On the convergence rates of a general class of weak
approximations of SDEs
D. Crisan∗,
S.Ghazali†
Abstract
In this paper, the convergence analysis of a class of weak approximations of solutions of stochastic differential equations is presented. This class includes recent
approximations such as Kusuoka’s moment similar families method and the LyonsVictoir cubature of Wiener Space approach. We show that the rate of convergence
depends intrinsically on the smoothness of the chosen test function. For smooth
functions (the required degree of smoothness depends on the order of the approximation), an equidistant partition of the time interval on which the approximation
is sought is optimal. For functions that are less smooth (for example Lipschitz
functions), the rate of convergence decays and the optimal partition is no longer
equidistant. Our analysis rests upon Kusuoka-Stroock’s results on the smoothness
of the distribution of the solution of a stochastic differential equation. Finally the
results are applied to the numerical solution of the filtering problem.
1
Introduction
Stochastic differential equations (SDEs) constitute an ideal mathematical model for a
multitude of phenomena and processes encountered in areas such as filtering, optimal
stopping, stochastic control, signal processes and mathematical finance, most notably in
option pricing (see for example Oksendal[34] and Kloeden & Platen[16]). Unlike their
deterministic counterparts, SDEs do not have explicit solutions, apart from in a few
exceptional cases, hence the necessity for a sound theory of their numerical approximation.
In this paper we will be concerned with SDEs written in Stratonovich form, in other
words, we will look at equations of the form,
t
k t
Xt = X0 +
V0 (Xs)ds +
Vj (Xs ) ◦ dWsj ,
(1)
0
j=1
0
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2BZ. E-mail:
[email protected].
†
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2BZ. E-mail:
[email protected].
∗
1
where the last term is a stochastic integral of Stratonovich type. There are two classes
of numerical methods for approximating SDEs. The objective of the first is to produce a
pathwise approximation of the solution (strong approximation). The second method involves approximating the distribution of the solution at a particular instance in time (weak
approximation). For example when one is only interested in the expectation E[ϕ(Xt )] for
some function ϕ, it is sufficient to have a good approximation of the distribution of the
random variable Xt rather than of its sample paths. This observation was first made by
Milstein [27] who showed that pathwise schemes and L2 estimates of the corresponding
errors are irrelevant in this context since the objective is to approximate the law of Xt .
This paper contains approximations that belong to this second class of algorithms.
Classical results in this area concentrate on solving numerically SDEs for which the socalled ‘ellipticity condition’, or more generally the ‘Uniform Hörmander condition’ (UH),
is satisfied. For a survey of such schemes see, for example, Kloeden & Platen[16] or
Burrage, Burrage & Tian [5]. Under this condition, for any bounded measurable function
ϕ, the semigroup of operators {Pt }t∈[0,∞) defined,
(Pt ϕ)(x) = E[ϕ(Xt (x))],
(2)
where X (x) = {Xt (x)}t∈[0,∞) solves (1) with initial condition X0 = x, is smooth for any
t > 0. It is this property upon which the majority of these schemes rely.
For example, the classical Euler-Maruyama scheme requires Pt ϕ to be four times differentiable in order to obtain the optimal rate of convergence. Talay([42],[43]) and, independently, Milstein[28] introduced the appropriate methodology to analyse this scheme.
They express the error as a difference including a sum of terms involving the solution of
a parabolic PDE. Their analysis also shows the relationship between the smoothness of ϕ
and the corresponding error. Talay & Tubaro[44] prove an even more precise result showing that, under the same conditions, the errors corresponding to the Euler-Maruyama and
many other schemes can be expanded in terms of powers of the discretisation step. Furthermore, Bally & Talay[1] show the existence of such an expansion under a much weaker
hypothesis on ϕ: that ϕ need only be measurable and bounded (even the boundedness
condition can be relaxed). Higher order schemes require additional smoothness properties
of Pt ϕ (see for example, Platen & Wagner[37]).
In the eighties, Kusuoka & Stroock([21], [22], [23]) studied the properties of Pt ϕ under
a weaker condition, the so-called UFG condition (see (5) in Section 2). Essentially, this
condition states that the Lie algebra generated by the vector fields {Vi }ki=1 ∈ Cb∞ (Rd ; Rd )
is finite dimensional as a Cb∞ (Rd )-module. Kusuoka & Stroock conclude Malliavin’s undertaking to recover Hörmander’s hypo-ellipticity theory for degenerate second order elliptic operators. They show that under the UFG condition, Pt ϕ retains certain regularity properties, in particular, that Vi1 Vi2 ...Vin Pt ϕ is well defined for any vector fields
Vir , where ir ∈ {1, ..., k}. They also give upper bounds for the supremum norm of
Vi1 Vi2 ...Vin Pt ϕ. Besides the UFG condition, our analysis rests upon a second assumption which we call the V0 condition. It states that V0 can be expressed in terms of
2
{V1 , ...Vk } ∪ {[Vi , Vj ] , 1 ≤ i < j ≤ k}. This premise is weaker than the ellipticity assumption and has been used, for example, by Jerison and Sánchez-Calle ([14],[38]) to obtain
estimates for the heat kernel. This second condition enables one to control the supremum
norm of Vi1 Vi2 ...Vin Pt ϕ for ir ∈ {0, 1, ..., k} (see Corollary 18 in the appendix).
A number of schemes have recently been developed to work under these weaker conditions rather than the ellipticity condition, their convergence depending intrinsically on the
above estimates of Vi1 Vi2 ...Vin Pt ϕ. A further advantage of this new generation of schemes
is a consequence
that the support of X (x) is the closure of
ϕ of the classical
result stating
d
ϕ
the set S = x : [0, ∞) → R where x solves the ODE,
xϕt
=x+
0
t
V0 (xϕs )ds
+
k j=1
0
t
Vj (xϕs )ϕ (s) ◦ ds,
and ϕ : [0, ∞) → Rd is an arbitrary smooth function (see Stroock & Varadhan [39],
[40], [41], Millet & Sanz-Sole[26]). These schemes attempt to keep the support of the
approximating process on the set S. In this way, stability problems that are known to
affect classical schemes can be avoided. For example, Ninomiya & Victoir[33] give an
explicit example where the Euler-Maruyama approximation fails whilst their algorithm
succeeds (see Example 5 below for the algorithm). Their example involves an SDE related
to the Heston stochastic volatility model in finance.
In this paper we give a general criterion for the convergence of a class of weak approximations incorporating this new category of schemes. This criterion is based upon
the stochastic Stratonovich-Taylor expansion of Pt ϕ and demonstrates how the rate of
convergence depends on the smoothness of the test function selected.
Our plan is as follows. In section 2 we set out some essential terminology, adopted
partly from Kusuoka[17], which is required to index the stochastic Stratonovich-Taylor
expansions that follow and to state the UFG and V0 conditions imposed on the given
vector fields (see (5) & (6) below). Section 3 contains our main results on the convergence analysis of a class of weak approximations of solutions of SDEs, characterised by
introducing the concept of an m-perfect family (Definition 2). The Lyons-Victoir and
Ninomiya-Victoir approximations are both members of this class. Although the Kusuoka
approximation is not within this family, it can effectively be categorised in the same way
(see Example 7 and the subsequent comment). In our main Theorem 8, we show that the
rate of convergence depends intrinsically on the smoothness of the chosen test function;
the higher the order of the approximation, the smoother the test function required.
For a smooth function, an equidistant partition of the time interval on which the
approximation is sought is optimal. For less smooth functions, this is no longer true.
We emphasise that the UFG+V0 conditions are not required for a smooth test function.
Furthermore, the Kusuoka approximation does not require the V0 condition. Finally, in
Section 4 we present an application of this theory to the numerical solution of the filtering
problem.
3
2
Preliminaries
Let (Ω, F, P) be a probability space satisfying the usual conditions and W = {Wt }t∈[0,∞)
be a k-dimensional Brownian motion defined on it. We also set Wt0 = t for t ∈ [0, ∞) . Let
Cb∞ (Rd , Rd ) denote the space of smooth functions ϕ : Rd → Rd with bounded derivatives,
that is, bounded partial derivatives (of all orders) of the d component functions {ϕi :
Rd → R}di=1 exist. We have a dual interpretation of this space, in that we also regard its
elements as vector fields. In other words, a smooth function V : Cb∞ (Rd ) → Cb∞(Rd ) is
equivalent to an operator on Cb∞ (Rd ) ≡ Cb∞ (Rd , R) defined by,
Vϕ=
d
i=1
Vi
∂ϕ
∂xi
where V i ∈ Cb∞ (Rd ) is the i-th component of L for i = 1, . . . , d.
