∈ 𝐵𝑀 𝑂 by
that the integrand 𝛽𝐿
𝑛
𝐿∗ )𝜎𝑛 ∈ 𝐵𝑀 𝑂 .
Lemma 5.11(i), we conclude that (𝑀
Fix
∗
𝑀𝐿
Proposition 6.4.
𝑝 → −∞,
∗ (𝑝)
𝑍𝐿
Proof.
Assume that 𝑆 and 𝐿(𝑝) are continuous for all 𝑝 < 0. As
→0
in 𝐿2𝑙𝑜𝑐 (𝑀 ) and 𝑁 𝐿
∗ (𝑝)
→0
2
.
in ℋ𝑙𝑜𝑐
𝑝0 < 0 and consider 𝑝 ∈ (−∞, 𝑝0 ]. Using Lemma 6.3, we
∗
𝑀 𝐿 (𝑝) ∈ ℋ2 and 𝜆 ∈ 𝐿2 (𝑀 ). Dene the
processes 𝑋 = 𝑋(𝑝) by
We x some
may assume by localization that
continuous
𝑋𝑡 (𝑝) := 𝑘2𝛽 𝜇∘ [𝑡, 𝑇 ] − 𝐿∗𝑡 (𝑝),
0 ≤ 𝑋(𝑝) ≤ (1∨𝑘2 )(1+𝑇 ) by (6.1). Fix 𝑝. We shall apply Itô's formula
Φ(𝑋), where
Φ(𝑥) := exp(𝑥) − 𝑥.
then
to
For
𝑥 ≥ 0, Φ
Φ(𝑥) ≥ 1,
satises
Φ′ (0) = 0,
Φ′ (𝑥) ≥ 0,
23
Φ′′ (𝑥) ≥ 1,
Φ′′ (𝑥) − Φ′ (𝑥) = 1.
∫𝑇
Φ′′ (𝑋) 𝑑⟨𝑋⟩ = Φ(𝑋𝑇 ) − Φ(𝑋0 ) − 0 Φ′ (𝑋) ( 𝑑𝑀 𝑋 + 𝑑𝐴𝑋 ). As
∗
Φ′ (𝑋) is like 𝑋 uniformly bounded and 𝑀 𝑋 = −𝑀 𝐿 ∈ ℋ2 , the stochastic
𝑋 is a true martingale and
integral wrt. 𝑀
]
]
[∫ 𝑇
[∫ 𝑇
[
]
Φ′ (𝑋) 𝑑𝐴𝑋 .
Φ′′ (𝑋) 𝑑⟨𝑋⟩ = 2𝐸 Φ(𝑋𝑇 ) − Φ(𝑋0 ) − 2𝐸
𝐸
1
2
We have
∫𝑇
0
0
0
Note that
∗
𝑑𝐴𝑋 = −𝑘2𝛽 𝑑𝜇 − 𝑑𝐴𝐿
, so that (6.2) yields
(
( )−1
(
)
∗ )⊤
∗
2 𝑑𝐴𝑋 = −𝑞 𝛽𝜆𝐿∗ + 2𝑍 𝐿
𝑑⟨𝑀 ⟩ 𝜆 − 𝑝 𝐿∗
𝑑⟨𝑁 𝐿 ⟩ + 2 𝐷𝛽 − 𝑘2𝛽 𝑑𝜇.
Letting
𝐿∗
𝑝 → −∞,
𝑞 → 1−
𝑝,
we have
and
𝛽 → 0+.
Hence, using that
𝑋
and
are bounded uniformly in
−𝑞𝛽𝐸
[∫
𝑇
]
Φ′ (𝑋)(𝜆𝐿∗ )⊤ 𝑑⟨𝑀 ⟩ 𝜆 → 0,
0
[∫
𝑇
(
) ]
Φ′ (𝑋) 𝐷𝛽 − 𝑘2𝛽 𝑑𝜇 → 0,
[ 0
]
𝐸 Φ(𝑋𝑇 ) − Φ(𝑋0 ) → 0,
𝐸
where the last convergence is due to Proposition 6.1 (and the subsequent
remark). If
𝑇
[∫
𝐸
0
𝑜
denotes the sum of these three expectations tending to zero,
]
Φ (𝑋) 𝑑⟨𝑋⟩
[∫ 𝑇
}]
{ ( ∗)
( ∗ )−1
𝐿∗
′
𝐿 ⊤
𝑑⟨𝑁 ⟩ + 𝑜.
=𝐸
Φ (𝑋) 2𝑞 𝑍
𝑑⟨𝑀 ⟩ 𝜆 + 𝑝 𝐿
′′
0
( ∗ )⊤
∗
∗
𝑑⟨𝑋⟩ = 𝑑⟨𝐿∗ ⟩ = 𝑍 𝐿
𝑑⟨𝑀 ⟩ 𝑍 𝐿 + 𝑑⟨𝑁 𝐿 ⟩. For the right hand side,
′
we use Φ (𝑋) ≥ 0 and ∣𝑞∣ < 1 and the Cauchy-Schwarz inequality to obtain
[∫ 𝑇
{( ∗ )
}]
′′
𝐿 ⊤
𝐿∗
𝐿∗
𝐸
Φ (𝑋) 𝑍
𝑑⟨𝑀 ⟩ 𝑍 + 𝑑⟨𝑁 ⟩
0
[∫ 𝑇
{( ∗ )
}]
( ∗ )−1
′
𝐿 ⊤
𝐿∗
⊤
𝐿∗
≤𝐸
Φ (𝑋) 𝑍
𝑑⟨𝑀 ⟩ 𝑍 + 𝜆 𝑑⟨𝑀 ⟩ 𝜆 + 𝑝 𝐿
𝑑⟨𝑁 ⟩ + 𝑜.
Note
0
We bring the terms with
𝑇
𝑍𝐿
∗
and
∗
𝑁𝐿
to the left hand side, then
]
}( 𝐿∗ )⊤
𝐿∗
𝐸
Φ (𝑋) − Φ (𝑋) 𝑍
𝑑⟨𝑀 ⟩ 𝑍
0
]
[∫ 𝑇
]
[∫ 𝑇
{ ′′
( ∗ )−1 }
′
𝐿∗
′
⊤
+𝐸
Φ (𝑋) − 𝑝Φ (𝑋) 𝐿
𝑑⟨𝑁 ⟩ ≤ 𝐸
Φ (𝑋)𝜆 𝑑⟨𝑀 ⟩ 𝜆 + 𝑜.
[∫
{
′′
′
0
As
Φ′ (0) = 0,
bound, hence
0
lim𝑝→−∞ Φ′ (𝑋𝑡 ) → 0 𝑃 -a.s.
𝜆 ∈ 𝐿2 (𝑀 ) implies that the right
we have
24
for all
𝑡,
with a uniform
hand side converges to
zero.
We recall
Φ′′ − Φ′ ≡ 1
and
( )−1
Φ′′ (𝑋) − 𝑝Φ′ (𝑋) 𝐿∗
≥ Φ′′ (0) = 1.
Whence both expectations on the left hand side are nonnegative and we can
conclude that they converge to zero; therefore,
and
𝐸[⟨𝑁
𝐿∗
⟩𝑇 ] → 0.
Proof of Theorem 3.1.
𝑝
[∫𝑇
0
∗
∗
(𝑍 𝐿 )⊤ 𝑑⟨𝑀 ⟩ 𝑍 𝐿
]
→0
In view of (4.2), part (i) follows from Proposition 6.1;
note that the convergence in
uniformly in
𝐸
ℛ𝑟𝑙𝑜𝑐
is immediate as
by Lemma 6.3 and (4.2).
and (6.2) that
𝜅
ˆ (𝑝)
is locally bounded
For part (ii), recall from (4.8)
∗
𝜋
ˆ = 𝛽(𝜆 + 𝑍 𝐿 /𝐿) = 𝛽𝜆 + 𝑍 𝐿 /𝐿∗
𝛽𝜆 → 0 in 𝐿2𝑙𝑜𝑐 (𝑀 ). By Lemma 6.3, 1/𝐿∗
is locally bounded uniformly in 𝑝, hence 𝜋
ˆ (𝑝) → 0 in 𝐿2𝑙𝑜𝑐 (𝑀 ) follows from
Proposition 6.4. As 𝜅
ˆ (𝑝) is locally bounded uniformly in 𝑝, Corollary A.4(i)
ˆ .
from the Appendix yields the convergence of the wealth processes 𝑋(𝑝)
for each
6.1
𝑝.
As
𝛽 → 0,
clearly
Convergence to the Exponential Problem
In this section, we prove Theorem 3.2 and establish the convergence of the
corresponding opportunity processes. We assume that there is no intermediate consumption, that
𝐵
𝑆
is locally bounded and satises (3.3), and that the
𝐵 later). Hence
ˆ
there exists an (essentially unique) optimal strategy 𝜗 ∈ Θ for (3.2). It is
ˆ does not depend on the initial capital 𝑥0 . If 𝜗 ∈ Θ,
easy to see that 𝜗
we denote by 𝐺(𝜗) = 𝜗 ∙ 𝑅 the corresponding gains process and dene
˜ = 𝐺𝑡 (𝜗)}. We consider the value process (from
Θ(𝜗, 𝑡) = {𝜗˜ ∈ Θ : 𝐺𝑡 (𝜗)
contingent claim
is bounded (we will choose a specic
initial wealth zero) of (3.2),
[
(
) ]
˜ ℱ𝑡 ,
𝑉𝑡 (𝜗) := ess sup𝜗∈Θ(𝜗,𝑡)
𝐸 − exp 𝐵 − 𝐺𝑇 (𝜗)
˜
0 ≤ 𝑡 ≤ 𝑇.
𝜗1 , 𝜗2 ∈ Θ ⇒ 𝜗1 1[0,𝑡] + 𝜗2 1(𝑡,𝑇 ] ∈ Θ. With
˜ = 𝐺𝑡 (𝜗) + 𝐺𝑡,𝑇 (𝜗1
˜ (𝑡,𝑇 ] ) for 𝜗˜ ∈ Θ(𝜗, 𝑡).
𝐺𝑇 (𝜗)
Note the concatenation property
𝐺𝑡,𝑇 (𝜗) :=
∫𝑇
𝑡
𝜗 𝑑𝑅,
we have
Therefore, if we dene the exponential opportunity process
[
(
) ]
˜ ℱ𝑡 ,
𝐿exp
:= ess inf 𝜗∈Θ
𝐸 exp 𝐵 − 𝐺𝑡,𝑇 (𝜗)
˜
𝑡
0 ≤ 𝑡 ≤ 𝑇,
(6.3)
then using standard properties of the essential inmum one can check that
𝑉𝑡 (𝜗) = − exp(−𝐺𝑡 (𝜗)) 𝐿exp
𝑡 .
𝐿exp is a reduced form of the value process, analogous to 𝐿(𝑝) for power
exp
utility. We also note that 𝐿𝑇
= exp(𝐵).
Thus
Lemma 6.5.
satisfying
𝐿exp
The exponential opportunity process 𝐿exp is a submartingale
≤ ∥ exp(𝐵)∥𝐿∞ (𝑃 ) and 𝐿exp , 𝐿exp
− > 0.
25
Proof.
The martingale optimality principle of dynamic programming is proved
here exactly as, e.g., in [27, Proposition A.2], and yields that
supermartingale for every
tingale if and only if
𝜗
𝜗 ∈ Θ
𝑉 (𝜗)
is a
𝐸[𝑉⋅ (𝜗)] > −∞ and a mar𝑉 (𝜗) = − exp(−𝐺(𝜗)) 𝐿exp , we
the choice 𝜗 ≡ 0. It follows that
such that
is optimal.
As
obtain the submartingale property by
∞
∞
𝐿exp ≤ ∥𝐿exp
𝑇 ∥𝐿 = ∥ exp(𝐵)∥𝐿 .
ˆ
The optimal strategy 𝜗 is optimal for all the conditional problems (6.3),
[
(
) ]
exp
ˆ 𝐿exp
ˆ ℱ𝑡 > 0. Thus 𝜉 := exp(−𝐺(𝜗))
hence 𝐿𝑡
= 𝐸 exp 𝐵 − 𝐺𝑡,𝑇 (𝜗)
is a positive martingale, by the optimality principle. In particular, we have
𝑃 [inf 0≤𝑡≤𝑇 𝜉𝑡 > 0] = 1,
and now the same property for
𝐿exp
follows.
