Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Optimal Stopping Rule under
Ambiguity
in Continuous Time
Weidong Tian
UNCC
(joint with Zengjing Chen, and Guoqing Zhao)
May, 2010
Chen, Tian & Zhao
Conclusion
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Motivation
Motivation
Optimal Stopping Decision
Find the best time based on what we know
Knowledge:
Payoff (outcome) structure
Distribution (or probability measure)
Knightian Uncertainty
What about loss of confidence on the distribution?
Decision maker has a set of probability measures
(priors), subject to a confidence level
Chen, Tian & Zhao
Conclusion
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
Classical
Classical Setting
Setting
Given a payoff process Xt , a probability measure P
The reward (value) process is
Vt = ess sup EP [Xτ |Ft ]
τ ≥t
Optimal stopping problem:
Characterize the value process
Describe the optimal stopping time.
Solution:
{Vt } is the smallest P-supermartingale that dominates
Xt
τ ∗ = inf{t : Vt = Xt } is the optimal stopping time
{Vt∧τ ∗ } is a P-martingale.
Chen, Tian & Zhao
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
Multiprior
Setting in Ambiguity
Setting
Consider a filtered probability space (Ω, (Ft )0≤t≤T , P)
there is a set of probability measures, Q on F = FT ,
there is a payout process Xt
Ambiguity
At any time, the decision maker has a set of beliefs
about the distribution of Xt
The reward process becomes ess infQ∈Q EQ [Xτ |Ft ],
the worst expected reward among all possible beliefs
A General Optimal Stopping Problem
Vt := ess supτ ≥t ess infQ∈Q EQ [Xτ |Ft ]
Characterize Vt and the optimal stopping time
Chen, Tian & Zhao
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
Multiprior
Main Insights
Differences with classical situation
Expectation EP [·] and a nonlinear expectation
EQ [·] := ess infQ∈Q EQ [·]
Conditional expectation EP [·|Ft ] and a nonlinear
conditional expectation EQ [·|Ft ] := ess infQ∈Q EQ [·|Ft ]
Possible Challenge
Expectation theory
Martingale theory
Important and unsolved question
Chen, Tian & Zhao
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
Multiprior
Discrete time (Riedel, 2009)
Assume Q is time consistent, then
there is a Q-martingale theory,
the optimal stopping problem can be solved similarly,
Vn = max {Xn , ess infQ∈Q EQ [Vn+1 |Fn ]}.
If the set of priors Q is not time consistent, the above
results are not hold anymore.
Q is time consistent, is equivalent to the law of
iterated nonlinear expectation
ess inf EQ ess inf EQ [Z|Ft+1 ]|Ft
Q∈Q
Q∈Q
= ess inf EQ [Z|Ft ]
Q∈Q
Chen, Tian & Zhao
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
Multiprior
Objective
We develop a theory of the optimal stopping problem
in a continuous time framework with ambiguity.
The theory is sufficiently general for economic theory
and computational tractable for applications.
Chen, Tian & Zhao
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
TC
Time Consistent
Time consistent emerges in “dynamic risk measures"
Artzner et al (2007), Riedel (2004), Follmer and
Schied (2004),
Time consistent is closely related to multiprior
Bayesian decision theory: Epstein and Schneider
(2003), Wang (2003)
Time consistent is related to optimal stopping under
constraint (Detemple, Tian and Xiong (2009))
Time consistent holds in Peng’s g-expectation and
G-expectation
Time consistent, however, doesn’t hold for capacity
though.
Chen, Tian & Zhao
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
TC
Time Consistent for continuous time
For Q that is absolutely continuous with respect to P,
Delbaen (2006) characterize a time consistent Q.
However, the corresponding martingale theory is
missing.
The question for the general Q is not solved yet.
Chen, Tian & Zhao
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
Continuous Time
Knightian Uncertainty in Continuous Time
Duffie and Epstein (1992a, 1992b), stochastic
recursive utility
Chen and Epstein (2002), continuous time with
ambiguity
Continuous time version of models in Maccheroni,
Marinacci and Rustichini (2006a, 2006b), Anderson,
Hansen and Sargent (2000). Multiplier preferences
and its variational preference
Chen, Tian & Zhao
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
Results
Contributions of this paper
Develop a general setting in continuous time to
encompasses those models
Characterize the value/reward and the optimal
stopping rule
Examine the affects of Knightian ambiguity and risk
averse
Develop a theory in terms of free boundary problem
(of PDE)
Chen, Tian & Zhao
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
g-expectation
Definition
(Ω, Ft , P), Ft = σ(Ws , 0 ≤ s ≤ t). C is a convex subset of
Rl . An adapted processes {ct }0≤t≤T that takes values in C.
A aggregator is a Boreal measurable
g(c, y, z, ω, t) : C × R × Rd × Ω × [0, T] → R
satisfies some standard conditions.