We next consider the Stratonovich SDE with drift vector {V0i (x)}di=1 and dispersion
d
∞
d
d
matrix {Vji (x)}d,k
i,j=1 for x ∈ R , for some V0 , . . . , Vk ∈ Cb (R , R ). This is written
componentwise as,
t
k t
i
i
i
Xt = X0 +
V0 (Xs )ds +
Vji (Xs ) ◦ dWsj
(3)
0
j=1
0
for i = 1, . . . , d.
The following essential terminology has been adopted from Kusuoka[17]. Let A be the
set of multi-indices,
∞
A = {∅} ∪ ∪ {0, 1, . . . , k}m
m=1
where k is the dimension of the Brownian Motion introduced above. This set will be used
to index the Stratonovich-Taylor expansions that follow. Let |·| and · be the following
two norms defined on A by
|∅| = 0, |α| = r if α = (i1 , . . . , ir ) ∈ {0, 1, . . . , k}r for r ∈ N
and α = |α| + card {1 ≤ j ≤ |α| : ij = 0} . Furthermore, let A0 = A\ {∅}, A1 =
A\ {∅, (0)} and correspondingly, A(m) = {α ∈ A : α ≤ m}, A0 (m) = {α ∈ A0 :
α ≤ m} and A1 (m) = {α ∈ A1 : α ≤ m} where the integer m ∈ N above corresponds to the level of the truncation in the Stratonovich-Taylor expansions that follow
(see (12)).
We also require the following concatenation on A,
(i1 , . . . , ir ) ∗ (j1 , . . . , js) = (i1 , . . . , ir , j1 , . . . , js )
and need to define a further operation on the vector fields {Vj ∈ Cb∞ (Rd , Rd )}kj=0 introduced above.
4
Definition 1 For α ∈ A , the vector field V[α] is defined inductively by,
V[φ] = 0; V[(j)] = Vj ; V[α∗(j)] = [Vα , Vj ]
for j = 0, 1, . . . , k where [Vi , Vj ] := Vi Vj − Vj Vi for Vi , Vj ∈ Cb∞ (Rd , Rd ).
In the following we will make use of the semi-norm,
ϕV,i =
i
u=1 α1 ,... ,αu ∈A0
α1 ∗...∗αu =i
V[α1 ] · · · V[αu ] ϕ .
∞
for i ∈ N. Furthermore, we introduce the semi-norm,
p
i ∇ ϕ
ϕp =
∞
(4)
i=1
for p ∈ N, ϕ ∈ Cbp (Rd ) where,
i ∇ ϕ =
∞
i
∂
ϕ
max
j1 ,... ,ji ∈{1,... ,d} ∂xj1 . . . ∂xji ∞
and note that it can easily be deduced from the chain rule, for α ∈ A0, ϕ ∈ Cb (Rd ), that
Vαϕ∞ ≤ C ϕ|α| and hence ϕV,i ≤ C ϕi for i ∈ N. Also let ·p,∞ be the norm
ϕp,∞ := ϕ∞ + ϕp for ϕ ∈ Cbp (Rd ). We define the space,
|α|
CbV,i (Rd ) = ϕ : ϕV,i < ∞
which appears in the definition of an m-perfect family below (see Definition 2).
We now introduce the two conditions on the vector fields {Vj ∈ Cb∞ (Rd , Rd )}kj=0
required to treat the case when the test function ϕ is not smooth. We emphasise that the
smooth case requires neither the UFG nor the V0 condition. Furthermore, the Kusuoka
approximation (Example 7) does not require the V0 condition as the corresponding error
is controlled in terms of {V[β] : β ∈ A1 } alone.
5
The UFG condition [Kusuoka & Stroock [17], [23]]. There exists some l ∈ N
such that for any α ∈ A1 we have,
V[α] =
ϕα,β V[β] .
(5)
where ϕα,β ∈ Cb∞(Rd ) for all β ∈ A1 (l).
β∈A1 (l)
The V0 condition. There exist ϕβ ∈ Cb∞ (Rd ), β ∈ A1 (2) such that,
V0 =
ϕβ V[β] .
β∈A1 (2)
(6)
As mentioned in the introduction, the UFG condition states that the Lie algebra
generated by the vector fields {Vi }ki=1 ∈ Cb∞ (Rd ; Rd ) is finite dimensional as a Cb∞ (Rd )module. It is implied by the Uniform Hörmander Condition which states that,
2
∃l ∈ N and c > 0 s.t.
V[α] (x), ξ ≥ c |ξ|2
(7)
α∈A1 (l)
for all x, ξ ∈ Rd , where V, ξ = di=1 V i ξ i for V ∈ Cb∞ (Rd , Rd ).
Under conditions (5)+(6), one can show that for any r ∈ Z+ , {αi ∈ A0 }ri=1 and
p = 1, ..., α1 ∗ . . . ∗ αr , there exists a constant CpT > 0 such that,
V[α1 ] . . . V[αr ] Pt ϕ ≤ CpT t(p−α1 ∗...∗αr )/2 ϕ ,
t ∈ [0, T ] .
(8)
p
∞
where {Pt }t∈[0,∞) is the semigroup associated with X defined in (2). We remark that
condition (5) will only give us (8) for αi ∈ A1 , i = 1, ...r. Under both (5)+(6) we have,
V0 Pt ϕ∞ ≤ Ct−1 ϕ∞ .
However, under condition (5) alone, V0 Pt ϕ∞ may be of higher order. Kusuoka has given
an explicit example in which,
ct− 2 ϕ∞ ≤ V0 Pt ϕ∞ ≤ Ct− 2 ϕ∞
l
l
for some constants c, C > 0 and where l is the constant appearing in (5) (see Proposition
14 and Proposition 16 in [19]). This coarser bound on V0 Pt ϕ∞ results in lower rates
of convergence. The authors believe that (6) is the most general condition required
to preserve the same rates of convergence as those obtained when {Vi }ki=1 satisfy the
ellipticity condition (for non-smooth test functions ϕ).
Result (8) is a corollary of a certain representation theorem proved in KusuokaStroock[23]. For completeness, we state the representation theorem (Theorem 16) in
the appendix where we also sketch a proof of inequality (8) in Corollary 18. We note
that the case p = 1 has been proved in [23] for {αi ∈ A1 }ri=1 . Inequality (8) proves to be
crucial in obtaining upper bounds on the error of of the class algorithms that we study
below (see Theorem 8).
6
3
The Main Theorem
In this section we introduce the concept of an m-perfect family. Such families correspond
to various weak approximations of SDEs, including the Lyons-Victoir and NinomiyaVictoir schemes. The main result appears in Theorem 8 and Corollary 9.
For α = (i1 , . . . , ir ) ∈ A0 and ϕ ∈ Cbr (Rd ) let fα,φ be defined as f(i1 ,...,ir ),ϕ :=
Vi1 ...Vir ϕ. We also need to define the iterated Stratonovich integral
sr−2
t s0
i1
Ifα,ϕ (t) :=
···
fα,ϕ (Xsr−1 ) ◦ dWsr−1 ◦ · · · ◦ dWsi1r−1 ◦ dWsi0r ,
0
0
0
for t ≥ 0. If i1 = 0 then Ifα,ϕ (t) is well defined for ϕ ∈ Cbr (Rd ). However, if i1 = 0 then
Ifα,ϕ (t) is well defined provided ϕ ∈ Cbr+2 (Rd ), since the semimartingale
sr−2 property of
fα,ϕ (X) is required in the definition of the first Stratonovich integral 0
fα,ϕ (Xsr−1 ) ◦
i1
dWsr−1 . Note that the Stratonovich integrals are evaluated innermost first. Finally let
Iα (t) :=
t
0
0
s0
···
0
sr−2
1◦
1
dWsir−1
◦ · · · ◦ dWsi1r−1 ◦ dWsi0r .
Let α = (i1 , . . . , ir ) ∈ A0 be an arbitrary multi-index such that α = m ∈ N (and
|α| = r ∈ N). If m is odd, then E[Iα(t)] = 0 and if m is even then
m
t2
if α ∈ Am,r
r− m
0
m
2
2
(
)!
,
(9)
E[Iα (t)] =
2
0
otherwise
where Am,r
is the set of multi-indices α = α1 ∗ · · · ∗ α m2 ∈ A0 (m) such that each αi = (0)
0
or (j, j) for some j ∈ {1, . . . , k}. Note that r − m2 is equal to the number of pairs of indices
(j, j) occurring in α. A proof of this result can be found in [12].