𝑆 is continuous and denote the KW decomposition of 𝐿exp
exp = 𝐿exp + 𝑍 𝐿exp ∙ 𝑀 + 𝑁 𝐿exp + 𝐴𝐿exp . Then the
with respect to 𝑀 by 𝐿
0 exp )
( exp 𝐿exp
triplet (ℓ, 𝑧, 𝑛) := 𝐿
,𝑍
, 𝑁𝐿
satises the BSDE
Assume that
𝑑ℓ𝑡 =
(
(
𝑧𝑡 ) ⊤
𝑧𝑡 )
1
ℓ𝑡− 𝜆𝑡 +
+ 𝑧𝑡 𝑑𝑀𝑡 + 𝑑𝑛𝑡
𝑑⟨𝑀 ⟩𝑡 𝜆𝑡 +
2
ℓ𝑡−
ℓ𝑡−
with terminal condition
ℓ𝑇 = exp(𝐵),
and the optimal strategy
(6.4)
𝜗ˆ is
exp
𝑍𝐿
𝜗ˆ = 𝜆 + exp .
𝐿−
(6.5)
This can be derived directly by dynamic programming or inferred, e.g., from
Frei and Schweizer [10, Proposition 1]. We will actually reprove the BSDE
later, but present it already at this stage for the following motivation.
We observe that (6.4) coincides with the BSDE (4.6), except that 𝑞 is
replaced by 1 and the terminal condition is exp(𝐵) instead of 𝐷𝑇 . From now
on we assume exp(𝐵) = 𝐷𝑇 , then one can guess that the solutions 𝐿(𝑝)
should converge to
𝐿exp
as
𝑞 → 1−,
or equivalently
𝑝 → −∞.
Let 𝑆 be continuous.
(i) As 𝑝 ↓ −∞, 𝐿𝑡 (𝑝) ↓ 𝐿exp
𝑃 -a.s. for all 𝑡, with a uniform bound.
𝑡
(ii) If 𝐿(𝑝) is continuous for each 𝑝 < 0, then 𝐿exp is also continuous and
the convergence 𝐿(𝑝) ↓ 𝐿exp is uniform in 𝑡, 𝑃 -a.s. Moreover,
Theorem 6.6.
(1 − 𝑝) 𝜋
ˆ (𝑝) → 𝜗ˆ
in 𝐿2𝑙𝑜𝑐 (𝑀 ).
We note that (ii) is also a statement about the rate of convergence for
𝜋
ˆ (𝑝) → 0 in Theorem 3.1(ii) for the case without intermediate consumption.
The proof occupies most of the remainder of the section. Part (i) follows from
the next two lemmata; recall that the monotonicity of
𝑝 7→ 𝐿𝑡 (𝑝) was already
established in Proposition 5.4 while the uniform bound is from Lemma 4.1.
Lemma 6.7.
We have 𝐿(𝑝) ≥ 𝐿exp for all 𝑝 < 0.
26
Proof.
𝐵 = 0 by a change of measure
𝑑𝑃 (𝐵) = (𝑒𝐵 /𝐸[𝑒𝐵 ]) 𝑑𝑃 . Let 𝑄𝐸 ∈ M 𝑒𝑛𝑡 be the measure with
minimal entropy 𝐻(𝑄∣𝑃 ); see, e.g., [19, Theorem 3.5]. Let 𝑌 be its 𝑃 -density
from
As is well-known, we may assume that
𝑃
to
process, then
𝐸
𝑄
− log(𝐿exp
𝑡 )=𝐸
[
]
]
[
log(𝑌𝑇 /𝑌𝑡 )ℱ𝑡 = 𝐸 (𝑌𝑇 /𝑌𝑡 ) log(𝑌𝑇 /𝑌𝑡 )ℱ𝑡 .
(6.6)
This is merely a dynamic version of the well-known duality relation stated,
e.g., in [19, Theorem 2.1] and one can retrieve this version, e.g., from [10,
Eq. (8),(10)]. Using the decreasing function
𝜙
from Lemma 5.5,
(
])
[
ℱ𝑡 = 𝜙(1)
𝐿exp
=
exp
−
𝐸
(𝑌
/𝑌
)
log(𝑌
/𝑌
)
𝑡
𝑡
𝑇
𝑇
𝑡
]1/𝛽
[
≤ 𝜙(𝑞) = 𝐸 (𝑌𝑇 /𝑌𝑡 )𝑞 ℱ𝑡
≤ 𝐿∗ (𝑝)1/𝛽 = 𝐿(𝑝),
where (4.4) was used for the second inequality.
Lemma 6.8.
Proof.
Fix
Let 𝑆 be continuous. Then lim sup𝑝→−∞ 𝐿𝑡 (𝑝) ≤ 𝐿exp
𝑡 .
𝑡 ∈ [0, 𝑇 ].
We denote
ℰ𝑡𝑇 (𝑋) := ℰ(𝑋)𝑇 /ℰ(𝑋)𝑡
and
𝑋𝑡𝑇 :=
𝑋𝑇 − 𝑋𝑡 .
𝜗 ∈ 𝐿(𝑅) be such that ∣𝜗 ∙ 𝑅∣ + ⟨𝜗 ∙ 𝑅⟩ is bounded by a constant.
Noting that 𝐿(𝑅) ⊆ 𝒜 because 𝑅 is continuous, we have from (3.1) that
]
]
[
[
𝑝
𝑝
𝐿𝑡 (𝑝) = ess inf 𝜋∈𝒜 𝐸 𝐷𝑇 ℰ𝑡𝑇
(𝜋 ∙ 𝑅)ℱ𝑡 ≤ 𝐸 𝐷𝑇 ℰ𝑡𝑇
(∣𝑝∣−1 𝜗 ∙ 𝑅)ℱ𝑡
[
(
) ]
= 𝐸 𝐷𝑇 exp − (𝜗 ∙ 𝑅)𝑡𝑇 + 1 ⟨𝜗 ∙ 𝑅⟩𝑡𝑇 ℱ𝑡 .
(i) Let
2∣𝑝∣
The expression under the last conditional expectation is bounded uniformly
[
(
) ]
𝑝, so the last line converges to 𝐸 exp 𝐵 − (𝜗 ∙ 𝑅)𝑡𝑇 ℱ𝑡 𝑃 -a.s.
𝑝 → −∞; recall 𝐷𝑇 = exp(𝐵). We have shown
[
(
) ]
lim sup 𝐿𝑡 (𝑝) ≤ 𝐸 exp 𝐵 − (𝜗 ∙ 𝑅)𝑡𝑇 ℱ𝑡 𝑃 -a.s.
in
when
(6.7)
𝑝→−∞
𝜗 ∈ 𝐿(𝑅) be such that exp(−𝜗 ∙ 𝑅) is of class (D). Dening
times 𝜏𝑛 = inf{𝑠 > 0 : ∣𝜗 ∙ 𝑅𝑠 ∣ + ⟨𝜗 ∙ 𝑅⟩𝑠 ≥ 𝑛}, we have
[
(
) ]
lim sup 𝐿𝑡 (𝑝) ≤ 𝐸 exp 𝐵 − (𝜗 ∙ 𝑅)𝜏𝑡𝑇𝑛 ℱ𝑡 𝑃 -a.s.
(ii) Let
stopping
the
𝑝→−∞
for each
𝑛,
and also
(iii)
] (D) property, the
[ 𝜗1((0,𝜏𝑛 ] . Using the
) class
𝐸 exp 𝐵 − (𝜗 ∙ 𝑅)𝑡𝑇 ℱ𝑡 in 𝐿1 (𝑃 ) as 𝑛 → ∞,
by step (i) applied to
right hand side converges to
𝑃 -a.s.
along a subsequence. Hence (6.7) again holds.
The previous step has a trivial extension: Let
𝑔𝑡𝑇 ∈ 𝐿0 (ℱ𝑇 )
𝑔𝑡𝑇 ≤ (𝜗 ∙ 𝑅)𝑡𝑇 for some 𝜗 as in (ii).
]
[
lim sup 𝐿𝑡 (𝑝) ≤ 𝐸 exp(𝐵 − 𝑔𝑡𝑇 )ℱ𝑡 𝑃 -a.s.
random variable such that
𝑝→−∞
27
Then
be a
(iv)
Let
𝑛
sequence 𝑔𝑡𝑇
𝜗ˆ ∈ Θ be the optimal strategy. We claim that there
∈ 𝐿0 (ℱ𝑇 ) of random variables as in (iii) such that
(
)
(
)
𝑛
ˆ in 𝐿1 (𝑃 ).
exp 𝐵 − 𝑔𝑡𝑇
→ exp 𝐵 − 𝐺𝑡,𝑇 (𝜗)
Indeed, we may assume
𝐵 = 0,
as in the previous proof.
exists a
Then our claim
follows by the construction of Schachermayer [30, Theorem 2.2] applied to
[𝑡, 𝑇 ]; recall[the denitions
[30, Eq. (4),(5)]. We conclude
(
) ]
ˆ ℱ𝑡 = 𝐿exp 𝑃 -a.s. by the
lim sup𝑝→−∞ 𝐿𝑡 (𝑝) ≤ 𝐸 exp 𝐵 − 𝐺𝑡,𝑇 (𝜗)
𝑡
𝐿1 (𝑃 )-continuity of the conditional expectation.
the time interval
that
ˆ exp is a martingale, hence of class (D).
exp(−𝐺(𝜗))𝐿
ˆ is
bounded away from zero, it follows that exp(−𝐺(𝜗))
Remark 6.9. Recall that
If
𝐿exp
is uniformly
already of class (D) and the last two steps in the previous proof are unnecessary. This situation occurs precisely when the right hand side of (6.6) is
bounded uniformly in 𝑡. In standard terminology, the latter condition states
that the reverse Hölder inequality R𝐿 log(𝐿) (𝑃 ) is satised by the density
process of the minimal entropy martingale measure.
Let 𝑆 be continuous and assume that 𝐿(𝑝) is continuous for
all 𝑝 < 0. Then 𝐿exp is continuous and 𝐿𝑡 (𝑝) → 𝐿exp
uniformly in 𝑡, 𝑃 -a.s.
𝑡
2 .
Moreover, 𝑍 𝐿(𝑝) → 𝑍 exp in 𝐿2𝑙𝑜𝑐 (𝑀 ) and 𝑁 (𝑝) → 𝑁 exp in ℋ𝑙𝑜𝑐
Lemma 6.10.
We have already identied the monotone limit
𝐿exp
= lim 𝐿𝑡 (𝑝).
𝑡
Hence,
by uniqueness of the KW decomposition, the above lemma follows from the
subsequent one, which we state separately to clarify the argument. The most
important input from the control problems is that by stopping, we can bound
𝐿(𝑝)
away from zero simultaneously for all
𝑝
(cf. Lemma 6.3).
Let
( 𝑆 be continuous
) and assume that 𝐿(𝑝) is continuous for
𝐿(𝑝)
˜ 𝑍,
˜ 𝑁
˜ ) of the
all 𝑝 < 0. Then 𝐿(𝑝), 𝑍 , 𝑁 (𝑝) converge to a solution (𝐿,
BSDE (6.4) as 𝑝 → −∞: 𝐿˜ is continuous and 𝐿𝑡 (𝑝) → 𝐿˜𝑡 uniformly in 𝑡,
˜ in 𝐿2 (𝑀 ) and 𝑁 (𝑝) → 𝑁
˜ in ℋ2 .
𝑃 -a.s.; while 𝑍 𝐿(𝑝) → 𝑍
𝑙𝑜𝑐
𝑙𝑜𝑐
Lemma 6.11.
Proof.
For notational simplicity, we write the proof for the one-dimensional
= 1). We x a sequence 𝑝𝑛 ↓ −∞ and corresponding 𝑞𝑛 ↑ 1. As
˜ 𝑡 := lim𝑛 𝐿𝑡 (𝑝𝑛 ) exists.
𝑝 7→ 𝐿𝑡 (𝑝) is monotone and positive, the 𝑃 -a.s. limit 𝐿
𝐿(𝑝
)
𝑛 of martingales is bounded in the Hilbert space ℋ2
The sequence 𝑀
𝐿(𝑝𝑛 ) ,
by Lemma 5.11(i). Hence it has a subsequence, still denoted by 𝑀
˜ ∈ ℋ2 in the weak topology of ℋ2 . If we denote
which converges to some 𝑀
˜ , we have by orthogonality that
˜=𝑍
˜ ∙𝑀 +𝑁
the KW decomposition by 𝑀
𝐿(𝑝
)
2
𝐿(𝑝
)
˜ weakly in 𝐿 (𝑀 ) and 𝑁 𝑛 → 𝑁
˜ weakly in ℋ2 . We shall use
𝑍 𝑛 →𝑍
case (𝑑
the BSDE to deduce a strong convergence.
