Jurgen Dippon · Daniel Schiemert

Stochastic Differential Equations Driven byGaussian Processes with Dependent Increments

Abstract A class of Gaussian processes, which includes fractional Brownian motion, is introducedand derivated in a distributional sense. This derivative is used to define a stochastic integral drivenby a Gaussian process with dependent increments. Several properties of this integral are proved. Achain rule for stochastic processes, which includes Ito’s rule, is shown and the existence and uniquenessof solutions of different types of stochastic differential equations driven by Gaussian processes withdependent increments are obtained.

Keywords Gaussian Process · Stochastic Integral Calculus · Wick Product

Mathematics Subject Classification (2000) 60H05 · 60H10 · 60H40

1 Introduction

Since fractional Brownian (fBM) motion has dependent increments, it is an interesting generalizationof ordinary Brownian motion to model the noise process in many applications such as finance (e.g.[Be], [HuOk], [HuOkSa], [So]), network simulations (e.g. [No]), and environmental processes. How-ever, covariance function of fBM depends only on a single parameter, the Hurst parameter. Hence,in comparison to Brownian motion, there is still not much more freedom in adapting to real worldproblems showing different covariance properties. For example, one cannot model a short range de-pendency with fBM. Therefore the authors considered a more general class of Gaussian processes withdependent increments containing fBM.

In the context of stochastic differential equations (SDE) it is desirable that a stochastic integral,driven by fBM, has expectation value 0. If the integral is defined by the use of the Wick product([Be], [HuOk]) the expectation value of the stochastic integral driven by fBM is zero. For this approachwhite noise distribution theory was employed ([HiKuPoSt], [Ku]). This theory offers the possibility toderivate the fBM in the Hida distribution sense, and has a lot of advantages in the treatment of theWick product. Therewith one can set up the stochastic differential equations driven by fBM, and solvethe bilinear SDE (e.g. [HuOk]).

Jurgen Dippon (corresponding author)Department of MathematicsUniversitat StuttgartPfaffenwaldring 5770569 Stuttgart GermanyE-mail: dippon@mathematik.uni-stuttgart.de

Daniel SchiemertDepartment of MathematicsUniversitat StuttgartPfaffenwaldring 5770569 Stuttgart GermanyE-mail: schiemert@mathematik.uni-stuttgart.de

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The authors of the present paper adopted this approach to set up a theory for stochastic differ-ential equations driven by Gaussian processes with dependent increments. In the next section theconstruction of the white noise space, which plays the role of the probability space here, is revisited. Inthe third section the class of Gaussian processes with dependent increments considered in this paperis introduced. In the fourth section the construction of the Hida test and distribution space, whichare suitable to derivate the Gaussian processes, is summarized. The necessary results of white noisedistribution theory including Wick product and S-transform are the content of the fifth section. Inthe sixth section the stochastic integral driven by a Gaussian process with dependent increments isdefined, the related bilinear SDE is solved, and some applications are given. In the seventh section thestochastic integral driven by a even more general stochastic process is discussed. Several versions ofa chain rule, which includes Ito’s rule in the special case of the ordinary Brownian motion, is provedin the eighth section. Existence and uniqueness of the solution of several types of SDEs, driven byGaussian processes with dependent increments, are shown in last section.

2 Construction of the white noise space

Let | · |0 denote the norm and 〈·, ·〉 the inner product of L2(R). If η ∈ C∞(R) and for all nonnegativeintegers n, k

|ukη(n)(u)| → 0 as |u| → ∞,

then η is called rapidly decreasing. The set S(R) denotes the set of rapidly decreasing functions andis called Schwartz space. Let S ′(R) denote the dual space of S(R), the space of tempered distributions.As a result of Hilbert space theory the inner product of L2(R) can be extended to the bilinear form〈ω, η〉, where ω ∈ S ′(R) and η ∈ S(R). There is another construction of the Schwartz space S(R) andits dual S ′(R) by the use of the operator

A := − d2

dx2+ x2 + 1

which is densely defined on L2(R). With the Hermite polynomial of degree n

Hn(x) := (−1)nex2(

d

dx

)n

e−x2

leten(x) :=

1√√π2nn!