The set of iterated Stratonovich integrals plays a central role in the theory of approximation of solutions of SDEs and there are numerous papers that study its structure. Here,
we adopt the hierarchical set approach introduced by Kloeden & Platen[15]. An alternative method can be found in Gaines[11] where it is shown how Lyndon words provide
a basis for iterated Stratonovich integrals and also how shuffle products may be used to
obtain moments of stochastic integrals. Pettersson[35] gives a notationally and computationally convenient Stratonovich-Taylor expansion. Furthermore, Burrage & Burrage[3]
use rooted-tree theory to describe the aforementioned set and Burrage & Burrage [4]
presents an approach based on B-series.
We state three further results in (10), (11) and (13). The proofs are all elementary
and can be found in [12]. The first two give an upper bound on the L2 norm of Ifα,ϕ (t) for
smooth ϕ. The third provides an explicit form for the remainder of ϕ(Xt ) when expanded
in terms of iterated integrals.
7
For ϕ ∈ Cb
have
||α||+2
(Rd ) and any multi-index α = (i1 , . . . , ir ) ∈ A0 such that i1 = 0, we
k
α+1
α
Ifα,ϕ (t) ≤ c1 fα,ϕ t 2 + c2
Vi fα,ϕ ∞ t 2
∞
2
(10)
i=1
for some constants c1 ≡ c1 (α), c2 ≡ c2 (α) ≥ 0. For ϕ ∈ Cb
α = (i1 , . . . , ir ) ∈ A0 such that i1 = 0 we have
||α||
(Rd ) and any multi-index
Ifα,ϕ (t) ≤ c1 fα,ϕ t α
2 .
∞
2
(11)
For m ∈ N, ϕ ∈ Cbm+3 (Rd ) and x ∈ Rd , we define the truncation,
ϕm
fα,ϕ (x)Iα(t).
t (x) := ϕ(x) +
(12)
α∈A0 (m)
Then for t ≥ 0 the remainder is
Rm,t,ϕ (x) := ϕ(Xt ) − ϕm
t (x) = (
α=m+1
+
α=m+2,α=0∗β,β=m
)Ifα,ϕ (t).
(13)
In the following, we define a class of approximations of X expressed in terms of certain
families of stochastic processes, X̄ (x) = {X̄t (x)}t∈[0,∞) for x ∈ Rd , which are explicitly
solvable. In particular, we can explicitly compute the operator,
(Qt ϕ)(x) = E[ϕ(X̄t (x))].
(14)
m
m
The semigroup PT will then be approximated by Qm
hn Qhn−1 . . . Qh1 where {hj := tj −
tj−1 }nj=1 and π n = {tj := ( nj )γ T }nj=0 for n ∈ N, is a sufficiently fine partition of the interval
[0, T ]. In particular hj ∈ [0, 1) for j = 1, ..., n. The underlying idea is that Qt ϕ will have
the same truncation as Pt ϕ.
So let X̄ (x) = {X̄t (x)}t∈[0,∞), where x ∈ Rd , be a family of progressively measurable
stochastic processes such that, limy→x0 X̄t0 (y) = X̄t0 (x0 ) P−almost surely, for any t0 ≥ 0
and x0 ∈ Rd . As a result, the operator Qt defined in (14) has the property that Qt ϕ ∈
Cb (Rd ) for any ϕ ∈ Cb (Rd ). In particular, Qt : Cb (Rd ) → Cb (Rd ) is a Markov operator.
Definition 2 For m ∈ N, the family X̄ (x) = {X̄t (x)}t∈[0,∞) where x ∈ Rd , is said to be
m-perfect for the process X if there exist constants C > 0 and M ≥ m + 1 such that for
ϕ ∈ CbV,M (Rd ),
sup |Qt ϕ(x) −
x∈Rd
E[ϕm
t (x)]|
8
≤C
M
i=m+1
ti/2 ϕV,i .
(15)
As we can see from (15), the quantity E[ϕm
t (x)] plays the same role as the classical
truncation in the standard Taylor expansion of a function. Using (9) we deduce that,
E[ϕ0t (x)] = ϕ(x)
E[ϕ2t (x)]
= ϕ(x) + V0 ϕ(x)t +
E[ϕ4t (x)] = E[ϕ2t (x)] +
+
k
i=1
k
i=1
k
i=1
t
Vi2 ϕ(x) = ϕ(x) + Lϕ(x)t
2
V02 ϕ(x)
Vi2 V0 ϕ(x)
k
t2 t2
+
V0 Vi2 ϕ(x)
2
4
i=1
k
t2
t2 2 2
+
Vj Vi ϕ(x)
4 i,j=1
8
t2
= ϕ(x) + Lϕ(x)t + L2 ϕ(x) ,
2
where L = V0 + 12 ki=1 Vi2 . Furthermore, since E[Iα (t)] = 0 for odd α, it follows that
E[ϕ1t (x)] = E[ϕ0t (x)], E[ϕ3t (x)] = E[ϕ2t (x)] and E[ϕ5t (x)] = E[ϕ4t (x)].
There now follow some examples of m−perfect families corresponding to {Pt }t∈[0,∞) as
described in (2), the Lyons-Victoir method and the Ninomiya-Victoir algorithm.
Example 3 The family of stochastic processes {Xt (x)}t∈[0,∞) , where x ∈ Rd , is m−perfect.
More precisely there exists a constant c3 > 0 such that for ϕ ∈ CbV,m+2 (Rd ),
sup |Pt ϕ(x) −
x
E[ϕm
t (x)]|
Proof. For ϕ ∈ CbV,m+3 (Rd ),
≤ c3
m+2
ti/2 ϕV,i ,
i=m+1
(16)
m
|Pt ϕ(x) − E[ϕt (x)]| = |E[Rm,t,ϕ (x)]| = E[(
+
)Ifα,ϕ (t)]
α=m+1 α=m+2,α=0∗β,β=m
Applying inequality (10) to the first sum,
α=m+1
Ifα,ϕ (t) ≤
2
{c1 (α) fα,ϕ ∞ t
α=m+1
≤ c4
m+2
ti/2 ϕV,i
i=m+1
9
m+1
2
k
m+2
+ c2 (α)
Vi fα,ϕ ∞ t 2 }
i=1
(17)
for some constant c4 > 0. Applying result (11) to the second sum,
Ifα,ϕ (t) ≤
c1 (α) fα,ψ ∞ t
2
α=m+2,α=0∗β,β=m
α=m+2,α=0∗β,β=m
≤ c5 ϕV,m+2 t
m+2
2
(18)
for some c5 > 0. The result for ϕ ∈ CbV,m+3 (Rd ) follows from combining (17) and (18).
Since none of the terms in (16) depend on partial derivatives of order m+3, the inequality
is also valid for any ϕ ∈ CbV,m+2 (Rd ) (a standard approximation method can be used).
In the following example, the family of processes X̄ (x) = {X̄t (x)}t∈[0,1] , where x ∈ Rd ,
corresponds to the Lyons-Victoir approximation (see [24]). The example involves a set of
l finite variation paths, ω 1 , . . . , ω l ∈ C00 ([0, 1], Rk ), for some l ∈ N, together with some
l
weights λ1 , . . . , λl ∈ R+ such that
λj = 1. These paths are said to define a cubature
j=1
formula on Wiener Space of degree m if, for any α ∈ A0 (m),
l
E[Iα(1)]= λj Iαωj (1)
j=1
where,
ω
I(i1j ,...,ir ) (1)
:=
1 s0
0
0
···(
sr−2
0
dω ij1 (sr−1 )) · · · dω ijr−1 (s1 )dω ijr (s0 ).
From the scaling properties of the Brownian motion we can deduce, for t ≥ 0,
l
E[Iα(t)]= λj Iαωt,j (t)
j=1
√
where ω t,1 , . . . , ω t,l ∈ C00 ([0, t], Rk ) is defined by ω t,j (s) = tω j st , s ∈ [0, t]. In other
words, the expectation of the iterated Stratonovich integrals Iα (t) is the same under the
Wiener measure as it is under the measure,
Qt :=
l
j=1
λj δ ωt,j .