𝑝𝑛 and in (6.4)
1 (
𝑧 )2
𝑓 (𝑡, 𝑙, 𝑧) := 𝑙 𝜆𝑡 +
2
𝑙
The drivers in the BSDE (4.6) corresponding to
𝑓 𝑛 (𝑡, 𝑙, 𝑧) := 𝑞𝑛 𝑓 (𝑡, 𝑙, 𝑧),
28
are
for (𝑡, 𝑙, 𝑧) ∈ [0, 𝑇 ] × (0, ∞) × ℝ.
(𝑙𝑚 , 𝑧𝑚 ) → (𝑙, 𝑧) ∈ (0, ∞) × ℝ, we
For xed
𝑡
and any convergent sequence
have
𝑓 𝑚 (𝑡, 𝑙𝑚 , 𝑧𝑚 ) → 𝑓 (𝑡, 𝑙, 𝑧) 𝑃 -a.s.
By Lemmata 6.7 and 6.5 we can nd a localizing sequence
1/𝑘 < 𝐿(𝑝)𝜏𝑘 ≤ 𝑘2
(𝜏𝑘 )
such that
𝑝 < 0,
for all
where the upper bound is from Lemma 4.1. For the processes from (2.6) we
𝜆𝜏𝑘 ∈ 𝐿2 (𝑀 ) and 𝑀 𝜏𝑘 ∈ ℋ2 for each 𝑘 .
𝑛
𝑛
𝐿(𝑝𝑛 ) , 𝑁 𝑛 = 𝑁 𝐿(𝑝𝑛 ) , and
To relax the notation, let 𝐿 = 𝐿(𝑝𝑛 ), 𝑍 = 𝑍
𝑀 𝑛 = 𝑀 𝐿(𝑝𝑛 ) = 𝑍 𝑛 ∙ 𝑀 + 𝑁 𝑛 . The purpose of the localization is that (𝑓 𝑛 )
𝑛
𝑛 𝜏
are uniformly quadratic in the relevant domain: As (𝐿 , 𝑍 ) 𝑘 takes values
in [1/𝑘, 𝑘2 ] × ℝ and
∣𝑓 𝑛 (𝑡, 𝑙, 𝑧)∣ ≤ 𝑙𝜆2𝑡 + 𝜆𝑡 𝑧 + 𝑧 2 /𝑙 ≤ (1 + 𝑙)𝜆2𝑡 + (1 + 1/𝑙)𝑧 2 ,
may assume that
we have for all
𝑚, 𝑛 ∈ ℕ
that
𝑛
∣𝑓 𝑚 (𝑡, 𝐿𝑛𝑡 , 𝑍𝑡𝑛 )∣𝜏𝑘 ≤ 𝜉𝑡 + 𝐶𝑘 (𝑍𝑡∧𝜏
)2 ,
𝑘
( )2
𝜉 := (1 + 𝑘2 ) 𝜆𝜏𝑘 ∈ 𝐿1𝜏𝑘 (𝑀 ),
where
(6.8)
𝐶𝑘 := 1 + 𝑘.
𝐿𝑟𝜏 (𝑀 ) := {𝐻 ∈ 𝐿2𝑙𝑜𝑐 (𝑀 ) : 𝐻1[0,𝜏 ] ∈ 𝐿𝑟 (𝑀 )} for a stopping time 𝜏 and
𝑟 ≥ 1. Similarly, we set ℋ𝜏2 = {𝑋 ∈ 𝒮 : 𝑋 𝜏 ∈ ℋ2 }. Now the following can
Here
be shown using a technique of Kobylanski [22].
For xed 𝑘,
˜ in ℋ𝜏2 ,
˜ in 𝐿2𝜏 (𝑀 ) and 𝑁 𝑛 → 𝑁
(i)
→𝑍
𝑘
𝑘
(ii) sup𝑡≤𝑇 ∣𝐿𝑛𝑡∧𝜏𝑘 − 𝐿˜𝑡∧𝜏𝑘 ∣ → 0 𝑃 -a.s.
Lemma 6.12.
𝑍𝑛
𝑘 , it follows
˜ 𝑡 ∣ → 0 𝑃 -a.s.
sup𝑡≤𝑇 ∣𝐿𝑛𝑡 − 𝐿
˜ 𝑍,
˜ 𝑁
˜ ) satises the
limit (𝐿,
The proof is deferred to Appendix B. Since (ii) holds for all
that
˜
𝐿
is continuous. Now Dini's Lemma shows
as claimed.
Lemma 6.12 also implies that the
[0, 𝜏𝑘 ] for all 𝑘 , hence on [0, 𝑇 ].
𝐿𝑛𝑇 = 𝐷𝑇 = exp(𝐵) for all 𝑛.
BSDE (6.4) on
satised as
The terminal condition is
To end the proof, note that the convergences hold for the original sequence
(𝑝𝑛 ),
rather than just for a subsequence, since
and since our choice of
(𝜏𝑘 )
𝑝 7→ 𝐿(𝑝) is monotone
does not depend on the subsequence.
We can now nish the proof of Theorem 6.6 (and Theorem 3.2).
Proof of Theorem 6.6.
Part (i) was already proved. For (ii), uniform conver-
gence and continuity were shown in Lemma 6.10. In view of (4.8) and (6.5),
the claim for the strategies is that
exp
(1 − 𝑝) 𝜋
ˆ (𝑝) = 𝜆 +
𝑍 𝐿(𝑝)
𝑍𝐿
→ 𝜆 + exp = 𝜗ˆ
𝐿(𝑝)
𝐿
29
in
𝐿2𝑙𝑜𝑐 (𝑀 ).
By a localization as in the previous proof, we may assume that 𝐿(𝑝) +
(𝐿(𝑝))−1 + 𝐿exp + (𝐿exp )−1 is bounded uniformly in 𝑝, and, by Lemma 6.10,
𝐿(𝑝) ∙ 𝑀 → 𝑍 exp ∙ 𝑀 in ℋ2 . We have
that 𝑍
𝐿(𝑝)
𝐿exp
𝑍
𝐿(𝑝) ∙ 𝑀 − 𝑍𝐿exp ∙ 𝑀 2
ℋ
(
) exp
exp )
1 ( 𝐿(𝑝)
1
𝐿
1
∙ 𝑀
∙ 𝑀
≤ 𝐿(𝑝) 𝑍
− 𝑍𝐿
+
−
𝑍
.
exp
𝐿
𝐿(𝑝)
2
2
ℋ
ℋ
𝐿exp
Clearly the rst norm converges to zero. Noting that 𝑍
𝐵𝑀 𝑂)
∙
𝑀∈
ℋ2 (even
due to Lemma 5.9, the second norm tends to zero by dominated
convergence for stochastic integrals.
The last result of this section concerns the convergence of the (normalized) solution
𝑌ˆ (𝑝)
after Remark 3.3.
of the dual problem (4.3); see also the comment
We recall the assumption (3.3) and that there is no
intermediate consumption.
measure which minimizes the relative entropy
𝑑𝑃 (𝐵) := (𝑒𝐵 /𝐸[𝑒𝐵 ]) 𝑑𝑃 .
𝑄𝐸 (𝐵) ∈ M be the
𝐻( ⋅ ∣𝑃 (𝐵)) over M , where
To state the result, let
For
𝐵 = 0
this is simply the minimal entropy
martingale measure, and the existence of
𝑄𝐸 (𝐵)
follows from the latter by
a change of measure.
Let 𝑆 be continuous and assume that 𝐿(𝑝) is
ˆ
for all 𝑝 < 0. Then 𝑌 (𝑝)/𝑌ˆ0 (𝑝) converges in the semimartingale
the density process of 𝑄𝐸 (𝐵) as 𝑝 → −∞.
Proposition 6.13.
Proof.
continuous
topology to
𝐿−1 ∙ 𝑁 → (𝐿exp )−1 ∙ 𝑁 exp in
𝑌ˆ /𝑌ˆ0 = ℰ(−𝜆 ∙ 𝑀 + 𝐿−1 ∙ 𝑁 ) by (4.9),
(
)
𝑌ˆ /𝑌ˆ0 → ℰ − 𝜆 ∙ 𝑀 + (𝐿exp )−1 ∙ 𝑁 exp in
We deduce from Lemma 6.10 that
2 , as in the previous proof. Since
ℋ𝑙𝑜𝑐
Lemma A.2(ii) shows that
the semimartingale topology. The right hand side is the density process of
𝑄𝐸 (𝐵);
7
this follows, e.g., from [10, Proposition 1].
The Limit
𝑝→0
In this section we prove Theorem 3.4, some renements of that result, as well
as the corresponding convergence for the opportunity processes and the dual
problem.
Due to substantial technical dierences, we consider separately
and from above. Recall the semimartingale
below
[ ∫ 𝑇 𝑝 → ∘0 from
]
ℱ𝑡 with canonical decomposition
𝐷
𝜇
(𝑑𝑠)
𝑠
𝑡
] ∫ 𝑡
[∫ 𝑇
𝜂𝑡 = (𝜂0 + 𝑀𝑡𝜂 ) + 𝐴𝜂𝑡 = 𝐸
𝐷𝑠 𝜇∘ (𝑑𝑠)ℱ𝑡 −
𝐷𝑠 𝜇(𝑑𝑠).
(7.1)
the limits
𝜂𝑡 = 𝐸
0
Clearly
𝜂
0
is a supermartingale with continuous nite variation part, and a
martingale in the case without intermediate consumption (𝜇
= 0).
From (2.4)
we have the uniform bounds
0 < 𝑘1 ≤ 𝜂 ≤ (1 + 𝑇 )𝑘2 .
30
(7.2)
7.1
The Limit
𝑝 → 0−
We start with the convergence of the opportunity processes.
As 𝑝 → 0−,
(i) for each 𝑡 ∈ [0, 𝑇 ], 𝐿∗𝑡 (𝑝) → 𝜂𝑡 𝑃 -a.s. and in 𝐿𝑟 (𝑃 ) for 𝑟 ∈ [1, ∞),
with a uniform bound.
(ii) if 𝔽 is continuous, then 𝐿∗𝑡 (𝑝) → 𝜂𝑡 uniformly in 𝑡, 𝑃 -a.s.; and in ℛ𝑟
for 𝑟 ∈ [1, ∞).
(iii) if 𝑢𝑝0 (𝑥0 ) < ∞ for some 𝑝0 ∈ (0, 1), then 𝐿∗𝑡 (𝑝) → 𝜂𝑡 uniformly in 𝑡,
𝑃 -a.s.; in ℛ𝑟 for 𝑟 ∈ [1, ∞); and prelocally in ℛ∞ .
The same assertions hold for 𝐿∗ replaced by 𝐿.
Proposition 7.1.
Proof.
We note that
𝐿 = (𝐿∗ )1/𝛽 ,
𝑝 → 0−
𝑞 → 0+ and 𝛽 → 1−. In view
𝐿∗ . From Lemma 4.1,
implies
of
it suces to prove the claims for
0 ≤ 𝐿∗𝑡 (𝑝) ≤ 𝜇∘ [𝑡, 𝑇 ]−𝛽𝑝 𝐸
[∫
𝑡
𝑇
]𝛽
𝐷𝑠 𝜇∘ (𝑑𝑠)ℱ𝑡 → 𝜂𝑡
To obtain a lower bound, we consider the density process
in
𝑌
ℛ∞ .
of some
(7.3)
𝑄 ∈ M.
(i) Using (4.4) we obtain
𝐿∗𝑡 (𝑝)
𝑇
∫
≥
]
[
𝐸 𝐷𝑠𝛽 (𝑌𝑠 /𝑌𝑡 )𝑞 ℱ𝑡 𝜇∘ (𝑑𝑠).
𝑡
𝐷𝑠𝛽 → 𝐷𝑠
ℛ∞
(𝑌𝑠 /𝑌𝑡 )𝑞 → 1 𝑃 -a.s. for 𝑞 → 0. We can argue
𝑞
1
as in Proposition 6.1: For 𝑠 ≥ 𝑡 xed, 0 ≤ (𝑌𝑠 /𝑌𝑡 ) ≤ 1 + 𝑌𝑠 /𝑌𝑡 ∈ 𝐿 (𝑃 )
]
[ 𝛽
𝑞
𝑞
yields 𝐸 𝐷𝑠 (𝑌𝑠 /𝑌𝑡 ) ℱ𝑡 → 𝐸[𝐷𝑠 ∣ℱ𝑡 ] 𝑃 -a.s. Since 𝑌 is a supermartingale,
]
[
0 ≤ 𝐸 𝐷𝑠𝛽 (𝑌𝑠 /𝑌𝑡 )𝑞 ℱ𝑡 ≤ 1 ∨ 𝑘2 , and we conclude for each 𝑡 that
Clearly
∫
𝑇
in
and
]
[
𝐸 𝐷𝑠𝛽 (𝑌𝑠 /𝑌𝑡 )𝑞 ℱ𝑡 𝜇∘ (𝑑𝑠) →
∫
𝑡
[ ]
𝐸 𝐷𝑠 ℱ𝑡 𝜇∘ (𝑑𝑠) = 𝜂𝑡 𝑃 -a.s.