Hn(x)e−x2

2 .

The functions en are the eigenfunctions of A corresponding to the eigenvalues λn := 2n + 2, n ∈ N0.The operator A−1 is bounded on L2(R), especially A−p is a Hilbert-Schmidt operator for any p > 1

2 .Let for each p ≥ 0, |f |p := |Apf |0. The norm is given by use of the eigenvalues as

|f |p =

( ∞∑n=0

(2n + 2)2p 〈f, en〉2)1/2

.

DefineSp(R) := {f : f ∈ L2(R), |f |p < ∞}.

The Schwartz space S(R) can be defined by S(R) := ∩p≥0Sp(R), too. Let S ′p(R) with the norm | · |−p

denote the dual of Sp(R). It follows that S ′(R) = ∪p≥0S ′p(R). By the properties of A it is ensured thatS(R) is a nuclear space with a locally convex topology. In the weak topology on S ′(R) the sequence{ωk}k ⊂ S ′(R) converges if and only if the sequence {〈ωk, η〉}k converges for all η ∈ S(R). Thereforethe Gel’fand triple S(R) ⊂ L2(R) ⊂ S ′(R) is declared and the continuous mappings hold

S(R) ⊂ Sp(R) ⊂ L2(R) ⊂ S ′p(R) ⊂ S ′(R).

Let B denote the Borel σ-algebra on S ′(R), i.e. the σ-algebra generated by the weak topology ofS ′(R).

3

Theorem 1 (Minlos theorem) A complex-valued function ζ on S(R) is the characteristic functionof a unique probability measure µ on (S ′(R),B), i.e.

ζ(η) =∫S′(R)

exp(i〈ω, η〉) dµ(ω)

with η ∈ S(R) if and only if the three conditions (1) ζ(0) = 1, (2) ζ is continuous and (3) ζ is positivedefinite are satisfied.

Further details can be found in the appendix of [HoOkUbZh]. By using Minlos theorem the followingresult is obtained.

Theorem 2 (Gaussian measure on S ′(R)) There exists a unique Gaussian measure µ on (S ′(R),B)whose characteristic function is

ζ(η) := exp(−|η|

20

2

)with η ∈ S(R).

The triple (S ′(R),B, µ) is called white noise space. Furthermore, (L2) denotes L2(S ′(R),B, µ) equippedwith norm ‖ · ‖0 and inner product 〈〈·, ·〉〉.

Lemma 1 The random variable 〈·, η〉 is in (L2) for all η ∈ S(R), where∫S′(R)

〈ω, η〉 dµ(ω) = 0

and ∫S′(R)

〈ω, η〉2 dµ(ω) = |η|20.

Proof The statements are proved by expanding both sides in∫S′(R)

exp(i〈ω, tη〉) dµ(ω) = exp(−|tη|

20

2

).

and comparison of the coefficients of t, t ∈ R. �

Lemma 2 For all f ∈ L2(R) the bilinear form 〈ω, f〉, ω ∈ S ′(R), is declared by the limit

limk→∞

〈·, ηk〉 = 〈·, f〉

with convergence in (L2), where ηk → f in L2(R) and {ηk} ⊂ S(R). Furthermore, E(〈·, f) = 0 andE(〈·, f〉2) = |f |20.

The lemma is an immediate consequence of lemma 1.

3 A class of Gaussian processes with dependent increments

This section introduces a class of Gaussian processes with dependent increments Bvt . The properties

of Bvt are based on a function m(·, ·).

Suppose m(·, ·) : R2 → R such that, for all t ∈ R, the function m(·, t) ∈ L2(R). Define

v(s, t) =∫

Rm(u, t)m(u, s) du.

Definition 1 The stochastic process Bvt is defined by

Bvt := 〈·,m(u, t)〉.

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Remark 1 (Notation) The notation 〈·,m(u, t)〉 is meant in the following way. The · stands for anelement ω ∈ S ′(R), such that it is a tempered distribution with respect to the variable u. Hence〈ω(u),m(u, t)〉 depends on the parameter t only and represents, for fixed ω, a path of the stochasticprocess Bv

t .