Example 4 If we choose X̄ to satisfy the evolution equation (3) but with the driving
Brownian motion replaced by the paths ω t,1 , . . . , ω t,l defined above then the family of
10
processes, {X t (x)}t∈[0,1] , with corresponding operator (Qt ϕ)(x) := EQt [ϕ(X t (x))],
is m−perfect. More precisely, there exists a constant c6 > 0 such that for ϕ ∈ CbV,m+2 (Rd ),
m+2
m
sup Qt ϕ(x) − E[ϕt (x)] ≤ c6
ti/2 ϕV,i
x
i=m+1
For example, if (λj , ω t,j ) are chosen such that for l = 2k the paths are ω t,j : t →
t(1, zj1 , .., zjk ) for j = 1, . . . , 2k with points zj ∈ {−1, 1}k and weights λj = 2−k , we obtain
a cubature formula of degree 3 and a corresponding 3-perfect family.
ω
ω
Proof. Let us first observe that Iα t,j (t) = t 2 Iα j (1) Hence, for ϕ ∈ CbV,m+2 (Rd ),
Qt ϕ(x) − E[ϕm
= |EQt [Rm,t,ϕ (x)]|
t (x)]
≤ (
+
) fα,ϕ ∞ EQt [Iα (t)]2
α=m+1
≤ (
≤ (
where kα =
l
j=1
ω
λj Iα j (1)2 .
α=m+1
α=m+1
|α|
+
+
α=m+2,α=0∗β,β=m
l
) fα,ϕ ∞
λj Iαωt ,j (t)]2
α=m+2,α=0∗β,β=m
j=1
)kα fα,ϕ ∞ t
α=m+2,α=0∗β,β=m
|α|
2
.
Remarks
(i) There has been no change to the underlying measure in the example above. Merely a
representation in terms of the measure Qt hasbeen introduced
to ease the computation
of Qt . More precisely, the family of processes X t (x) t∈[0,1] where x ∈ Rd is constructed
as follows. We take,
X 0 (x) = x
and then randomly choose a path ωt,r from the set {ω t,1 , . . . , ω t,l } with corresponding
probabilities (λ1 , . . . , λl ). Each process then follows a deterministic trajectory driven by
the solution of the ordinary differential equation,
dX t = V0 (X t )dt +
11
k
j=1
Vj (X t )dω jt,k
for some V0 , . . . , Vk ∈ Cb∞ (Rd , Rd ) as in (3). We can therefore compute the expected
values of a functional of Xt (x) as integrals on the path space with respect to the Radon
measure Qt . Hence the identities,
Qt ϕ(x) = E ϕ(X t (x)) = EQt ϕ(X t (x))
(ii) The approach adopted by Lyons and Victoir to construct the above approximation
resembles the ideas developed by Clark and Newton in a series of papers ([8],[9],[29],[30]).
Heuristically, Clark and Newton constructed strong approximations of SDEs using flows
driven by vector fields which were measurable with respect to the filtration generated by
the driving Wiener process. In a similar vein, Castell & Gaines[7] provide a method of
strongly approximating the solution of an SDE by means of exponential Lie series.
For the following example, we will denote by exp(V t)f the value at time t of the
solution of the ODE y ′ = V (y) , y (0) = f where V ∈ Cb∞ (Rd , Rd ). In particular,
exp(V t) (x) is exp(V t)f for f being the identity function. The family of processes
Y (x) = {Yt (x)}t∈[0,1] below corresponds to the Ninomiya-Victoir approximation (see
[33]).
Example 5 Let Λ and Z be two independent random variables such that Λ is Bernoulli
distributed P(Λ = 1) = P(Λ = −1) = 21 and Z = (Z i )ki=1 is a standard normal k−
dimensional random variable. Consider the family of processes Y (x) = {Yt (x)}t∈[0,1]
defined by

k


if Λ = 1
 exp( V20 t) exp(Z i Vi t1/2 ) exp( V20 t)(x)
i=1
Yt (x) =
k


 exp( V20 t) exp(Z k+1−i Vk+1−i t1/2 ) exp( V20 t)(x) if Λ = −1
i=1
with the corresponding operator (Qt ϕ)(x) := E[ϕ(Yt (x))] . Then there exists a constant
c7 > 0 such that for ϕ ∈ CbV,8 (Rd )
sup Qt ϕ(x) − E[ϕ5t (x)] ≤ c7 t3 ϕV,6
x
Hence {Yt (x)}t∈[0,1] is 5-perfect.
k+1
Proof. We first consider the case Λ = 1. Let {Y i }i=0
be defined,
dϕ 0
(Y )
ds s
dϕ 1
(Y )
ds s
dϕ i
(Y )
ds s
dϕ k+1
(Y
)
ds s
t
= V0 ϕ(Ys0 ) for s ∈ [0, ], Y00 = x
2
√
1
1
= Z V1 ϕ(Ys ) for s ∈ [0, t], Y01 = Y t0
= Z i Vi ϕ(Ysi ) for s ∈ [0,
√
2
t], Y0i = Y√i−1
, i = 2, . . . , k
t
t
= V0 ϕ(Ysk+1 ) for s ∈ [0, ], Y0k+1 = Y√kt
2
12
It follows from the definition of the algorithm and by Itô’s Formula that,
t/2 s1 s2
2
k+1
k
k t
2
k t
√
√
√
ϕ(Y t ) = ϕ(Y t ) + V0 ϕ(Y t ) + V0 ϕ(Y t ) +
V03 ϕ(Ysk+1
)ds3 ds2 ds1 .
3
2
2
8
0
0
0
(19)
We need to expand the right hand side of (19) and divide the resulting expansion into two
parts: the required truncation and a remainder whose expected value should be bounded
by Ct3 ϕV,6 . The final term in (19) belongs to the remainder and indeed,
t/2 s1 s2
3 (t/2)3
t3
3
k+1
E V0 ϕ(Ys3 )ds3 ds2 ds1 ≤ V0 ϕ∞
≤
ϕV,6
(20)
0
6
48
0
0
Expanding the third term in (19),
√t
V02 ϕ(Y√kt ) = V02 ϕ(Y√k−1
Z k Vk V02 ϕ(Ysk )ds
)+
t
0
=
V02 ϕ(Y√k−1
)
t
=
V02 ϕ(Y t0 )
= ...
2
+
So
V02 ϕ(Y√kt )
=
i=1
+Z
√
k t
0
k √
0
t
2
0
2
Zi
2
Zi
t+
√
t s1
0
0
k 2 2 2
Z
Vk V0 ϕ(Ysk2 )ds2 ds1
k
√
√
) t
t+
Z i Vi V02 ϕ(Y√i−1
t
i=2
Vi2 V02 ϕ(Ysi2 )ds2 ds1
V03 ϕ(Ys0 )ds
t s1
0
Vk V02 ϕ(Y√k−1
)
t
V1 V02 ϕ(Y t0 )
0
+
i=1
1
s1
V02 ϕ(x)
+
+Z
k
√
2
+Z
1
k
√
√
t+
Z i Vi V02 ϕ(Y√i t ) t
V1 V02 ϕ(Y t0 )
2
i=2
Vi2 V02 ϕ(Ysi2 )ds2 ds1
√
√
Now the last four terms are all O(t) since E Z 1 V1 V02 ϕ(Y t0 ) t = E Z i Vi V02 ϕ(Y√i−1
)
t =
t
2
t
0 because Z is normal, E 02 V03 ϕ(Ys0 )ds ≤ V03 ϕ∞ 2t
√ t s
2
and finally E 0 0 1 (Z i ) Vi2 V02 ϕ(Ysi2 )ds2 ds1 ≤ Vi2 V02 ϕ∞ 2t .
So for the third term in (19) we have established,
k
2
2
3
3 2 2 t
t
t
2
E V02 ϕ(Y√k )
Vi V0 ϕ
−
V
ϕ(x)
≤
V
ϕ
+
0
0
t
∞
∞
8
8
16
i=1
≤
13
t3
ϕV,6 .
16
(21)
Similarly, for the second term on the RHS of (19),
k
t
t 2
t
2
k
√
Vi V0 ϕ(x)
E V0 ϕ(Y t ) − V0 ϕ(x) + V0 ϕ(x) +
2
2
2
t t3
≤ ϕV,6
2
8
i=1
Finally, for the first term on the RHS of (19),
2
2
ϕ(x)
+ V0 ϕ(x) 2t + V02 ϕ(x)"t8 + ki=1 Vi4 ϕ(x) t4!
!
k
2
E ϕ(Y√t ) −
t2
2 2
+ ki=1 Vi2 ϕ(x) 2t + V0 Vi2 ϕ(x) t4 + ki=2 i−1
j=1 Vj Vi ϕ(x) 4
(22)
t3
ϕV,6
(23)
16
Substituting (20), (21), (22) and (23) in (19) gives the bound for the case Λ = 1. An
analogous bound is then established for the case Λ = −1. The final result is obtained by
taking the average of the two cases. See [12] for details.