𝑡
𝐿∗𝑡 (𝑝) → 𝜂𝑡 𝑃 -a.s.
Hence
𝑇
and the convergence in
𝐿𝑟 (𝑃 )
follows by the
bound (7.3).
(ii)
Assume that
𝔽
is continuous.
Our argument will be similar to
Proposition 6.1, but the source of monotonicity is dierent.
[0, 𝑇 ] × Ω
(𝑠, 𝜔) ∈
and consider
] 1
[
𝑔𝑞 (𝑡) := 𝐸 (𝑌𝑠 /𝑌𝑡 )𝑞 ℱ𝑡 1−𝑞 (𝜔),
Then
Fix
𝑔𝑞 (𝑡)
is continuous in
Dini's lemma yields
𝑔𝑞 → 1
𝑡
and decreasing in
uniformly on
31
[0, 𝑠],
𝑡 ∈ [0, 𝑠].
𝑞
by virtue of Lemma 5.5.
hence
]
[
𝐸 (𝑌𝑠 /𝑌𝑡 )𝑞 ℱ𝑡 → 1
𝑡.
uniformly in
formly in
𝑡
We deduce that
]
[
𝐸 𝐷𝑠𝛽 (𝑌𝑠 /𝑌𝑡 )𝑞 ℱ𝑡 (𝜔) → 𝐸[𝐷𝑠 ∣ℱ𝑡 ](𝜔)
uni-
since
[
]
𝐸 𝐷𝑠𝛽 (𝑌𝑠 /𝑌𝑡 )𝑞 ℱ𝑡 − 𝐸[𝐷𝑠 ∣ℱ𝑡 ]
] [
]
[
≤ 𝐸 ∣𝐷𝑠𝛽 − 𝐷𝑠 ∣(𝑌𝑠 /𝑌𝑡 )𝑞 ℱ𝑡 + 𝐸 𝐷𝑠 {(𝑌𝑠 /𝑌𝑡 )𝑞 − 1}ℱ𝑡 [
]
]
[
≤ ∥𝐷𝑠𝛽 − 𝐷𝑠 ∥𝐿∞ (𝑃 ) 𝐸 (𝑌𝑠 /𝑌𝑡 )𝑞 ℱ𝑡 + ∥𝐷𝑠 ∥𝐿∞ (𝑃 ) 𝐸 (𝑌𝑠 /𝑌𝑡 )𝑞 ℱ𝑡 − 1
[
]
≤ ∥𝐷𝑠𝛽 − 𝐷𝑠 ∥𝐿∞ (𝑃 ) + 𝑘2 𝐸 (𝑌𝑠 /𝑌𝑡 )𝑞 ℱ𝑡 − 1.
The rest of the argument is like the end of the proof of Proposition 6.1.
(iii) Let
𝑢𝑝0 (𝑥0 ) < ∞
for some
𝑝0 ∈ (0, 1).
Then we can take a dierent
approach via Proposition 5.1, which shows that
𝐿∗𝑡 (𝑝)
≥ 𝐸
[∫
𝑡
for all
𝑝 < 0,
𝑇
]1−𝑞/𝑞0 (
)𝑞/𝑞0
𝐷𝑠𝛽 𝜇∘ (𝑑𝑠)ℱ𝑡
𝑘1𝛽−𝛽0 𝐿∗𝑡 (𝑝0 )
where we note that
𝑞0 < 0.
Using that almost every path of
𝐿∗ (𝑝
0 ) is bounded and bounded away from zero (Lemma 4.1), the right hand
∫𝑇
𝑃 -a.s. tends to 𝜂𝑡 = 𝐸[ 𝑡 𝐷𝑠 𝜇∘ (𝑑𝑠)∣ℱ𝑡 ] uniformly in 𝑡 as 𝑞 → 0. Since
𝐿∗ (𝑝0 ) is prelocally bounded, the prelocal convergence in ℛ∞ follows in the
side
same way.
Remark 7.2. One can ask when the convergence in Proposition 7.1 holds
even in
ℛ∞ .
The following statements remain valid if
𝐿∗
replaced by
𝐿.
𝑢𝑝0 (𝑥0 ) < ∞ for some 𝑝0 ∈ (0, 1), and in addition
∗
that 𝐿 (𝑝0 ) is (locally) bounded. Then the argument for Proposi∗
∞ (ℛ∞ ).
tion 7.1(iii) shows 𝐿 (𝑝) → 𝜂 in ℛ
𝑙𝑜𝑐
∗
∞ (ℛ∞ ) implies that 𝐿∗ (𝑝) is (locally)
(ii) Conversely, 𝐿 (𝑝) → 𝜂 in ℛ
𝑙𝑜𝑐
bounded away from zero for all 𝑝 < 0 close to zero, because 𝜂 ≥ 𝑘1 > 0.
(i) Assume again that
As we turn to the convergence of the martingale part
𝑀 𝐿(𝑝) ,
a suitable
localization will again be crucial.
Lemma 7.3.
(
Proof.
Let 𝑝1 < 0. There exists a localizing sequence (𝜎𝑛 ) such that
)𝜎
𝐿(𝑝) −𝑛 > 1/𝑛
simultaneously for all 𝑝 ∈ [𝑝1 , 0).
This follows from Proposition 5.4 and Lemma 4.1.
Next, we state a basic result (i) for the convergence of
𝑀 𝐿(𝑝)
in
2
ℋ𝑙𝑜𝑐
and
stronger convergences under additional assumptions (ii) and (iii), for which
Remark 7.2(i) gives sucient conditions.
32
Assume that 𝑆 is continuous. As 𝑝 → 0−,
2 .
(i)
→
in ℋ𝑙𝑜𝑐
𝐿(𝑝) → 𝑀 𝜂 in 𝐵𝑀 𝑂 .
(ii) if 𝐿(𝑝) → 𝜂 in ℛ∞
𝑙𝑜𝑐
𝑙𝑜𝑐 , then 𝑀
(iii) if 𝐿(𝑝) → 𝜂 in ℛ∞ , then 𝑀 𝐿(𝑝) → 𝑀 𝜂 in 𝐵𝑀 𝑂.
Proposition 7.4.
𝑀 𝐿(𝑝)
Proof.
𝑀𝜂
𝑋 = 𝑋(𝑝) = 𝜂 − 𝐿(𝑝).
𝑋 is bounded uniformly in 𝑝
𝑀 𝑋(𝑝) → 0. Lemma 5.9 applied to
∥𝜂∥∞ −𝜂 shows that 𝑀 𝜂 ∈ 𝐵𝑀 𝑂. We may restrict our attention to 𝑝 in some
𝐿(𝑝) ∥
interval [𝑝1 , 0) and Lemma 5.11 shows that sup𝑝∈[𝑝 ,0) ∥𝑀
𝐵𝑀 𝑂 < ∞.
1
𝐿
𝐿
𝐿
∙
Due to the orthogonality of the sum 𝑀
= 𝑍
𝑀 + 𝑁 , we have in
Set
Then
by Lemma 4.1 and our aim is to prove
particular that
sup ∥𝑍 𝐿 (𝑝) ∙ 𝑀 ∥𝐵𝑀 𝑂 < ∞.
(7.4)
𝑝∈[𝑝1 ,0)
Under the condition of (iii),
to zero since
𝜂 ≥ 𝑘 1 > 0;
𝐿(𝑝)
is bounded away from zero for all
moreover,
𝜆
∙
𝑀 ∈ 𝐵𝑀 𝑂
(i) and (ii) we may assume by a localization as in Lemma 7.3 that
bounded away from zero uniformly in
assume that
𝜆 ∙ 𝑀 ∈ 𝐵𝑀 𝑂,
𝑝.
Since
𝑀
𝑝
close
by Corollary 5.12. For
𝐿− (𝑝)
is
is continuous, we may also
by another localization.
Using the formula (4.7) for
𝐴𝐿
and the decomposition (7.1) of
𝜂,
the
𝑋 is continuous and
nite variation part 𝐴
{
}
2 𝑑𝐴𝑋 = 2 (1 − 𝑝)𝐷𝛽 𝐿𝑞− − 𝐷 𝑑𝜇
(7.5)
}
{
)
(
𝐿 ⊤
𝑑⟨𝑀 ⟩ 𝑍 𝐿 .
− 𝑞 𝐿− 𝜆⊤ 𝑑⟨𝑀 ⟩ 𝜆 + 2𝜆⊤ 𝑑⟨𝑀 ⟩ 𝑍 𝐿 + 𝐿−1
− 𝑍
In particular, we note that
𝑋
[𝑀 ] = [𝑋] −
𝑋02
2
=𝑋 −
𝑋02
∫
−2
𝑋− 𝑑𝑋.
(7.6)
𝑋02 → 0 and 𝐸[𝑋𝑇2 ] → 0 by Proposition 7.1 (Remark 6.2
∞ by assumption and under (ii)
applies). In case (iii) we have 𝑋 → 0 in ℛ
]
[ 2
1
2
the same holds after a localization. If we denote 𝑜𝑡 := 𝐸 𝑋𝑇 − 𝑋𝑡 ℱ𝑡 , we
1
1
∞ in cases (ii) and
therefore have that 𝑜0 → 0 in case (i) and 𝑜 → 0 in ℛ
]
[
∫
𝑇
2
𝛽 𝑞
(iii). Denote also 𝑜𝑡 := 2𝐸
𝑡 𝑋− {(1 − 𝑝)𝐷 𝐿− − 𝐷} 𝑑𝜇 ℱ𝑡 . Recalling
𝑞
𝛽
that 𝑝 → 0− implies 𝑞 → 0+ and 𝛽 → 1−, we have (1 − 𝑝)𝐷 𝐿− − 𝐷 → 0 in
2
∞
ℛ∞ and since 𝑋− is bounded uniformly
follows that 𝑜 → 0 in ℛ .
∫ in 𝑝, it 𝑋
𝑋
As 𝑀
∈ 𝐵𝑀 𝑂 and 𝑋− is bounded, 𝑋− 𝑑𝑀 is a martingale and (7.6)
For case (i) we have
yields
[
]
]
[
[
𝐸 [𝑀 𝑋 ]𝑇 − [𝑀 𝑋 ]𝑡 ℱ𝑡 = 𝐸 𝑋𝑇2 − 𝑋𝑡2 ℱ𝑡 − 2𝐸
∫
𝑡
33
𝑇
]
𝑋− 𝑑𝐴𝑋 ℱ𝑡 .
Using (7.5) and the denitions of
𝑜1
and
𝑜2 ,
we can rewrite this as
]
[
𝐸 [𝑀 𝑋 ]𝑇 − [𝑀 𝑋 ]𝑡 ℱ𝑡 − 𝑜1𝑡 + 𝑜2𝑡
[∫ 𝑇
{
( 𝐿 )⊤
} ]
𝐿 𝑋− 𝐿− 𝜆⊤ 𝑑⟨𝑀 ⟩ 𝜆 + 2𝜆⊤ 𝑑⟨𝑀 ⟩ 𝑍 𝐿 + 𝐿−1
𝑍
= 𝑞𝐸
𝑑⟨𝑀
⟩
𝑍
ℱ𝑡 .
−
𝑡
Applying the Cauchy-Schwarz inequality and using that
𝑋− , 𝐿− , 𝐿−1
−
are
𝑝, it follows that
]
[
𝐸 [𝑀 𝑋 ]𝑇 − [𝑀 𝑋 ]𝑡 ℱ𝑡 − 𝑜1𝑡 + 𝑜2𝑡
]
[∫ 𝑇
𝑋− (1 + 𝐿− )𝜆⊤ 𝑑⟨𝑀 ⟩ 𝜆ℱ𝑡
≤ 𝑞𝐸
𝑡
]
[∫ 𝑇
( 𝐿 )⊤
𝑋− (1 + 𝐿−1
𝑑⟨𝑀 ⟩ 𝑍 𝐿 ℱ𝑡
+ 𝑞𝐸
− ) 𝑍
𝑡
(
)
∙
≤ 𝑞𝐶 ∥𝜆 𝑀 ∥𝐵𝑀 𝑂 + ∥𝑍 𝐿(𝑝) ∙ 𝑀 ∥𝐵𝑀 𝑂 ,
bounded uniformly in
where
𝐶 > 0 is a constant independent of 𝑝 and 𝑡. In view of
𝑞𝐶 ′ with a constant 𝐶 ′ > 0 and we
]
[
𝐸 [𝑀 𝑋 ]𝑇 − [𝑀 𝑋 ]𝑡 ℱ𝑡 ≤ 𝑞𝐶 ′ + 𝑜1𝑡 − 𝑜2𝑡 .
hand side is bounded by
(7.4), the right
have
For (i) we only have to prove the convergence to zero of the left hand side
for
𝑡 = 0 and so this ends the proof. For (ii) and (iii) we use [𝑀 𝑋 ]𝑡 =
+ (Δ𝑀𝑡𝑋 )2 and ∣Δ𝑀 𝑋 ∣ = ∣Δ𝑋∣ ≤ 2∥𝑋∥ℛ∞ to obtain
]
[
sup 𝐸 [𝑀 𝑋 ]𝑇 − [𝑀 𝑋 ]𝑡− ℱ𝑡 ≤ 𝑞𝐶 ′ + ∥𝑜1 ∥ℛ∞ + ∥𝑜2 ∥ℛ∞ + 4∥𝑋∥2ℛ∞
[𝑀 𝑋 ]𝑡−
𝑡≤𝑇
and we have seen that the right hand side tends to
7.2
The Limit
0
as
𝑝 → 0−.