In the following it will be assumed that the function m(u, t) is differentiable in the sense of tempereddistributions with respect to t, i.e.

d m(·, t)dt

∈ S ′(R)

for all t ∈ R.

Proposition 1 The stochastic process Bvt is a Gaussian process with mean 0 and covariance function∫

S′(R)

Bvt (ω)Bv

s (ω) dµ(ω) = v(t, s)

and

Bvt =

∫R

m(u, t) dBu

with an ordinary Brownian motion Bu on the real line.

Proof It follows from Lemma 2 that the process Bvt has mean 0 and that E((Bv

t )2) = |m(·, t)|20. Theprocess Bv

t is therefore normally distributed. The Wiener integral∫

R m(u, t)dBu is normally distributedwith the same mean and variance as Bv

t , consequently

Bvt =

∫R

m(u, t) dBu

holds a.s. and for all t ∈ R. The covariance function of∫

R m(u, t) dBu is∫S′(R)

∫R

m(u, t) dBu

∫R

m(u, s) dBu dµ =∫

Rm(u, t)m(u, s) du = v(t, s) (1)

by use of the Ito isometry. �

In the following there are some special instances of Bvt .

Example 1 (Ordinary Brownian motion) The stochastic process Bvt = Bt is the ordinary Brownian

motion if m(u, t) = 1[0,t](u), where 1[0,t] is the indicator function of the intervall [0, t]. Then this processhas the covariance function v(s, t) = min(t, s). This example is further discussed in [Ku], Chapter 3.1.

Example 2 (Fractional Brownian motion) The following operator MH± is used to introduce fractional

Brownian motion and later on its derivative. Define MH± with H ∈ (0, 1) for η ∈ S(R) as

(MH+ η)(t) :=

KH

Γ (H−1/2)

∫ t

−∞ η(s)(t− s)H−3/2 ds for H > 1/2η(t) for H = 1/2KH(H−1/2)Γ (1/2−H) lim

ε→0

∫∞ε

η(t)−η(t−s)(s)3/2−H ds for H < 1/2

and

(MH− η)(t) :=

KH

Γ (H−1/2)

∫ t

−∞ η(s)(t− s)H−3/2 ds for H > 1/2η(t) for H = 1/2KH(H−1/2)Γ (1/2−H) lim

ε→0

∫∞ε

η(t)−η(t+s)(s)3/2−H ds for H < 1/2

with

KH = Γ (H + 1/2)(

2HΓ (3/2−H)Γ (H + 1/2)Γ (2− 2H)

)1/2

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The operator MH± is essentially the Riemann-Liouville fractional integral for H > 1/2 and the Mar-

chaud fractional derivative for H < 1/2. For further information about these operators see [SaKiMa],Chapter 6, and [Be], Chapter 1.6. The operator MH

± is just applied on functions η ∈ S(R), and forindicator functions 1[0,t] for which MH

± is defined; see Chapter 1 of [Be]. For the fractional Brownianmotion BH

t it holds that

BHt =

∫R(MH

− 1(0,t))(s) dBs

up to modification. Another representation of the fractional Brownian motion up to modification isBH

t = 〈·,MH− (1(0,t))〉. The covariance function is therefore v(s, t) = 1/2(|t|2H + |s|2H − |s− t|2H). This

example is further discussed in [Be].

Remark 2 Note that the fractional Brownian motion is not a semimartingale for H 6= 1/2, so the classof Gaussian processes Bv

t has elements to which the semimartingale calculus does not suit.

Example 3 (A Gaussian process with short range dependency) Let Bvt , t ∈ R, be a centered Gaussian

process with covariance function v(s, t), such that v(s, ·) has a global maximum and limt→∞ v(s, t) = 0for all fixed s ∈ R. Then Bv

t is said to be a short range Brownian motion. Let m(u, t) = t2 exp(−(u−t)2),hence v(s, t) =

√πt2s2 exp(−(t− s)2/2). This process Bv

t is a short range Brownian motion.