≤
The following Lemma is required to prove the main theorem below.
Lemma 6 For 0 < s ≤ t ≤ 1 and any m−perfect family {X t (x)}t∈(0,1] with corresponding
operator Q = {Qt }t∈(0,1] we have,
Pt (Ps ϕ) − Qt (Psϕ)∞ ≤ c8 ϕp
M
tj/2
j=m+1 s
,
j−p
2
(24)
where ϕ ∈ Cbp (Rd ) for 0 ≤ p < ∞ and some constant c8 > 0. In particular, for ϕ ∈
CbM (Rd ),
Pt (Ps ϕ) − Qt (Psϕ)∞ ≤ c8 ϕp t
m+1
2
.
(25)
Proof. Since Cb∞ (Rd ) is dense in Cbp (Rd ) in the topology generated by the norm
||·||p,∞ it suffices to prove (24) and (25) only for a function ϕ ∈ Cb∞ (Rd ). By Corollary 18
in the appendix,
Pt ϕV,j =
≤
j
i=1 α1 ,... ,αi ∈A0
α1 ∗...∗αi =j
j
i=1 α1 ,... ,αi ∈A0
α1 ∗...∗αi =j
V[α1 ] · · · V[αi ] Pt ϕ
∞
CpT ϕp
t(α1 ∗...∗αi −p)/2
≤
c9 ϕp
t
j−p
2
for some c9 ≡ c9 (j, p) ≥ 0. Then (24) and (25) follow from the definition of an m−perfect
family.
The family of processes X̄ (x) = {X̄t (x)}t∈[0,∞) below corresponds to the Kusuoka
approximation. We recall that Kusuoka’s result requires only the UFG condition.
14
Example 7 A family of random variables {Zα : α ∈ A0 } is said to be m-moment
similar if E[ |Zα |r ] < ∞ for any r ∈ N, α ∈ A0 and Z(0) = 1 with,
E[Zα1 . . . Zαj ] = E[Iα1 . . . Iαj ]
for any j = 1, . . . , m and α1 , . . . , αj ∈ A0 such that α1 + · · · + αj ≤ m where Iα is
defined as above.
Let {Zα : α ∈ A0 } be a family of m−moment similar random variables and let X̄ (x) =
{X̄t (x)}t∈[0,∞) be the family of processes,
m
1
X̄t (x) =
j!
j=0
α1 ,... ,αj ∈A0 ,
α1 +···+αj ≤m
t
αj α1 +···+
2
(Pα01 . . . Pα0j )(V[α1 ] . . . V[αj ] H)(x)
where H : Rd → Rd is defined H(x) = x and
Pα0
:= |α|
−1
|α|
(−1)j+1
j=0
j
β 1 ∗...∗β j =α
(26)
Zβ 1 . . . Zβ j
with the corresponding operator Q = {Qt }t∈(0,1] in Cb (Rd ) defined by,
Qt ϕ(x) = E[ϕ(X̄t (x))]
for ϕ ∈ Cb∞ (Rd ) Then,
Pt+s ϕ − Qt Ps ϕ∞ ≤ c10 ∇ϕ∞
m+1
m
tj/2
j=m+1 s
j−1
2
(27)
for some constant c10 > 0.
Proof. See Definition 1, Theorem 3 and Lemma 18 in Kusuoka[18] for (27).
The family X̄ (x) , x ∈ Rd as defined in (26) is not m-perfect. However, inequality
(27) is a particular case of (24) where p = 1 and M = mm+1 . Since (24) is the only result
required to obtain (28), we deduce from the proof of Theorem 8 that (28), with p = 1,
holds for Kusuoka’s method as well. Similarly part (ii) of Corollary 9 holds for Kusuoka’s
method.
The set of vector fields appearing in (26) belong to the Lie algebra generated by the
original vector fields {V0 , V1 , ..., Vk } . Ben Arous[2] and Burrage & Burrage[3] employ the
same set of vector fields to produce strong approximations of solutions of SDEs. Notably,
the same ideas appear much earlier in Magnus[25], in the context of approximations of the
solution of linear (deterministic) differential equations. Castell[6] also gives an explicit
15
formula for the solution of an SDE in terms of Lie brackets and iterated Stratonovich
integrals.
We now prove our main result on m-perfect families, the gist of which can be conveyed
by the concept of local and global order of an approximation. Local order measures
how close an approximation is to the exact solution on a sub-interval of the integration,
given an exact initial condition at the start of that subinterval. The global order of an
approximation looks at the build up of errors over the entire integration range. The
theorem below states that, in the best possible case, the global order of an approximation
obtained using an m-perfect family is one less than the local order. More precisely, for a
suitable partition, the global error is of order m−1
whilst the local error is of order m+1
.
2
2
Let us define the function,
# − 1 min(γp,(m−1))
n 2
if γp = m − 1
p
Υ (n) =
−(m−1)/2
n
ln n
for γp = m − 1
In the following,
m
m E γ,n (ϕ) := PT ϕ − Qm
hn Qhn−1 . . . Qh1 ϕ ∞
for γ ∈ R, n ∈ N.
Theorem 8 Let T, γ > 0 and π n = {tj = ( nj )γ T }nj=0 be a partition of the interval [0, T ]
where n ∈ N is such that {hj = tj − tj−1 }nj=1 ⊆ (0, 1]. Then for any m-perfect family
{X t (x)}t∈[0,T ] with corresponding operator Q = {Qt }t∈(0,1] we have, for ϕ ∈ Cbp (Rd ) where
p = 1, ..., m,
E γ,n (ϕ) ≤ c11 Υp (n) ϕp + Ph1 ϕ − Qm
ϕ
(28)
h1
∞
for some constant c11 ≡ c11 (γ, M, T ) > 0 where M ≥ m + 1, as in Definition 2. In
particular, if γ ≥ m−1
then,
p
Proof. We have,
E γ,n (ϕ) ≤
c11
n
m−1
2
ϕp + Ph1 ϕ − Qm
h1 ϕ ∞
E γ,n (ϕ) = Phn (PT −hn ϕ) − Qm
hn (PT −hn ϕ)
n−1
m
m
+
Qm
hn . . . Qhj+1 (PT −hj+1 −···−hn ϕ − Qhj PT −hj −···−hn ϕ)
j=1
= Phn (Ptn−1 ϕ) − Qm
hn (Ptn−1 ϕ)
n−1
m
m
+
Qm
.
.
.
Q
(P
P
ϕ
−
Q
P
ϕ
)
h
t
t
hn
hj+1
hj
j
j−1
j−1
j=1
16
By Lemma 6, there exists a constant c8 > 0 such that,
Phn (Ptn−1 ϕ) − Qm (Ptn−1 ϕ)
hn
∞
≤ c8 ϕp
M
l/2
hn
l−p
2
l=m+1 tn−1
Since P is a semigroup and Qm
hj is a Markov operator for j = 2, . . . , n − 1,
m
m
m
Qhn . . . Qhj+1 (Phj Ptj−1 ϕ − Qhj Ptj−1 ϕ )
m
≤ Phj Ptj−1 ϕ − Qhj Ptj−1 ϕ ∞
≤ c12 ϕp
l/2
M
hj
∞
l−p
2
l=m+1 tj−1
for some c12 > 0. Finally, since Qm
hj is a Markov operator, it follows from (32) that,
m
m
m Qh . . . Qm
h2 (Ph1 ϕ − Qh1 ϕ) ∞ ≤ Ph1 ϕ − Qh1 ϕ ∞
n
Combining these last four results gives,
E
γ,n
l/2
n
M
hj
m
m m (ϕ) = PT ϕ − Qhn . . . Qh1 ϕ ∞ ≤ Ph1 ϕ − Qh1 ϕ ∞ + c12 ||ϕ||p
l−p
2
j=2 l=m+1 tj−1
It follows, almost immediately from the definition of hj that,
γT (j − 1)γ−1
hj =
nγ
j
j−1
u
j−1
γ−1
du
j γ−1
u γ−1
but for j ∈ {2, . . . , n}, ( j−1
)
≤ max[( j−1
) , 1] ≤ max[2γ−1 , 1]. Hence for l = m +
1, . . . , M ,
l/2
hj
(l−p)/2
tj−1
( γT (j−1)
max[2γ−1 , 1])l/2
nγ
≤
j−1 γ (l−p)/2
T
n
(γ−1)l γ(l−p)
γp−l
T l (l−p)
T p
≤ c13 ( γ ) 2 − 2 (j − 1) 2 − 2 = c13 ( γ ) 2 (j − 1) 2
n
n
γ−1
where c13 ≡ c13 (γ, M) = max[1, (γ max[2γ−1 , 1])M/2 ]. It follows that,
M
l/2
hj
(l−p)/2
l=m+1 tj−1
M
γp−l
1 γp ≤ c14 ( ) 2
(j − 1) 2
n
l=m+1
17
γp−l
2
where c14 ≡ c14 (γ, M, T ) = T p/2 c13 and since M
=
l=m+1 (j − 1)
γp−(m+1) M −(m+1)
γp−(m+1)
l
−
(j − 1) 2
(j − 1) 2 ≤ (j − 1) 2 M we have,
l=0
l/2
hj
M
(l−p)/2
l=m+1 tj−1
γp−(m+1)
1 γp
≤ c14 M( ) 2 (j − 1) 2
n
(29)
We now !
consider" (29) for three different ranges of γ.