𝑝 → 0+
𝐿(𝑝) for 𝑝 → 0+ is meaningless without supposing
𝑢𝑝0 (𝑥0 ) < ∞ for some 𝑝0 ∈ (0, 1), so we make this a standing assumption
We notice that the limit of
that
for the entire Section 7.2. We begin with a result on the integrability of the
tail of the sequence.
Lemma 7.5.
that
Let 1 ≤ 𝑟 < ∞. There exists a localizing sequence (𝜎𝑛 ) such
ess sup
𝐿𝑡∧𝜎𝑛 (𝑝)
𝑡∈[0,𝑇 ], 𝑝∈(0,𝑝0 /𝑟]
Proof.
is in 𝐿𝑟 (𝑃 ) for all 𝑛.
𝑝1 = 𝑝0 /𝑟 and 𝜎𝑛 = inf{𝑡 > 0 : 𝐿𝑡 (𝑝1 ) > 𝑛} ∧ 𝑇 , then by
𝑟
Corollary 5.2(ii), sup𝑡 𝐿𝑡∧𝜎𝑛 (𝑝1 ) ≤ 𝑛 + Δ𝐿𝜎𝑛 (𝑝1 ) ∈ 𝐿 (𝑃 ). But 𝐿(𝑝) ≤
𝐶𝐿(𝑝1 ) by Corollary 5.2(i), so (𝜎𝑛 ) already satises the requirement.
Let
34
As 𝑝 → 0+,
Proposition 7.6.
𝐿∗ (𝑝) → 𝜂,
uniformly in 𝑡, 𝑃 -a.s.; in ℛ𝑟𝑙𝑜𝑐 for 𝑟 ∈ [1, ∞); and prelocally in ℛ∞ . Moreover, the convergence takes place in ℛ∞ (in ℛ∞
𝑙𝑜𝑐 ) if and only if 𝐿(𝑝1 ) is
(locally) bounded for some 𝑝1 ∈ (0, 𝑝0 ). The same assertions hold for 𝐿∗
replaced by 𝐿.
Proof.
implies
𝑝 ∈ (0, 𝑝0 ) in this proof and recall that 𝑝 → 0+
𝛽 → 1−. Since 𝐿 = (𝐿∗ )1/𝛽 , it suces to prove the
We consider only
𝑞 → 0− and
𝐿∗ . Using Lemma
claims for
𝐿∗𝑡 (𝑝)
∘
−𝛽𝑝
≥ 𝜇 [𝑡, 𝑇 ]
4.1,
𝐸
[∫
𝑇
𝑡
]𝛽
𝐷𝑠 𝜇∘ (𝑑𝑠)ℱ𝑡 → 𝜂𝑡
in
ℛ∞ .
(7.7)
Conversely, by Proposition 5.1,
𝐿∗𝑡 (𝑝)
≤ 𝐸
[∫
𝑇
𝐷𝑠𝛽
𝑡
]1−𝑞/𝑞0 (
)𝑞/𝑞0
𝜇 (𝑑𝑠)ℱ𝑡
𝑘1𝛽−𝛽0 𝐿∗𝑡 (𝑝0 )
.
∘
𝐿∗ (𝑝0 ) is
𝑡 as 𝑞 → 0−.
Since almost every path of
tends to
𝜂𝑡
uniformly in
bounded, the right hand side
By localizing
𝐿∗ (𝑝0 )
∗
We have proved that 𝐿 (𝑝)
𝑃 -a.s.
to be prelocally
bounded, the same argument shows the prelocal convergence in
Lemma 7.5, the convergence in
(7.8)
ℛ∞ .
→ 𝜂 uniformly in 𝑡, 𝑃 -a.s. In view
ℛ𝑟𝑙𝑜𝑐 follows by dominated convergence.
of
For the second claim, note that the if statement is shown exactly like
ℛ∞ convergence and the converse holds by boundedness of 𝜂 .
Of course, if 𝐿(𝑝1 ) is (locally) bounded for some 𝑝1 ∈ (0, 𝑝0 ), then in fact
𝐿(𝑝) has this property for all 𝑝 ∈ (0, 𝑝1 ], by Corollary 5.2(i).
the prelocal
We turn to the convergence of the martingale part. The major diculty
will be that
𝐿(𝑝)
may have unbounded jumps; i.e., we have to prove the
convergence of quadratic BSDEs whose solutions are not locally bounded.
Assume that 𝑆 is continuous. As 𝑝 → 0+,
2 .
(i)
→
in ℋ𝑙𝑜𝑐
(ii) if there exists 𝑝1 ∈ (0, 𝑝0 ] such that 𝐿(𝑝1 ) is locally bounded, then
𝑀 𝐿(𝑝) → 𝑀 𝜂 in 𝐵𝑀 𝑂𝑙𝑜𝑐 .
(iii) if there exists 𝑝1 ∈ (0, 𝑝0 ] such that 𝐿(𝑝1 ) is bounded, then 𝑀 𝐿(𝑝) → 𝑀 𝜂
in 𝐵𝑀 𝑂.
Proposition 7.7.
𝑀 𝐿(𝑝)
𝑀𝜂
The following terminology will be useful in the proof. We say that real
numbers
(𝑥𝜀 )
converge to
𝑥
linearly
as
𝜀→0
if
lim sup 1𝜀 ∣𝑥𝜀 − 𝑥∣ < ∞.
𝜀→0+
35
Let 𝑥𝜀 → 𝑥 linearly and 𝑦𝜀 → 𝑦 linearly. Then
(i) lim sup𝜀→0 1𝜀 ∣𝑥𝜀 − 𝑦𝜀 ∣ < ∞ if 𝑥 = 𝑦,
(ii) 𝑥𝜀 𝑦𝜀 → 𝑥𝑦 linearly,
(iii) if 𝑥 > 0 and 𝜑 is a real function with 𝜑(0) = 1 and dierentiable at 0,
then (𝑥𝜀 )𝜑(𝜀) → 𝑥 linearly.
Lemma 7.8.
Proof.
from
(i) This is immediate from the triangle inequality. (ii) This follows
∣𝑥𝜀 𝑦𝜀 − 𝑥𝑦∣ ≤ ∣𝑥𝜀 ∣∣𝑦𝜀 − 𝑦∣ + ∣𝑦∣∣𝑥𝜀 − 𝑥∣ because convergent sequences are
bounded. (iii) Here we use
∣(𝑥𝜀 )𝜑(𝜀) − 𝑥∣ ≤ ∣𝑥𝜀 ∣∣(𝑥𝜀 )𝜑(𝜀)−1 − 1∣ + ∣𝑥𝜀 − 𝑥∣;
𝑥𝜀 → 𝑥 linearly, the question is reduced to the
−1
𝜑(𝜀)−1
boundedness of 𝜀
∣(𝑥𝜀 )
− 1∣. Fix 0 < 𝛿1 < 𝑥 < 𝛿2 and set 𝜚(𝛿, 𝜀) :=
∣𝛿 𝜑(𝜀)−1 − 1∣. For 𝜀 small enough, 𝑥𝜀 ∈ [𝛿1 , 𝛿2 ] and then
as
{𝑥𝜀 }
is bounded and
𝜚(𝛿1 , 𝜀) ∧ 𝜚(𝛿2 , 𝜀) ≤ ∣(𝑥𝜀 )𝜑(𝜀)−1 − 1∣ ≤ 𝜚(𝛿1 , 𝜀) ∨ 𝜚(𝛿2 , 𝜀).
′
−1 ∣𝜚(𝛿, 𝜀)∣ = 𝑑 𝛿 𝜑(𝜀) ∣
For 𝛿 > 0 we have lim𝜀 𝜀
𝜀=0 = ∣ log(𝛿)𝜑 (0)∣ < ∞.
𝑑𝜀
Hence the upper and the lower bound above converge to 0 linearly.
Proof of Proposition 7.7.
We rst prove (ii) and (iii), i.e, we assume that
𝐿(𝑝) ≥ 𝑘1 from Lemma 4.1.
𝐶 > 0 independent of 𝑝 such that
Hence 𝐿(𝑝) is bounded uniformly in
𝐿(𝑝1 ) is locally bounded (resp. bounded).
Recall
By Corollary 5.2(i) there exists a constant
𝐿(𝑝) ≤ 𝐶𝐿(𝑝1 ) for all 𝑝 ∈ (0, 𝑝1 ].
𝑝 ∈ (0, 𝑝1 ] in the case (iii) and for (ii) this holds after a localization. Now
𝐿(𝑝) ∥
Lemma 5.11(ii) implies sup𝑝∈(0,𝑝 ] ∥𝑀
𝐵𝑀 𝑂 < ∞ and we can proceed
1
exactly as in the proof of items (ii) and (iii) of Proposition 7.4.
(i)
This case is more dicult because we have to use prelocal bounds
and Lemma 5.11(ii) does not apply.
Again, we want to imitate the proof
of Proposition 7.4(i), or more precisely, the arguments after (7.6). We note
2 -convergence those estimates are required only at
ℋ𝑙𝑜𝑐
𝐵𝑀 𝑂-norms can be replaced by ℋ2 -norms. Inspecting
that for the claimed
𝑡 = 0
and so the
that proof in detail, we see that we can proceed in the same way once we
establish:
∙
There exists a localizing sequence
all
(𝜎𝑛 )
and constants
𝐶𝑛
such that for
𝑛,
(b)
(𝐻1[0,𝜎𝑛 ] ) ∙ 𝑀 𝐿(𝑝) is a martingale for all 𝐻 predictable and bounded,
and all 𝑝 ∈ (0, 𝑝0 ),
(
)
sup𝑝∈(0,𝑝0 ] 𝐿− (𝑝) + 𝐿−1
− (𝑝) ≤ 𝐶𝑛 on [0, 𝜎𝑛 ],
(c)
lim sup𝑝→0+ ∥𝑍 𝐿(𝑝) 1[0,𝜎𝑛 ] ∥𝐿2 (𝑀 ) ≤ 𝐶𝑛 .
(a)
36
We may assume by localization that
(𝜎𝑛 )
𝑝 ∈ (0, 𝑝0 ).
instead of indicating
(a)
Fix
𝜆 ∙ 𝑀 ∈ ℋ2 .
We now prove (a)-(c);
explicitly, we write by localization. . . as usual.
By Lemma 4.1 and Lemma 5.2(ii),
𝐿 = 𝐿(𝑝) is
𝐿 =
𝐿 is a
and 𝑀
a supermartingale of class (D). Hence its Doob-Meyer decomposition
𝐿0 + 𝑀 𝐿 + 𝐴𝐿
is such that
𝐴𝐿
is decreasing and nonpositive,
true martingale. Thus
0 ≤ 𝐸[−𝐴𝐿
𝑇 ] = 𝐸[𝐿0 − 𝐿𝑇 ] < ∞.
After localizing as in Lemma 7.5 (with
𝐿
Hence sup𝑡 ∣𝑀𝑡 ∣
≤ sup𝑡 𝐿𝑡 −
𝐴𝐿
𝑇
∈
𝑟 = 1),
𝐿1 (𝑃 ).
we have
sup𝑡 𝐿𝑡 ∈ 𝐿1 (𝑃 ).
Now (a) follows by the BDG
inequalities exactly as in the proof of Lemma 5.9.
(b) We have 𝐿− (𝑝) ≥ 𝑘1 by Lemma 4.1. Conversely, by Corollary 5.2(i),
𝐿− (𝑝) ≤ 𝐶𝐿− (𝑝0 ) for 𝑝 ∈ (0, 𝑝0 ] with some universal constant 𝐶 > 0, and
𝐿− (𝑝0 ) is locally bounded by left-continuity.