Chen, Tian & Zhao
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
g-expectation
BSDE
Given a aggregator g(c, y, z, ω, t), there exists unique
solution of the following backward stochastic differential
equation (BSDE), for τ ∈ S and ξτ ∈ (Ω, Fτ , P),
Z τ
Z τ
yt = ξτ −
g(cs , ys , zs , ω, s)ds −
zs · dWs , 0 ≤ t ≤ τ. (1)
t
t
has a unique solution (yt , zt )0≤t≤τ . See Karoui et al (1997)
and Peng (2004) for a basic theory of BSDE.
Chen, Tian & Zhao
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
g-expectation
Examples
When g = 0, Eg [Z|Ft ] = EP [Z|Ft ].
When g(y, z, s) = bs · z, where (bs ) is a uniformly bounded
adapted process. Let
Z t
Z
1 t
2
Qt := exp
bs · dWs −
|bs | ds
(2)
2 0
0
Then
dyt = −bt · zt dt + zt dWt
By Girsanov’s theorem, Eg [Z|Ft ] = EQ [Z|Ft ].
Chen, Tian & Zhao
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
g-expectation
Eg[·] and Eg[·|Ft ]
Given υ ∈ S, τ ∈ Sυ and a random payoff ξτ ∈ L2 (Ω, Fτ , P),
g
define Eg [ξτ |Fυ ] := Eυ,τ
[ξτ ] := yυ the g-expectation of ξτ
generated by the aggregator g on the time period [υ, τ ],
where (yt , zt )0≤t≤τ is the unique solution of the BSDE (1).
yυ describes the conditional expectation of ξτ under the
sigma-algebra Fυ regard to the nonlinear expectation Eg
in the sense that
Eg [1A ξτ ] = Eg [1A yυ ], ∀A ∈ Fυ .
(3)
Time Consistent:
Eg [Eg [Z|Fr ]|Ft ] = Eg [Z|Ft ], ∀t ≤ r.
Chen, Tian & Zhao
(4)
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
g-expectation
Theory of g-expectation (Peng, El Karouri
etc)
g-martingale,
Doob-Meyer decomposition of g-supermartingale
Chen, Tian & Zhao
Conclusion
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
Optimal Stopping Problem
Problem
For any stopping time υ, define the maximum expected
reward from the perspective at υ,
Vυ := ess sup Eg [Xτ |Fυ ], for υ ∈ S.
τ ∈Sυ
Chen, Tian & Zhao
(5)
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
Optimal Stopping Problem
Solution: Value Process
The process {Vt }0≤t≤T is the Snell envelope under
g-expectation of the process {Xt }0≤t≤T . Furthermore,
{Vt }0≤t≤T has a RCLL modification {Vt0 }0≤t≤T with Vτ = Vτ0 ,
a.s., for all τ ∈ S.
Chen, Tian & Zhao
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
Optimal Stopping Problem
Solution: Stopping Rule
Assume Eg [sup0≤t≤T Xt ] < ∞, and {Xt }0≤t≤T has continuous
paths. Then
τ ∗ = inf t ∈ [0, T] : Vt0 = Xt
0
is the optimal stopping time. Moreover, {Vt∧τ
}
is a
∗ 0≤t≤T
g-martingale.
Chen, Tian & Zhao
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
Optimal Stopping Problem
EXAMPLE 1: Multiprior Model
The set of priors is
dQθ
Θ
θ
θ
P = Q : θ ∈ Θ,
|F = zt , for all t ,
dP t
where
zθt
Z
Z t
1 t
2
|θs | ds −
θs · dWs , 0 ≤ t ≤ T
≡ exp −
2 0
0
and Θ, the set of generator {θt }0≤t≤T satisfies standard
assumptions.
minQ∈P Θ EQ [ζ|Ft ] is the same as the g-expectation Eg [ζ|Ft ]
where g(z, ω, t) = maxθ∈Θ θt (ω) · z is defined over
Rd × Ω × [0, T].
Chen, Tian & Zhao
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
Optimal Stopping Problem
EXAMPLE 2: Chen-Epstein (2001), Duffie
and Epstein (1992)
Given a function
f (c, v) : C × R → R,
and a set of density generators Θ as in EXAMPLE 1, there
exists an unique process, namely the stochastic utility
process, of the consumption process {ct }0≤t≤T , such that
dyt = −f (ct , yt ) + max θt · σt dt + σt · dWt , yT = 0.
θ∈Θ
Then g(c, y, z, ω, t) = −f (c, y) + maxθ∈Θ θt (ω) · z. The Eg [·]
expectation represents the multiprior stochastic recursive
utility.
Chen, Tian & Zhao
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
Optimal Stopping Problem
EXAMPLE 3: Model with Penalty Terms
Consider the optimal stopping time problem in the
continuous time version of Maccheroni, Marinacci and
Rustichini (2006):
Vυ = ess sup inf {EQ [u(Xτ )|Fυ ] + αυ,τ (Q|Fυ )}.
τ ∈Sυ Q∈P Θ
(p1). αυ,τ (P|Fυ ) = 0, αυ,υ (Q|Fυ ) = 0
(p2).
αυ,τ (Q|Fυ ) = αυ,δ (Q|Fυ ) + EQ [αδ,τ (Q|Fδ )|Fυ ], a.s.