γp−(m+1)
γp−(m+1)
2
For γ ∈ 0, m−1
, nj=2 (j − 1) 2
≤ ∞
and since the series on the
j=2 (j − 1)
p
RHS is convergent, we have,
n
− γp
2
n
γp−(m+1)
γp
(j − 1) 2
≤ c15 n− 2
j=2
for some constant
(γ, M ) > 0.
n c15 ≡ c15−1
m−1
For γ = p ,
≤ c16 ln n for some constant c16 ≡ c16 (γ, M) > 0 so we have,
j=2 (j − 1)
n
− γp
2
n
j=2
(j − 1)
γp−(m+1)
2
≤ c16 n−
(m−1)
2
ln n
1 γp−(m+1)
1 −1+ γp−(m−1)
n j−1 γp−(m+1) 1
2
2
2
For γ > m−1
dx
=
c
x
dx ≤ c18 (like
,
(
)
≤
c
x
17
17
j=2
0
p
n
n
0
a Riemann integral) for some constants c17 ≡ c17 (γ, M), c18 ≡ c18 (γ, M ) so,
n
− γp
2
n
j=2
(j − 1)
γp−(m+1)
2
=n
− m−1
2
n j −1
j=2
n
γp−(m+1)
2
m−1
1
≤ c18 n− 2
n
We observe that the rate of convergence is the controlled by the maximum between
k1 ,k2
Υ (n) and the rate at which Ph1 ϕ − Qm
(n) =
h1 ϕ ∞ convergences to 0. We define Ῡ
Υk1 (n) + n−
γk2
2
. Hence we have the following corollary:
Corollary 9
(i) For any ϕ ∈ CbM (Rd ),
E γ,n (ϕ) ≤ c19 Ῡm+1,m+1 (n) ϕM .
for some constant c19 > 0. In particular, if γ ≥ 1, then E γ,n (ϕ) ≤
(ii) For any ϕ ∈ Cb1 (Rd ),
E γ,n (ϕ) ≤ c21 Ῡ1,1 (n) ϕ1
18
n
c19
m−1
2
ϕM .
for some constant c21 > 0, if there exists a constant c20 > 0 independent of t such that,
√
sup X̄t (x) − x ≤ c20 t.
(30)
x∈Rd
In particular, if γ ≥ m − 1, then E γ,n (ϕ) ≤
(iii) For any ϕ ∈ Cbl (Rd ) where 1 < l < M,
n
c21
m−1
2
ϕ1 .
E γ,n (ϕ) ≤ c24 Ῡl,c23 (n) ϕl
for some constant c24 > 0, if there exist two constants c22 , c23 > 0 independent of t such
that,
Pt ϕ − Qm
t ϕ∞ ≤ c22 t
In particular, if γ ≥ m − 1, then E γ,n (ϕ) ≤
n
c24
m−1
2
c23
2
ϕl .
(31)
ϕl .
Proof.
(i) The result follows from the theorem and the definition of an m−perfect family.
(ii) If ϕ ∈ Cb (Rd ) is Lipschitz then,
√
(32)
|Qt ϕ(x) − ϕ (x)| ≤ c20 ||∇ϕ||∞ t
hence,
√
Ph1 ϕ − Qm
t.
h1 ϕ ∞ ≤ c20 ||ϕ||1
(iii) The result follows from the theorem and (31).
Finally we define µt to be the law of Xt :
µt (ϕ) = E [ϕ (Xt )] for ϕ ∈ Cb (Rd ).
We also define µN
t to be the probability measure defined by,
m m
N
m
m
m
µt (ϕ) = E Qhn Qhn−1 . . . Qh1 ϕ (X0 ) =
Qm
hn Qhn−1 . . . Qh1 ϕ (x) µ0 (dx)
Rd
for ϕ ∈ Cb (Rd ) and need to introduce the family of norms on the set of signed measures:
l
d
|µ|l = sup |µ (ϕ)| , ϕ ∈ Cb (R ), ϕl,∞ ≤ 1 , l ≥ 1.
Obviously, |µ|l ≤ |µ|l′ if l ≤ l′ . In other words, the higher the value of l, the coarser the
norm. We have the following.
19
Corollary 10
µt − µN
≤ c19 Ῡm+1,m+1 (n). In particular, if γ ≥ 1, then
(i)
For
l
≥
M,
we
have
t
l
19
µt − µN ≤ cm−1
.
t l
n 2
1,1
(ii)
If
(30)
is
satisfied then µt − µN
(n). In particular, if γ ≥ m − 1, then
t l ≤ c21 Ῡ
c21
µt − µN
t l ≤ m−1 .
n 2
l,c23
µt − µN
(iii)
If
(31)
is
satisfied
then
(n). In particular, if γ ≥ m − 1, then
t l ≤ c24 Ῡ
24
µt − µN ≤ cm−1
.
t l
n
2
Throughout, the constants c19 , c21 , c23 , c24 > 0 correspond to those found in Corollary 9.
Remark We deduce that there is a payoff between the rate of convergence and the
coarseness of the norm employed: the finer the norm the slower the rate of convergence.
Hence intermediate results such as part (iii) of Corollaries 9 and 10 may prove useful in
subsequent applications. The additional constraint (31) holds, for example, for the LyonsVictoir method, as a cubature formula of degree m is also a cubature formula of degree m′
for m′ ≤ m. Similarly, it holds for Kusuoka’s approximation since an m−similar family
is also m′ −similar for any m′ ≤ m.
4
An Application to Filtering
We begin with a short description of the filtering problem. Let (X, Y ) be a system of
partially observed random processes. The process X satisfies the stochastic differential
equation (1) and is the unobserved component. The process Y is the observable component and satisfies the evolution equation,
t
Yt =
h(Xs )ds + Bt ,
0
l
where
{Bt }t∈[0,∞) is an l-dimensional Brownian motion independent of X and h = (hi )i=1 ∈
Cb∞ Rd , Rl . Let (Yt )t≥0 be the filtration generated by Y , Yt = σ (Ys , 0 ≤ s ≤ t). The
problem of stochastic filtering for the partially observed system (X, Y ) involves the construction of π t (ϕ), where π = {π t , t ≥ 0} is the conditional distribution of Xt given Yt and
ϕ belongs to a suitably large class of functions. If ϕ is square integrable with respect to
the law of Xt then,
π t (ϕ) = E [ϕ (Xt ) |Yt ] , P − almost surely.
Using Girsanov’s theorem, one can find a new probability measure P̃ absolutely continuous
with respect to P (and vice versa), so that Y is a Brownian motion under P̃, independent
of X and, almost surely,
π t (ϕ) =
ρt (ϕ)
,
ρt (1)
20
(33)
where,
ρt (ϕ) = Ẽ ϕ(Xt ) exp
l 0
i=1
t
l
1
hi (Xs )dYsi −
2 i=1
0
t
hi (Xs )2 ds Yt
(34)
and Ẽ is the expectation with respect to P̃. The measure ρt is called the unnormalised
conditional distribution of the signal. The identity (33) is called the Kallianpur-Striebel
Formula. In the following, we will denote by ||·||p , the Lp -norm with respect to the prob1
ability measure P̃, |ξ|p = Ẽ [|ξ|p ] p , for any random variable ξ. The laws of the families
X (x) = {Xt (x)}t∈[0,∞) , x ∈ Rd and X̄ (x) = {X̄t (x)}t∈[0,∞), x ∈ Rd are not affected by
the change of measure, hence, to avoid working with both P and P̃ we can write,
(Pt ϕ)(x) = Ẽ[ϕ(Xt (x))], (Qt ϕ)(x) = Ẽ[ϕ(X̄t (x))].