(c) We shall use the rate of convergence obtained for 𝐿(𝑝) and the
𝐿 contained in 𝐴𝐿 via the Bellman BSDE. We may
information about 𝑍
assume by localization that (a) and (b) hold with 𝜎𝑛 replaced by 𝑇 . Thus
it suces to show that
√
𝑍 𝐿(𝑝) √
lim sup 𝐿
(𝑝)𝜆
+
< ∞.
−
𝐿− (𝑝) 𝐿2 (𝑀 )
𝑝→0+
Suppressing again
𝑝
in the notation, (a) and the formula (4.7) for
𝐴𝐿
imply
𝐸[𝐿0 − 𝐿𝑇 ] = 𝐸[−𝐴𝐿
𝑇]
∫ 𝑇
[
] 𝑞 [∫ 𝑇
(
(
𝑍 𝐿 )⊤
𝑍 𝐿 )]
𝛽 𝑞
.
= 𝐸 (1 − 𝑝)
𝐷 𝐿− 𝑑𝜇 − 𝐸
𝐿− 𝜆 +
𝑑⟨𝑀 ⟩ 𝜆 +
2
𝐿−
𝐿−
0
0
Recalling that
√
1
2
𝐿𝑇 = 𝐷𝑇 ,
this yields
[∫ 𝑇
(
(
𝑍 𝐿 )]
𝑍𝐿 𝑍 𝐿 )⊤
1
𝐿− 𝜆 + √ 2
= 2𝐸
𝐿− 𝜆 +
𝑑⟨𝑀 ⟩ 𝜆 +
𝐿−
𝐿−
𝐿− 𝐿 (𝑀 )
0
(
∫ 𝑇
[
])
𝛽 𝑞
1
= ∣𝑞∣ 𝐸[𝐿0 − 𝐿𝑇 ] − 𝐸 (1 − 𝑝)
𝐷 𝐿− 𝑑𝜇
0
(
∫ 𝑇
[
])
𝛽 𝑞
1
= ∣𝑞∣ 𝐿0 − 𝐸 𝐷𝑇 + (1 − 𝑝)
𝐷 𝐿− 𝑑𝜇
0
=
1
∣𝑞∣ (𝐿0
− Γ0 ),
∫𝑇
Γ0 = Γ0 (𝑝) = 𝐸[𝐷𝑇 + (1 − 𝑝) 0 𝐷𝛽 𝐿𝑞− 𝑑𝜇]. We know that
[∫𝑇
]
Γ0 converge to 𝜂0 = 𝐸 0 𝐷𝑠 𝜇∘ (𝑑𝑠) as 𝑝 → 0+ (and hence
where we have set
𝐿0 and
𝑞 → 0−). However,
both
we are asking for the stronger result
1
∣𝐿0 (𝑝) − Γ0 (𝑝)∣ < ∞.
lim sup ∣𝑞∣
𝑝→0+
37
𝐿0 (𝑝) → 𝜂0 linearly and Γ0 (𝑝) → 𝜂0
𝑡 = 0 read
]1/𝛽+𝑝/𝑞0 (
)𝑞/𝑞0
1−𝛽 /𝛽
.
𝐷𝑠𝛽 𝜇∘ (𝑑𝑠)
𝑘1 0 𝐿0 (𝑝0 )
By Lemma 7.8(i), it suces to show that
linearly. Using
𝐿∗ = 𝐿𝛽 ,
inequalities (7.7) and (7.8) evaluated at
𝜇∘ [0, 𝑇 ]−𝑝 𝜂0 ≤ 𝐿0 (𝑝) ≤ 𝐸
[∫
𝑇
0
Recalling the bound (2.4) for 𝐷 , items (ii) and (iii) of Lemma 7.8 yield that
𝐿0 (𝑝) → 𝜂0 linearly. The second claim, that Γ0 (𝑝) → 𝜂0 linearly, follows from
the denitions of Γ0 (𝑝) and 𝜂0 using again (2.4) and the uniform bounds for
𝐿− from (b). This ends the proof.
7.3
Proof of Theorem 3.4 and Other Consequences
Assume that 𝑆 is continuous and that there exists 𝑝0 > 0 such
that 𝑢𝑝0 (𝑥0 ) < ∞. As 𝑝 → 0,
Lemma 7.9.
𝑍 𝐿(𝑝)
𝑍𝜂
→
𝐿− (𝑝)
𝜂−
in 𝐿2𝑙𝑜𝑐 (𝑀 ) and
1
𝐿− (𝑝)
∙
𝑁 (𝑝) →
1
𝜂−
∙
𝑁𝜂
2
.
in ℋ𝑙𝑜𝑐
(7.9)
For a sequence 𝑝 → 0− the convergence 𝐿𝑍− (𝑝) → 𝑍𝜂− in 𝐿2𝑙𝑜𝑐 (𝑀 ) holds also
without the assumption on 𝑝0 .
Proof. By localization we may assume that 𝐿− (𝑝) is bounded away from zero
𝐿(𝑝)
and innity, uniformly in
𝑝
𝜂
(Lemma 7.3 and Lemma 4.1 and the preceding
proof ); we also recall (7.2). We have
𝑍 𝐿(𝑝)
) (
) 𝑍 𝜂 𝑍 𝜂 1 ( 𝐿(𝑝)
−
𝑍
− 𝑍 𝜂 + 𝜂− − 𝐿− (𝑝)
≤
.
𝐿− (𝑝) 𝜂−
𝐿− (𝑝)
𝐿− (𝑝)𝜂−
𝑢𝑝0 (𝑥0 ) < ∞. The rst part of (7.9) follows from the 𝐿2𝑙𝑜𝑐 (𝑀 ) and
∞ convergences mentioned in Propositions 7.4, 7.7 and Proposiprelocal ℛ
Let
tions 7.1, 7.6, respectively. The proof of the second part of (7.9) is analogous.
𝑢𝑝0 (𝑥0 ) < ∞ and consider a sequence
𝐿𝑡 (𝑝𝑛 ) → 𝜂𝑡 𝑃 -a.s. for each 𝑡,
rather than the convergence of 𝐿𝑡− (𝑝𝑛 ) to 𝜂𝑡− . Consider the optional set
∩
Λ := 𝑛 {𝐿− (𝑝𝑛 ) = 𝐿(𝑝𝑛 )} ∩ {𝜂 = 𝜂− }. Because 𝐿(𝑝𝑛 ) and 𝜂 are càdlàg,
{𝑡 : (𝜔, 𝑡) ∈ Λ𝑐 } ⊂ [0, 𝑇 ] is countable 𝑃 -a.s. and as 𝑀 is continuous is follows
∫𝑇
𝑐
that
0 1Λ 𝑑⟨𝑀 ⟩ = 0 𝑃 -a.s. Now dominated convergence for stochastic
𝜂
𝜂
integrals yields that {(𝜂− − 𝐿− (𝑝𝑛 ))𝑍 } ∙ 𝑀 = {(𝜂 − 𝐿(𝑝𝑛 ))1Λ 𝑍 } ∙ 𝑀 → 0
2
in ℋ𝑙𝑜𝑐 and the rest is as before.
Now drop the assumption that
𝑝𝑛 → 0−.
Then Proposition 7.1 only yields
Proof of Theorem 3.4 and Remark 3.5.
The convergence of the optimal con-
sumption is contained in Propositions 7.1 and 7.6 by the formula (4.2). The
convergence of the portfolios follows from Lemma 7.9 in view of (4.8).
𝑝 ∈ (0, 𝑝0 ] we have the uniform bound 𝜅
ˆ (𝑝) ≤ (𝑘2 /𝑘1 )𝛽0 by Lemma 4.1
(4.2); while for 𝑝 ∈ [𝑝1 , 0), 𝜅
ˆ (𝑝) is prelocally uniformly bounded by
For
and
Lemma 7.3 and (4.2). Hence the convergence of the wealth processes follows
from Corollary A.4(i).
38
We complement the convergence in the primal problem by a result for
the solution
𝑌ˆ (𝑝)
of the dual problem (4.3).
Assume that 𝑆 is continuous and that there exists 𝑝0 > 0
such that 𝑢𝑝0 (𝑥0 ) < ∞ holds. Moreover, assume that there exists 𝑝1 ∈ (0, 𝑝0 ]
such that 𝐿(𝑝1 ) is locally bounded. As 𝑝 → 0,
Proposition 7.10.
𝜂0 (
1
𝑌ˆ (𝑝) →
ℰ −𝜆∙𝑀 +
𝑥0
𝜂−
∙
𝑁𝜂
)
𝑟
in ℋ𝑙𝑜𝑐
for all 𝑟 ∈ [1, ∞).
If 𝜂 and 𝐿(𝑝) are continuous for 𝑝 < 0, the convergence for a limit 𝑝 → 0−
holds in the semimartingale topology without the assumptions on 𝑝0 and 𝑝1 .
Proof. (i) If 𝐿(𝑝1 ) is locally bounded, then 𝐿(𝑝) → 𝜂 in ℛ∞
𝑙𝑜𝑐 by Remark 7.2
𝑀 𝐿(𝑝) → 𝑀 𝜂 in 𝐵𝑀 𝑂𝑙𝑜𝑐 by Propositions 7.4
→ 𝑁 𝜂 in 𝐵𝑀 𝑂𝑙𝑜𝑐 by orthogonality of the KW
and Proposition 7.6. Moreover,
and 7.7. This implies
𝑁 𝐿(𝑝)
decompositions. It follows that
−𝜆 ∙ 𝑀 +
1
𝐿− (𝑝)
∙
𝑁 𝐿(𝑝) → −𝜆 ∙ 𝑀 +
1
𝜂−
∙
𝑁𝜂
in
𝐵𝑀 𝑂𝑙𝑜𝑐 .
This implies that the corresponding stochastic exponentials converge in
for
𝑟 ∈ [1, ∞)
𝑟
ℋ𝑙𝑜𝑐
(see Theorem 3.4 and Remark 3.7(2) in Protter [28]). In view
of the formula (4.9) for
𝑌ˆ (𝑝),
this ends the proof of the rst claim.
(ii) Using Lemma 7.9 and Lemma A.2(ii), the proof of the second claim
is similar.
Note that in the standard case
sition 7.10 is
ℰ(−𝜆
∙
𝑀 ),
𝐷 ≡ 1
the normalized limit in Propo-
i.e., the minimal martingale density (cf. [31]).
We conclude by an additional statement concerning the convergence of the
wealth processes in Theorem 3.4.
Let the conditions of Theorem 3.4(ii) hold and assume
in addition that there exists 𝑝1 ∈ (0, 𝑝0 ] such that 𝐿(𝑝1 ) is locally bounded.
Then the convergence of the wealth processes in Theorem 3.4(ii) takes place
𝑟 for all 𝑟 ∈ [1, ∞).
in ℋloc
Proof. Under the additional assumption, the results of this section yield the
Proposition 7.11.
ˆ (𝑝) ∙ 𝑀 in 𝐵𝑀 𝑂𝑙𝑜𝑐
𝜅
ˆ (𝑝) in ℛ∞
𝑙𝑜𝑐 and the convergence of 𝜋
𝜔
ℋ𝑙𝑜𝑐 ) by the same formulas as before. Corollary A.4(ii) yields
convergence of
(and hence in
the claim.
A
Convergence of Stochastic Exponentials
This appendix provides some continuity results for stochastic exponentials of
continuous semimartingales in an elementary and self-contained way. They
are required for the main results of Section 3 because our wealth processes
are exponentials.
We also use a result from the (much deeper) theory of
ℋ𝜔 -dierentials; but this is applied only for renements of the main results.
39
Let 𝑋 𝑛 = 𝑀 𝑛 +𝐴𝑛 , 𝑛 ≥ 1 be continuous semimartingales
with
∑
continuous canonical ∫decompositions and assume that 𝑛 ∥𝑋 𝑛 ∥ℋ2 < ∞.
Then 𝑀 𝑛 , [𝑀 𝑛 ] and ∣𝑑𝐴𝑛 ∣ are locally bounded uniformly in 𝑛.
Lemma A.1.
Proof.
𝑀𝑡𝑛★
𝜎𝑘 = inf{𝑡 > 0 : sup𝑛 ∣𝑀𝑡𝑛 ∣ > 𝑘} ∧ 𝑇 . We use the notation
= sup𝑠≤𝑡 ∣𝑀𝑠𝑛 ∣, then the norms ∥𝑀𝑇𝑛★ ∥𝐿2 and ∥𝑀 𝑛 ∥ℋ2 are equivalent
Let
by the BDG inequalities. Now
[
]
∑
∥𝑀𝑇𝑛★ ∥𝐿2
𝑃 sup 𝑀𝑇𝑛★ > 𝑘 ≤ 𝑘 −2
𝑛
𝑛
]
∑
𝑃 [𝜎𝑘 < 𝑇 ] → 0. Similarly, 𝑃 sup𝑛 [𝑀 𝑛 ]𝑇 > 𝑘 ≤ 𝑘 −1 𝑛 ∥𝑀 𝑛 ∥ℋ2
[
]
∫𝑇
∑
𝑃 sup𝑛 0 ∣𝑑𝐴𝑛 ∣ > 𝑘 ≤ 𝑘 −2 𝑛 ∥𝐴𝑛 ∥ℋ2 yield the other claims.