(p3). αυ,τ (Q|Fυ ) ≥ 0.
Then there exists a aggregator g(z, ω, t) such that
Vυ = ess supρ∈Sυ Eg [u(Xρ )|Fυ ].
Chen, Tian & Zhao
(6)
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Knightian Uncertainty Varies
Increase Knightian uncertainty
An increase in Knightian uncertainty reduces the
reward process. Hence the optimal stopping time
decreases.
Intuition: When people loss confidence in their
forecast on the outcome distribution, they generally
prefer certainty to uncertainty.
Extends Nishimura and Ozaka (2004)’s search
model.
Chen, Tian & Zhao
Conclusion
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
Knightian Uncertainty Varies
Continuity
If the aggregators {gn }n≥1 converge to an aggregator
g, the optimal stopping times for gn converges to the
optimal time for the decision maker with aggregator g.
Intuition: When the k-ignorance is very small, the
Knightian uncertainty is eliminated, then the optimal
stopping time is very close to the best time in the
absence of Knightian uncertainty.
Chen, Tian & Zhao
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
Risk Aversion Varies
Risk aversion Increases
Increase risk aversion leads to early optimal stopping.
An aggregative agent, since she want to take
advantage of the risk, tends to exercise at a late time.
If the risk aversion increases, the more concern on
the risk, the earlier the agent decides to make stop
decision.
Intuition: Risk averse decision maker prefer riskless
to risk.
Chen, Tian & Zhao
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
Model Framework
Framework
Consider a joint-Markov process {ct , St }, where {ct } is an
auxiliary process such as vector of factors, consumption
processes or cash flow processes, and {St } is a value
processes of some securities or project. Precisely, for any
time period t ≤ s ≤ T,
dcs = µc (cs )ds + sc (cs )dWs , s ≥ t; ct = c, c ∈ C,
(7)
and
dSs = b(cs , Ss )ds + σ(cs , Ss )dWs , s ≥ t; St = x, x ∈ Rp , (8)
where the coefficients µc (·), σ c (·), b(·, ·) and σ(·, ·) are
deterministic functions and satisfy standard regularity
conditions.
Chen, Tian & Zhao
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
Characterization
Continuous Version of Recursive Structure
The value function is
u(t, x, c) := sup Eg [Φ(τ, Sτ )|St = x, ct = c].
τ ∈St
In the continuation region,
∂t u(t, x, c) + Au(t, x, c) − g(c, φ, σ∂x u, t) = 0
where A := b(c, x)∂x +
Chen, Tian & Zhao
σ(c,x)σ T (c,x) 2
∂x .
2
(9)
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
Characterization
Smooth fit condition
On the boundary of the continuation region,
∂x u(t, x, c) = ∂x Φ(t, x).
Chen, Tian & Zhao
(10)
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
Example
Firm Value Process
Consider one firm with operating profit process St .
dSt = bSt dt + σSt dWt
The firm is not sure about the expected return. The set of
priors P Θ is defined by Θ = {(θt ) : |θt | ≤ κ}. Hence, the
firm value is
Z ∞
−β(s−t)
Vt := inf EQ
e
Ss ds|Ft .
(11)
Q∈P Θ
t
It can be check that
Vt =
where λ = β − (b − κσ).
Chen, Tian & Zhao
St
,
λ
(12)
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Conclusion
Example
Optimal Decision Problem
The firm can exit by selling the technology or the real
asset, obtain $I dollars. Hence, the optimal exit decision
is characterized by the following optimal stopping time
problem
u(x) := ess sup ess inf EQ e−β(τ −t) (I − Vτ )+ |St = x . (13)
τ ∈St
Chen, Tian & Zhao
Q∈P Θ
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Example
Solution
(
(I −
u(x) =
(I −
x0
)( xx0 )π1
λ
x0 +
)
λ
, x > x0 ;
, 0 < x ≤ x0 .
1
where x0 = πλIπ
. The optimal exit time is the first hitting
1 −1
time when {St }t≥0 hits x0 , and
r
1
1 1 2
πi = 2 [ σ − b − κσ ± ( σ 2 − b − κσ)2 + 2βσ 2 ],
σ 2
2
where π1 < 0 < π2 .
Chen, Tian & Zhao
Conclusion
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
Example
Optimal Decision Problem
Value Function of Enter/Exit
0
4
u(x)
0.1
2
0.2
0
0
0.3
1
0.4
2
κ−Ignorance
3
4
5
x
Chen, Tian & Zhao
0.5
Conclusion
Introduction
Contributions
Continuous Time
Implications
Markovian Setting
summary
Summary
We develop an optimal stopping time theory in
continuous time with ambiguity
The ambiguity in continuous time is represented by
Eg -expectation.
The theory extends the classical theory and the
theory in discrete time with ambiguity.
We reduce it to a free boundary problem.
Risk averse makes stopping early.
Knightian uncertainty yields stopping sooner rather
later.
Chen, Tian & Zhao
Conclusion