In the following, we will only consider equidistant partitions and smooth functions.
The method of approximation and the results closely follow the application of the classical
Euler method as described in Picard [36] and Talay [42].
Let yr , r = 1, ..., n be the observation process increments yr = Y (r+1)t − Y rtn and
n
hr ∈ Cb∞ Rd , r = 0, ..., n − 1, be the (observation dependent) functions defined by
2
t
hr = li=1 (hi yri − 2n
(hi ) ). Let Rsr , R̄sr : Cb∞ Rd → Cb∞ Rd , r = 0, 1, ..., n be the
following operators,
Rsn ϕ (x) = Ps ϕ (x) , R̄sn ϕ (x) = Qsϕ (x) for ϕ ∈ Cb∞ Rd , x ∈ Rd
and, for r = 0, 1, ..., n − 1, and for ϕ ∈ Cb∞ Rd , x ∈ Rd ,
Rsr ϕ (x) = Ẽ [ ϕ (Xs (x)) exp (hr (Xs (x)))| Ys ] = Ps ϕr (x)
R̄sr ϕ (x) = Ẽ ϕ X̄s (x) exp hr X̄s (x) Ys = Qs ϕr (x) ,
where ϕr = ϕ exp (hr ) and s ∈ [0, 1].
Firstly, one approximates ρ by replacing the (continuous)
observation
path with a
it
discrete version. We choose the equidistant partition n , i = 0, 1, ...n of the interval
[0, t] and consider only the observation data {yr , r = 0, 1, ..., n}. We define the measure,
n−1
" !
ρnt (ϕ) = Ẽ ϕ(Xt ) exp
hi X it
(35)
Yt
n
i=0
0 1
n
= Ẽ R t R t ...R t ϕ (X0 ) Yt for ϕ ∈ Cb Rd .
n
n
n
Following Theorem 1 from Picard [36], for any ϕ ∈ Cb∞ (Rd ) there is a constant c ≡ c (t, ϕ)
such that,
c
||ρt (ϕ) − ρnt (ϕ)||2 ≤
n
21
The next step is to approximate ni=0 Rit with ni=0 R̄it . For this we need to adapt the
n
n
definition of an m−perfect family so that we may use functions which are parametrized
bythe observation
path Y . Let CbY,∞ (Rd ) be the set of measurable functions, f : Rd ×
C [0, T ], Rl → R with the following properties:
i. for any y ∈ C [0, T ], Rl the function x → f (x, y) belongs to Cb∞(Rd ).
ii. for any multi-index α ∈ A, any x ∈ Rd and p ≥ 1, |Dα f (x, Y )|p < ∞.
iii. for any multi-index α ∈ A and p ≥ 1, |||Dα f|||p,∞ = supx∈Rd |Dαf (x, Y )|p < ∞.
For f ∈ CbY,∞ (Rd ) we define the norm |||f |||m
|||Dαf |||p,∞ . We note that if
p =
α∈A(m)
f : Rd × C [0, T ], Rl → R is constant
y variable, then ||Dα f (x, Y )||p =
1 in the m
m
|Dαf (x, Y )| and |||f |||p = ||f ||∞ + ∇ f ∞ + ... ∇ f ∞ .
We now consider an m-perfect family X̄ (x) that satisfies the equivalent of (15) extended to functions in CbY,∞ (Rd ); more precisely we will assume that for any f ∈ CbY,∞ (Rd ),
m
Qt f − Ẽ[ft |Y ]
p,∞
≤C
M
ti/2 |||f |||ip ,
i=m+1
(36)
for some constants C > 0 and M ≥ m + 1, where ftm is the truncation defined in (12).
Note that the original definition (15) implies (36) due to the inequality ϕV,i ≤ C ϕi
for i ∈ N first established in Section 2. Indeed, if f ∈ Cb∞(Rd ), in other words it is
constant in the y variable, then (15) and (36) actually coincide. The original Markov
family X (x), the family generated by the Lyons-Victoir method and the one generated
by the Ninomiya-Victoir algorithm satisfy (36).
The following two lemmas are required to prove the main theorem below.
Lemma 11 Let X̄ (x) be an m-perfect family X̄ (x) that satisfies (36). Then there is a
constant c25 = c25 (t, m, p) > 0 such that for any ϕ ∈ Cb∞ (Rd ) and r = 1, ..., n, we have,
r−1 r r+1
n
r−1 r
r+1
n ≤ c25 n−(m+1)/2 ||ϕ||M .
R̄ t R t R t ...R t ϕ − R t R t R t ...R t ϕ
n
n
n
n
n
n
p,∞
n
n
Proof. Again, using the variational argument in Friedman[10] p.122 (5.17) one can
check that for any ϕ ∈ Cb∞ (Rd ) and r = 1, ..., n the functions,
exp (hr−1 ) Rrt Rr+1
...Rnt ϕ ∈ CbY,∞ (Rd ).
t
n
n
n
Moreover there is a constant c26 ≡ c26 (M, p) such that for any ϕ ∈ Cb∞ (Rd ) and r =
1, ..., n,
M
...Rnt ϕ ≤ c26 ϕM
(37)
exp (hr−1 ) Rrt Rr+1
t
n
n
n
22
p
Let X̄ (x) be an m-perfect family that satisfies (36). Then since X (x) also satisfies (36),
it follows by the triangle inequality that for s ∈ [0, 1] ,
|||Qs f − Ps f |||p,∞ ≤ Cs(m+1)/2 |||f |||M
p
Hence,
r+1
r+1
r
n
r
n Q nt exp (hr−1 ) R t R t ...R t ϕ − P nt exp (hr−1 ) R t R t ...R t ϕ
n
n
n
n
n
n
p,∞
(m+1)/2 M
t
n ≤C
...R
ϕ
exp (hr−1 ) Rrt Rr+1
t
t
n
n
n
n
p
(38)
The result now follows from (37) and (38) and the fact that R̄r−1
Rrt Rr+1
...Rnt ϕ =
t
t
n
n
n
n
Q nt exp (hr−1 ) Rrt Rr+1
...Rnt ϕ and Rr−1
Rrt Rr+1
...Rnt ϕ = P nt exp (hr−1 ) Rrt Rr+1
...Rnt ϕ.
t
t
t
t
n
n
n
n
n
n
n
n
n
n
Lemma 12 Let X̄ (x) be an m-perfect family that satisfies (36). Then there is a constant
c27 ≡ c27 (t, m, p) > 0 such that for any ϕ ∈ Cb∞ (Rd ) we have,
0 1
n
0
1
n ≤ c27 n−(m−1)/2 ||ϕ||M .
R t R t ...R t ϕ − R̄ t R̄ t ...R̄ t ϕ
n
n
n
n
n
p,∞
n
Proof. Let us observe that,
R0t R1t ...Rnt ϕ − R̄0t R̄1t ...R̄nt ϕ = R0t R1t ...Rnt ϕ − R̄0t R1t ...Rnt ϕ
n
n
n
n
n
n
n
+
n
n−1
j=1
n
n
R̄ t ...R̄ t
j−1
0
n
n
n
n
R̄ t R t ...R t ϕ − R t R t ...R t ϕ
!
j
j+1
n
n
n
n
j
j+1
n
n
n
n
!
"
n
n
R̄
+R̄0t ...R̄n−1
ϕ
−
R
ϕ
t
t
t
n
n
n
n
Also note that for p ≥ 1 and r = 1, ..., n,
!
"p
0
j−1
j
j+1
n
n
R̄jt Rj+1
...R
R
...R
ϕ
−
R
ϕ
R̄ t ...R̄ t
t
t
t
t
t
n
n
n
n
n
n
n
n
p,∞
p r−1 r r+1
r−1 r
r+1
n
n ≤ c28 Ẽ R̄ t R t R t ...R t ϕ − R t R t R t ...R t ϕ
≤ c28 n
n
−(m+1)/2
n
n
n
||ϕ||M .
n
n
n
n
p,∞
where c28 ≡ c28 (t, m, p) > 0 is a constant independent of ϕ and r. The claim follows.
Finally let us define now the measures,
ρ̄nt (ϕ) = Ẽ R̄0t R̄1t ...R̄nt ϕ (X0 ) Yt for ϕ ∈ Cb Rd .
n
n
n
and let π̄nt = ρ̄nt /ρ̄nt (1) be its normalized version.
23
"
Theorem 13 Let X̄ (x) be an m-perfect family that satisfies (36). Then there is a constant c29 ≡ c29 (t, m, p) > 0 such that for any ϕ ∈ Cb∞ (Rd ) we have,
||ρnt (ϕ) − ρ̄nt (ϕ)||p ≤ c29 n−(m−1)/2 ||ϕ||M .