[
shows
and
We sometimes write in
𝒮 0
to indicate convergence in the semimartingale
topology.
Let 𝑋 𝑛 = 𝑀 𝑛 + 𝐴𝑛 , 𝑛 ≥ 1 and 𝑋 = 𝑀 + 𝐴 be continuous
semimartingales with continuous canonical decompositions.
∑
2 .
(i) 𝑛 ∥𝑋 𝑛 − 𝑋∥ℋ2 < ∞ implies ℰ(𝑋 𝑛 ) → ℰ(𝑋) in ℋ𝑙𝑜𝑐
2 implies ℰ(𝑋 𝑛 ) → ℰ(𝑋) in 𝒮 0 .
(ii) 𝑋 𝑛 → 𝑋 in ℋ𝑙𝑜𝑐
(iii) 𝑋 𝑛 → 𝑋 in 𝒮 0 implies ℰ(𝑋 𝑛 ) → ℰ(𝑋) in 𝒮 0 .
Lemma A.2.
Proof.
(i)
in
ℋ2 .
𝑛.
Note
𝑀
and
∫
and, by Lemma A.1, that
independent of
∫
∣𝑑𝐴∣ are bounded
∣𝑀 𝑛 ∣ and ∣𝑑𝐴𝑛 ∣ are bounded by a constant 𝐶
𝑛
2
𝑛
that 𝑋 → 𝑋 in ℋ ; we shall show ℰ(𝑋 ) → ℰ(𝑋)
By localization we may assume that
Since this is a metric space, no loss of generality is entailed by passing
to a subsequence. Doing so, we have
uniformly in time,
𝑃 -a.s.
𝑀 𝑛 → 𝑀 , [𝑀 𝑛 ] → [𝑀 ],
and
𝐴𝑛 → 𝐴
In view of the uniform bound
(
)
𝑌 𝑛 := ℰ(𝑋 𝑛 ) = exp 𝑋 𝑛 − 21 [𝑀 𝑛 ] ≤ 𝑒2𝐶
we conclude that
of the stochastic
∥𝑌
∙
𝑌 𝑛 → 𝑌 := ℰ(𝑋) = exp(𝑋 − 12 [𝑀 ]) in ℛ2 . By denition
𝑛 = 𝑌 ∙ 𝑋 − 𝑌 𝑛 ∙ 𝑋 𝑛 , where
exponential we have 𝑌 − 𝑌
𝑋 − 𝑌 𝑛 ∙ 𝑋 𝑛 ∥ℋ2 ≤ ∥(𝑌 − 𝑌 𝑛 ) ∙ 𝑋∥ℋ2 + ∥𝑌 𝑛 ∙ (𝑋 − 𝑋 𝑛 )∥ℋ2 .
The rst norm tends to zero by dominated convergence for stochastic inte-
∣𝑌 𝑛 ∣ ≤ 𝑒2𝐶 and 𝑋 𝑛 → 𝑋 in ℋ2 .
𝑛
Consider a subsequence of (𝑋 ). After passing to another subse2
(i) shows the convergence in ℋ𝑙𝑜𝑐 and Proposition 2.2 yields (ii).
grals and for the second we use that
(ii)
quence,
(iii) This follows from (ii) by using Proposition 2.2 twice.
We return to the semimartingale
𝑅
of asset returns, which is assumed to
be continuous in the sequel. We recall the structure condition (2.6) and dene
𝐿𝜔 (𝑀 ) := {𝜋 ∈ 𝐿(𝑀 ) : ∥𝜋∥𝐿𝜔 (𝑀 ) < ∞}, where ∥𝜋∥𝐿𝜔 (𝑀 ) := ∥𝜋
𝜔
and ℋ was introduced at the end of Section 2.2.
40
∙
𝑀 ∥ℋ𝜔
Lemma A.3.
𝜋𝑛
→𝜋
Proof.
in
Let 𝑅 be continuous, 𝑟 ∈ {2, 𝜔}, and 𝜋, 𝜋𝑛 ∈ 𝐿𝑟𝑙𝑜𝑐 (𝑀 ). Then
𝑟 .
if and only if 𝜋𝑛 ∙ 𝑅 → 𝜋 ∙ 𝑅 in ℋ𝑙𝑜𝑐
𝐿𝑟𝑙𝑜𝑐 (𝑀 )
By (2.6) we have
∫
𝜋 ∙ 𝑅 = 𝜋 ∙ 𝑀 + 𝜋 ⊤ 𝑑⟨𝑀 ⟩ 𝜆.
Let
𝜒 :=
∫
𝜆⊤ 𝑑⟨𝑀 ⟩ 𝜆
denote the mean-variance tradeo process. The inequality
𝐸
𝑇
[( ∫
)2 ]
[( ∫
∣𝜋 ⊤ 𝑑⟨𝑀 ⟩ 𝜆∣
≤𝐸
𝑇
𝜋 ⊤ 𝑑⟨𝑀 ⟩ 𝜋
)( ∫
)]
𝜆⊤ 𝑑⟨𝑀 ⟩ 𝜆
0
0
0
𝑇
∥𝜋 ∙ 𝑀 ∥ℋ2 ≤ ∥𝜋 ∙ 𝑅∥ℋ2 ≤ (1 + ∥𝜒𝑇 ∥𝐿∞ )∥𝜋 ∙ 𝑀 ∥ℋ2 .
bounded due to continuity, this yields the result for 𝑟 = 2.
𝑟 = 𝜔 is similar.
implies
As
𝜒
is locally
The proof for
Let 𝑅 be continuous and (𝜋, 𝜅), (𝜋𝑛 , 𝜅𝑛 ) ∈ 𝒜.
(i) Assume that 𝜋𝑛 → 𝜋 in 𝐿2𝑙𝑜𝑐 (𝑀 ), that (𝜅𝑛 ) is prelocally bounded uniformly in 𝑛, and that 𝜅𝑛𝑡 → 𝜅𝑡 𝑃 -a.s. for each 𝑡 ∈ [0, 𝑇 ]. Then
𝑋(𝜋 𝑛 , 𝜅𝑛 ) → 𝑋(𝜋, 𝜅) in the semimartingale topology.
𝑛 𝑛
(ii) Assume 𝜋𝑛 → 𝜋 in 𝐿𝜔𝑙𝑜𝑐 (𝑀 ) and 𝜅𝑛 → 𝜅 in ℛ∞
𝑙𝑜𝑐 . Then 𝑋(𝜋 , 𝜅 ) →
𝑟
𝑋(𝜋, 𝜅) in ℋ𝑙𝑜𝑐 for all 𝑟 ∈ [1, ∞).
Corollary A.4.
Proof.
𝜇(𝑑𝑠)𝑡 = 𝜅𝑛𝑠 ∙ 𝜇(𝑑𝑠)𝑡− for all 𝑡. After
∫𝑇 𝑛
localization, bounded convergence yields
0 ∣𝜅𝑡 −𝜅𝑡 ∣ 𝜇(𝑑𝑡) → 0 𝑃 -a.s. and in
𝐿2 (𝑃 ). Using Lemma A.3, we have 𝜋 𝑛 ∙ 𝑅 + 𝜅𝑛 ∙ 𝜇(𝑑𝑡) → 𝜋 ∙ 𝑅 + 𝜅 ∙ 𝜇(𝑑𝑡)
2
in ℋ𝑙𝑜𝑐 . In view of (2.2) we conclude by Lemma A.2(ii).
𝑛
𝑛
(ii) With Lemma A.3 we obtain 𝜋 ∙ 𝑅 + 𝜅 ∙ 𝜇(𝑑𝑡) → 𝜋 ∙ 𝑅 + 𝜅 ∙ 𝜇(𝑑𝑡)
𝑟
𝜔
in ℋ𝑙𝑜𝑐 . Thus the stochastic exponentials converge in ℋ𝑙𝑜𝑐 for all 𝑟 ∈ [1, ∞)
(i)
By continuity of
𝜇, 𝜅𝑛𝑠
∙
(see Theorem 3.4 and Remark 3.7(2) in [28]).
B
Proof of Lemma 6.12
In this section we give the proof of Lemma 6.12. As mentioned above, the
argument is adapted from the Brownian setting of [22, Proposition 2.4].
We use the notation introduced before Lemma 6.12, in particular, re-
𝑘 throughout and let 𝜏 := 𝜏𝑘 . For xed integers 𝑚 ≥ 𝑛
𝛿𝐿 = 𝐿𝑛 − 𝐿𝑚 , moreover, 𝛿𝑀 , 𝛿𝑍 , 𝛿𝑁 have the analogous
meaning. Note that 𝛿𝐿 ≥ 0 as 𝑚 ≥ 𝑛. The technique consists in applying
Itô's formula to Φ(𝛿𝐿), where, with 𝐾 := 6𝐶𝑘 ,
call (6.8). We x
we abbreviate
)
1 ( 4𝐾𝑥
𝑒
− 4𝐾𝑥 − 1 .
2
8𝐾
Φ(𝑥) =
On
ℝ+
this function satises
Φ(0) = Φ′ (0) = 0,
Moreover,
Φ′′ ≥ 0
Φ′ ≥ 0,
Φ ≥ 0,
and hence
ℎ(𝑥) :=
nonnegative and nondecreasing.
41
1 ′′
2 Φ (𝑥)
1 ′′
2Φ
− 2𝐾Φ′ ≡ 1.
− 𝐾Φ′ (𝑥) = 1 + 𝐾Φ′ (𝑥)
is
(i) By Itô's formula we have
∫ 𝜏
[
]
𝑚
Φ′ (𝛿𝐿𝑠 ) 𝑓 𝑛 (𝑠, 𝐿𝑛𝑠 , 𝑍𝑠𝑛 ) − 𝑓 𝑚 (𝑠, 𝐿𝑚
Φ(𝛿𝐿0 ) = Φ(𝛿𝐿𝜏 ) −
𝑠 , 𝑍𝑠 ) 𝑑⟨𝑀 ⟩𝑠
0
∫ 𝜏
∫ 𝜏
′′
1
−
Φ′ (𝛿𝐿𝑠 ) 𝑑𝛿𝑀𝑠 .
2 Φ (𝛿𝐿𝑠 ) 𝑑 ⟨𝛿𝑀 ⟩𝑠 −
0
0
𝑚
By elementary inequalities we have for all
and
𝑛
that
)
˜ 2 + ∣𝑍∣
˜ 2 𝜏,
∣𝑓 𝑛 (𝑡, 𝐿𝑛 , 𝑍 𝑛 ) − 𝑓 𝑚 (𝑡, 𝐿𝑚 , 𝑍 𝑚 )∣𝜏 ≤ 𝜉 + 𝐾 ∣𝑍 𝑛 − 𝑍 𝑚 ∣2 + ∣𝑍 𝑛 − 𝑍∣
(
where the index
𝑡
was omitted. Hence
[
𝜏
(
)]
Φ′ (𝛿𝐿𝑠 ) 𝜉𝑠 + 𝐾 ∣𝛿𝑍𝑠 ∣2 + ∣𝑍𝑠𝑛 − 𝑍˜𝑠 ∣2 + ∣𝑍˜𝑠 ∣2 𝑑⟨𝑀 ⟩𝑠
Φ(𝛿𝐿0 ) ≤ Φ(𝛿𝐿𝜏 ) +
0
∫ 𝜏
∫ 𝜏
1 ′′
Φ′ (𝛿𝐿𝑠 ) 𝑑𝛿𝑀𝑠 .
−
2 Φ (𝛿𝐿𝑠 ) 𝑑 ⟨𝛿𝑀 ⟩𝑠 −
∫
0
0
The expectation of the stochastic integral vanishes since
𝛿𝐿
is bounded and
𝛿𝑀 ∈ ℋ2 . We deduce
∫ 𝜏
∫ 𝜏
[ 1 ′′
]
′
2
1 ′′
𝐸
Φ
(𝛿𝐿
)
−
𝐾Φ
(𝛿𝐿
)
∣𝛿𝑍
∣
𝑑⟨𝑀
⟩
+
𝐸
𝑠
𝑠
𝑠
𝑠
2
2 Φ (𝛿𝐿𝑠 ) 𝑑⟨𝛿𝑁 ⟩𝑠 (B.1)
0
0
∫ 𝜏
˜𝑠 ∣2 𝑑⟨𝑀 ⟩𝑠 + Φ(𝛿𝐿0 )
−𝐸
𝐾Φ′ (𝛿𝐿𝑠 )∣𝑍𝑠𝑛 − 𝑍
(B.2)
0
∫ 𝜏
[
]
[
]
˜𝑠 ∣2 𝑑⟨𝑀 ⟩𝑠 .