Proof. We have for p ≥ 1,
||ρnt (ϕ)
−
ρ̄nt
(ϕ)||p
hence the result.
p 1
!
"
p
0 1
n
0 1
n
≤ Ẽ Ẽ R t R t ...R t ϕ − R̄ t R̄ t ...R̄ t ϕ (X0 ) Yt
n
n
n
n
n
n
0 1
n
0 1
n ≤ R t R t ...R t ϕ − R̄ t R̄ t ...R̄ t ϕ ,
n
n
n
n
n
n
p,∞
Corollary 14 Let X̄ (x) be an m-perfect family that satisfies (36) with m ≥ 3 and assume
that X0 has all moments finite. Then for any ϕ ∈ Cb∞ (Rd ) there is a constant c30 ≡
c30 (t, m, p, ϕ) > 0 such that,
Proof. Since,
||π nt (ϕ) − π̄ nt (ϕ)||p ≤
|π t (ϕ) − π̄ nt (ϕ)| ≤
c30 (ϕ)
.
n
1
||ϕ||
|ρt (ϕ) − ρ̄nt (ϕ)| +
|ρ (1) − ρ̄nt (1)|
ρt (1)
ρt (1) t
the Corollary follows by applying Theorem 13 and the fact that ||ρt (1)−1 ||p is finite for
any p ≥ 0.
5
Appendix
The main aim of the Appendix is to deduce inequality (41). In the following we will adopt
the framework of Kusuoka-Stroock[23]. Let (Θ, B, W) be the standard Wiener space with
continuous paths θ : [0, ∞) → Rd satisfying θ(0) = 0. Then Θ with the topology of
uniform convergence on compact intervals is a Polish space. Also let H ⊂ Θ be the Hilbert
∞
space of absolutely continuous functions h ∈ Θ such that hH = ( 0 |h′ (t)|2 dt)1/2 < ∞.
Let W 1 (R) denote the space of measurable Φ : Θ → R with the following two properties:
(i) For all h ∈ H, there exists a measurable function Φh : Θ → R such that Φh =
Φ, W − a.s. and t ∈ R → Φh(θ + th) ∈ R is strictly absolutely continuous for all θ.
(ii) There exist a measurable map, DΦ : Θ → L(H; R) such that, for all h ∈ H and ε > 0,
Φ(θ + th) − Φ(t)
limW θ : − DΦ(θ)(h) ≥ ε = 0.
|t|↓0
t
24
On W 1 (R) the norm ·q;R is defined as follows,
(n)
(n) Φq;R := ΦR (n)
Lq (W)
where ΦR =
(n)
n
,
D m ΦHm (R) for q ∈ [2, ∞) . Note that D 0 Φ = Φ with D 0 ΦH0 (R) =
m=0
(n)
Φq;R .
|Φ| so E[|Φ|] ≤
Finally let G(L) be the set of all Φ ∈ W 1 (R) to which D and the
Ornstein-Uhlenbeck operator L as defined in [21] can be applied infinitely often.
Definition 15 We say that f ∈ η r (Rd ; R) for r ∈ Z if f is a measurable map from
(0, ∞) × Rd × Ω into R such that,
(i) f (t, ·, ω) : Rd → R is smooth for each t ∈ (0, ∞) and W − a.e ω ∈ Ω.
(ii) f (·, x, ·) : (0, ∞) × Ω → R is progressively measurable for each x ∈ Rd .
∂α
d
(iii) ∂x
α f (t, x, ·) ∈ G(L) and is continuous in t ∈ (0, ∞) for any α ∈ A1 , x ∈ R
(k)
1 ∂α
(iv) sup sup tr/2
f (t, x, ω)q;R < ∞ for any α ∈ A1 , k ∈ N, T > 0 and 2 ≤ q < ∞.
∂xα
0<t≤T x∈Rd
For a proof of the following Theorem see Kusuoka-Stroock [23], Theorem 2.15 (p. 405).
Theorem 16 (Kusuoka-Stroock) Under the UFG condition, for any Φ ∈ η r (Rd ; R), r ∈ Z
and α ∈ A1 there exists Φ̄α ∈ η r−α (Rd ; R) such that,
V[α] E[Φ(t, x)f (Xt (x))] = E[Φ̄α (t, x)f (Xt (x))]
(39)
for any f ∈ Cb∞ (Rd ), t > 0, x ∈ Rd .
The following immediate extension of Theorem 16 holds under the additional condition
V0.
Lemma 17 Under the UFG+V0 condition, for any Φ ∈ η r (Rd ; R) there exists Φ̄α ∈
η r−2 (Rd ; R) such that,
V0 E[Φ(t, x)f (Xt (x))] = E[Φ̄0 (t, x)f (Xt (x))]
for any f ∈ Cb∞ (Rd ), t > 0, x ∈ Rd .
(40)
Proof. From Theorem 16 we get that for any Φ ∈ η r (Rd ; R), r ∈ Z and β ∈ A1 (2)
there exists Φ̄β ∈ η r−β (Rd ; R) such that,
V[β] E[Φ(t, x)f(Xt (x))] = E[Φ̄β (t, x)f (Xt (x))]
25
for any f ∈ Cb∞(Rd ), t > 0, x ∈ Rd . Hence,
V0 E[Φ(t, x)f (Xt (x))] =
where Φ̄0 (t, x) =
β∈A1 (2) ϕβ Φ̄β (t, x)
β∈A1 (2)
ϕβ V[β] E[Φ(t, x)f (Xt (x))]
= E[Φ̄0 (t, x)f (Xt (x))]
∈ η r−2 (Rd ; R).
The following Corollary is a generalization of Corollary 2.19 (p.407) in Kusuoka-Stroock[23].
Corollary 18 For any r ∈ Z+ , {αi ∈ A}ri=1 and α1 ∗ . . . ∗ αr , there exists a constant
CpT > 0 such that,
V[α1] . . . V[αr ] Pt ϕ ≤ CpT t(p−α1∗...∗αr )/2 ϕ ,
t ∈ [0, T ] .
(41)
p
∞
for ϕ ∈ Cb∞ (Rd ) where ·p for p ∈ N is defined in (4).
Proof.
For the case p = 1, let Y (x) be the matrix valued process Yti,j (x) := ∂x∂ j (X i (x)) where
i, j = 1, ..., d. (see Ikeda & Watanabe[13], Chapter V, for details). Then for any i, j =
1, ..., d we have Y i,j (t, x) ∈ η 0 (Rd ; R), in particular,
d
sup sup E[
t∈[0,T ]x∈Rd
i=1
Y i,j (x) ≤ c(T ).
Differentiating under the integral sign (see Friedman[10], p.122 (5.17)) gives,
d
∂
∂ϕ
i,j
E[ϕ(Xt (x))] =
E
(Xt (x))Yt (x) .
∂xj
∂x
i
i=1
(42)
(43)
Hence,
d
∂ϕ
i
(Xt (x))Φ (t, x)
V[αr ] Pt ϕ (x) =
E
∂x
i
i=1
with Φi (t, x) = dj=1 V[αj r ] (x) Yti,j (x) ∈ η 0 (Rd ; R) for i = 1, ..., d. Then by Theorem 16
& Lemma 17 there exists Φ̄iα ∈ η −(α1 ∗···∗αr−1 ) (Rd ; R), i = 1, ..., d such that, for any
ϕ ∈ Cb1 (Rd ),
d
∂ϕ
i
V[α1 ] . . . V[αr−1 ] V[αr ] Pt ϕ (x) =
V[α1] . . . V[αr−1 ] E
(Xt (x))Φ (t, x)
∂x
i
i=1
d
∂ϕ
i
=
E Φ̄α (t, x)
(Xt (x))
∂x
i
i=1
26
for any ϕ ∈ Cb∞ (Rd ). Hence,
d
V[α1 ] . . . V[αr ] Pt ϕ (x) ≤ ||ϕ||
E[|Φ̄iα (t, x)|]
1
i=1
and (41) follows from the uniform bound (in (t, x)) on the L1 -norm of Φ̄iα(t, x), i = 1, ..., d.
For p > 1 one can obtain an analogue of (43) with higher derivatives of Pt ϕ in terms
of derivatives of ϕ and a set of processes analogous to Y . The proof follows in a similar
manner.
Acknowledgments The authors wish to thank the anonymous referee for his comments.
We would also like to thank Alan Bain for carefully reading the paper and his many
pertinent remarks.
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