Φ′ (𝛿𝐿𝑠 ) 𝜉𝑠 + 𝐾∣𝑍
(B.3)
≤ 𝐸 Φ(𝛿𝐿𝜏 ) + 𝐸
0
We let
for all
𝑛
˜
𝑚 tend to innity, then 𝛿𝐿𝑡 = 𝐿𝑛𝑡 − 𝐿𝑚
𝑡 converges to 𝐿𝑡 − 𝐿𝑡 𝑃 -a.s.
𝑡 and with a uniform bound, so (B.3) converges to
∫ 𝜏
[
]
[
]
𝑛
˜
˜ 𝑠 ) 𝜉𝑠 + 𝐾∣𝑍
˜𝑠 ∣2 𝑑⟨𝑀 ⟩𝑠 ;
𝐸 Φ(𝐿𝜏 − 𝐿𝜏 ) + 𝐸
Φ′ (𝐿𝑛𝑠 − 𝐿
0
while (B.2) converges to
∫
𝜏
−𝐸
˜ 𝑠 )∣𝑍𝑠𝑛 − 𝑍
˜𝑠 ∣2 𝑑⟨𝑀 ⟩𝑠 + Φ(𝐿𝑛0 − 𝐿
˜ 0 ).
𝐾Φ′ (𝐿𝑛𝑠 − 𝐿
0
We turn to (B.1). The continuous function
in the rst integrand.
Φ′′
𝑚,
and note that
decreasing in
We recall that
ℎ
ℎ(𝑥) = 12 Φ′′ (𝑥) − 𝐾Φ′ (𝑥)
is nonnegative and nondecreasing
has the same properties.
𝑛 ˜
ℎ(𝛿𝐿𝑠 ) = ℎ(𝐿𝑛𝑠 −𝐿𝑚
𝑠 ) ↑ ℎ(𝐿𝑠 − 𝐿𝑠 );
occurs
Moreover, as
𝐿𝑚
𝑡
is monotone
′′
𝑛 ˜
Φ′′ (𝛿𝐿𝑠 ) = Φ′′ (𝐿𝑛𝑠 −𝐿𝑚
𝑠 ) ↑ Φ (𝐿𝑠 − 𝐿𝑠 )
𝑃 -a.s. for all 𝑠. Hence we have for any xed 𝑚0 ≤ 𝑚 that
∫ 𝜏
∫ 𝜏
𝑛
𝑚
𝑛
𝑚
𝑛
𝑚
0
𝐸
ℎ(𝐿𝑠 − 𝐿𝑠 )∣𝑍𝑠 − 𝑍𝑠 ∣ 𝑑⟨𝑀 ⟩𝑠 ≥ 𝐸
ℎ(𝐿𝑛𝑠 − 𝐿𝑚
𝑠 )∣𝑍𝑠 − 𝑍𝑠 ∣ 𝑑⟨𝑀 ⟩𝑠 ;
0
0
∫ 𝜏
∫ 𝜏
′′
𝑛
𝑚
𝑛
𝑚
𝑛
𝑚
0
𝐸
Φ (𝐿𝑠 − 𝐿𝑠 ) 𝑑⟨𝑁 − 𝑁 ⟩𝑠 ≥ 𝐸
Φ′′ (𝐿𝑛𝑠 − 𝐿𝑚
𝑠 ) 𝑑⟨𝑁 − 𝑁 ⟩𝑠 .
0
0
42
The right hand sides are convex lower semicontinuous functions of
𝐿2 (𝑀 ) and
𝑍𝑚 ∈
𝑁𝑚
∈ ℋ2 , respectively, hence also weakly lower semicontinuous.
𝑚 →𝑍
˜ and 𝑁 𝑚 → 𝑁
˜ that
We conclude from the weak convergences 𝑍
𝜏
∫
𝑛
˜𝑚
ℎ(𝐿𝑛𝑠 − 𝐿𝑚
𝑠 )∣𝑍𝑠 − 𝑍𝑠 ∣ 𝑑⟨𝑀 ⟩𝑠
∫ 𝜏
𝑛
0
˜
ℎ(𝐿𝑛𝑠 − 𝐿𝑚
≥𝐸
𝑠 )∣𝑍𝑠 − 𝑍𝑠 ∣ 𝑑⟨𝑀 ⟩𝑠 ;
lim inf 𝐸
𝑚→∞
0
0
𝜏
∫
lim inf 𝐸
Φ
𝑚→∞
′′
(𝐿𝑛𝑠
𝑛
− 𝐿𝑚
𝑠 ) 𝑑⟨𝑁
∫
𝑚
0
𝑚0 .
for all
𝜏
− 𝑁 ⟩𝑠 ≥ 𝐸
𝑛
0
˜
Φ′′ (𝐿𝑛𝑠 − 𝐿𝑚
𝑠 ) 𝑑⟨𝑁 − 𝑁 ⟩𝑠
0
We can now let
𝑚0 tend to∫innity, then by monotone convergence
𝜏
˜ 𝑠 )∣𝑍 𝑛 − 𝑍
˜𝑠 ∣ 𝑑⟨𝑀 ⟩𝑠 and the
𝐸 0 ℎ(𝐿𝑛𝑠 − 𝐿
𝑠
the rst right hand side tends to
second one tends to
𝜏
∫
𝐸
∫
˜ 𝑠 ) 𝑑⟨𝑁 𝑛 − 𝑁
˜ ⟩𝑠 ≥ 2𝐸
Φ′′ (𝐿𝑛𝑠 − 𝐿
0
𝜏
[
]
˜ ⟩𝑠 = 2𝐸 ⟨𝑁 𝑛 − 𝑁
˜ ⟩𝜏 ,
𝑑⟨𝑁 𝑛 − 𝑁
0
where we have used that
˜ ≥ 0
𝐿𝑛 − 𝐿
and
Φ′′ (𝑥) = 2𝑒4𝐾𝑥 ≥ 2
for
𝑥 ≥ 0.
Altogether, we have passed from (B.1)(B.3) to
∫
𝐸
𝜏
(
)
[
]
˜ ⟩𝜏
˜ 𝑠 ) ∣𝑍𝑠𝑛 − 𝑍
˜𝑠 ∣2 𝑑⟨𝑀 ⟩𝑠 + 𝐸 ⟨𝑁 𝑛 − 𝑁
− 2𝐾Φ′ (𝐿𝑛𝑠 − 𝐿
0
∫ 𝜏
[
]
˜ 𝜏 ) − Φ(𝐿𝑛 − 𝐿
˜0 ) + 𝐸
˜ 𝑠 ) 𝜉𝑠 + 𝐾∣𝑍
˜𝑠 ∣2 𝑑⟨𝑀 ⟩𝑠 .
≤ 𝐸Φ(𝐿𝑛𝜏 − 𝐿
Φ′ (𝐿𝑛𝑠 − 𝐿
0
1 ′′
2Φ
0
As
1 ′′
2Φ
we let
𝑛
− 2𝐾Φ′ ≡ 1,
the rst integral reduces to
𝐸
∫𝜏
0
∣𝑍𝑠𝑛 − 𝑍𝑠 ∣2 𝑑⟨𝑀 ⟩𝑠 .
If
tend to innity, the right hand side converges to zero by dominated
convergence, so that we conclude
∫
𝜏
˜𝑠 ∣2 𝑑⟨𝑀 ⟩𝑠 → 0;
∣𝑍𝑠𝑛 − 𝑍
𝐸
[
]
˜ ⟩𝜏 → 0
𝐸 ⟨𝑁 𝑛 − 𝑁
0
as claimed.
(ii) For all
∣𝐿𝑛𝑡∧𝜏
−
𝐿𝑚
𝑡∧𝜏 ∣
The sequence
𝑚
and
we have
−
𝐿𝑚
𝜏 ∣
∫
𝜏
𝑚
∣𝑓 𝑛 (𝑠, 𝐿𝑛𝑠 , 𝑍𝑠𝑛 ) − 𝑓 𝑚 (𝑠, 𝐿𝑚
𝑠 , 𝑍𝑠 )∣ 𝑑⟨𝑀 ⟩𝑠
𝑡∧𝜏
𝑛
𝑚 + (𝑀𝜏𝑛 − 𝑀𝜏𝑚 ) − (𝑀𝑡∧𝜏
− 𝑀𝑡∧𝜏
).
(B.4)
≤
∣𝐿𝑛𝜏
𝑛
𝑀 𝑚 = 𝑍𝑚
subsequence, still denoted
+
𝑀 + 𝑁 𝑚 is Cauchy in ℋ𝜏2 . We pick a fast
𝑚
𝑚 − 𝑀 𝑚+1 ∥
−𝑚 .
by 𝑀 , such that ∥𝑀
ℋ𝜏2 ≤ 2
∙
This implies that
𝑀 ∗ := sup ∣𝑀 𝑚 ∣ ∈ ℋ𝜏2 ;
𝑚
𝑍 ∗ := sup ∣𝑍 𝑚 ∣ ∈ 𝐿2𝜏 (𝑀 )
𝑚
43
𝑚
𝑍 𝑚 converges 𝑃 ⊗⟨𝑀 𝜏 ⟩-a.e. to 𝑍˜. Therefore, lim𝑛 𝑓 𝑚 (𝑡, 𝐿𝑚
𝑡 , 𝑍𝑡 ) =
𝜏
𝑚
𝑚
𝑚
𝜏
∗
2
˜ 𝑡 , 𝑍˜𝑡 ) 𝑃 ⊗⟨𝑀 ⟩-a.e. Moreover, ∣𝑓 (𝑡, 𝐿 , 𝑍 ) ∣ ≤ 𝜉𝑡 +𝐶∣𝑍 ∣ and this
𝑓 (𝑡, 𝐿
𝑡
𝑡
𝑡
1
bound is in 𝐿𝜏 (𝑀 ). Passing to a subsequence if necessary, we have
∫ 𝜏
𝑚
∣𝑓 𝑛 (𝑠, 𝐿𝑛𝑠 , 𝑍𝑠𝑛 ) − 𝑓 𝑚 (𝑠, 𝐿𝑚
lim
𝑠 , 𝑍𝑠 )∣ 𝑑⟨𝑀 ⟩𝑠
𝑚→∞ 0
∫ 𝜏
˜𝑠 , 𝑍
˜𝑠 )∣ 𝑑⟨𝑀 ⟩𝑠 𝑃 -a.s.
∣𝑓 𝑛 (𝑠, 𝐿𝑛𝑠 , 𝑍𝑠𝑛 ) − 𝑓 (𝑠, 𝐿
=
and that
0
As
˜
𝑀𝑚 → 𝑀
picking a
𝑚→∞
[
]
𝑚 −𝑀
˜𝑡∧𝜏 ∣ → 0 and, after
ℋ𝜏2 , we have 𝐸 sup𝑡≤𝑇 ∣𝑀𝑡∧𝜏
𝑚
˜𝑡∧𝜏 ∣ → 0 𝑃 -a.s. We can now take
subsequence, sup𝑡≤𝑇 ∣𝑀𝑡∧𝜏 − 𝑀
in
in (B.4) to obtain
˜ 𝑡∧𝜏 ∣ ≤ ∣𝐿𝑛𝜏 − 𝐿
˜𝜏 ∣ +
sup ∣𝐿𝑛𝑡∧𝜏 − 𝐿
𝑡≤𝑇
∫
𝜏
˜𝑠 , 𝐿
˜ 𝑠 )∣ 𝑑⟨𝑀 ⟩𝑠
∣𝑓 𝑛 (𝑠, 𝐿𝑛𝑠 , 𝑍𝑠𝑛 ) − 𝑓 (𝑠, 𝐿
0
𝑛
˜𝜏 ) − (𝑀𝑡∧𝜏
˜𝑡∧𝜏 ).
+ sup (𝑀𝜏𝑛 − 𝑀
−𝑀
𝑡≤𝑇
With exactly the same arguments, extracting another subsequence if necessary, the right hand side converges to zero
𝑛
shown that lim𝑛 sup𝑡≤𝑇 ∣𝐿𝑡∧𝜏
˜ 𝑡∧𝜏 ∣ = 0,
−𝐿
𝑃 -a.s.
as
𝑛 → ∞.
along a subsequence.
monotonicity, we conclude the result for the whole sequence.
We have
But by
□
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