Our definition of Bvt was formulated in terms of a function m(u, t). Then the covariance function

v(t, s) was computed from m(u, t). However, as the following proposition shows, it is also under certaincircumstances possible to derive m(u, t) from a given covariance function v(t, s).

Proposition 2 Let v(·, ·) : R2 → R, such that

v(t, s) =∞∑

k=1

αk(t)αk(s) < ∞

for all t, s ∈ R with real-valued functions αk(·). Let further {φk} be an orthonormal base of L2(R).Then

m(u, t) =∞∑

k=1

αk(t)φk(u)

with convergence in L2(R) with respect to u and pointwise in t.

Proof The function m(u, t) satisfies

∫R

m(u, t)m(u, s) du =∞∑

j,k=1

αk(t)αj(s)∫

Rφj(u)φk(u) du

=∞∑

k=1

αk(t)αk(s) = v(t, s).

�

Note that the function m(u, t) is with respect to u not unique in L2(R), it depends on the choice ofthe orthonormal base {φk}. This is no further problem, because the centered Gaussian process Bv

t isuniquely determined by its covariance function v(t, s).

Remark 3 The Gaussian process Bvt is stationary if v(t, s) depends only on t − s. E.g., let m(u, t) =

exp(−(u− t)2) so v(t, s) =√

π exp(−(t− s)2/2).

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4 Construction of the Hida test and distribution space

In this section we sketch the chaos decomposition theorem and the related calculus, which is used todefine the Hida test and distribution space.

Definition 2 The trace operator τ ∈ (S ′(R))⊗2 is given by

〈τ, η1 ⊗ η2〉 := 〈η1, η2〉

for η1, η2 ∈ S(R).

Definition 3 The Wick tensor : ω⊗n : of an ω ∈ S ′(R) for n ∈ N0 is defined by

: ω⊗n : :=n/2∑k=0

(n

2k

)(2k − 1)!!(−1)kω⊗(n−2k)⊗τ⊗k, (2)

where the upper limit of the summation index is the integer part of n/2, ⊗ denotes the symmetrictensor product, and (2k − 1)!! = (2k − 1) · (2k − 3) · · · 3 · 1 with (−1)!! = 1.

Thus the wick tensor : ω⊗k : is an element of (S ′(R))⊗k. Formula (2) can be transformed to

ω⊗n :=n/2∑k=0

(n

2k

)(2k − 1)!! : ω⊗(n−2k) : ⊗τ⊗k,

as can be seen in [Ku], Chapter 5.1.

Lemma 3 For all ω ∈ S ′(R) and for all f ∈ L2(R) it is true that

⟨: ω⊗n :, f⊗n

⟩=

n/2∑k=0

(n

2k

)(2k − 1)!!(−|f |20)k〈ω, f〉n−2k (3)

and‖〈: ·⊗n :, f⊗n〉‖0 =

√n!|f |n0 . (4)

The proof can be obtained by using equation (2), see [Ku], Chapter 5.1. For f ∈ L2(Rn) let the multipleWiener Integral with respect to the ordinary Brownian motion

In(f) :=∫

Rn

f(t1, t2, t3, . . . , tn) dBt1dBt2 . . . dBtn

with n ∈ N0 and I0 denotes a constant c. A function f : Rn → C with n ∈ N is said to be sym-metric, if the function f is invariant under permutation of the variables. The function f denotes thesymmetrization of the function f and is defined by

f(t1, . . . , tn) :=1n!

∑σ

f(tσ(1), . . . , tσ(m)),

where σ runs over all permutations of {1, . . . , n}. Let L2(Rn) be the space of symmetric complex-valuedL2 functions on Rn.

Proposition 3 If f ∈ L2(Rn), then it holds a.s. that

In(f)(ω) = 〈: ω⊗n :, f〉.

For a proof see Theorem 5.4 in [Ku].The following outline of the construction of the Hida test and distribution space is taken essentially

from [Ku] Chapter 3.3. The next proposition is known as the chaos decompostion of (L2). Here, witha slight abuse of the notation, the | · |0 denotes the norm in L2(Rn), too.

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Proposition 4 For all F ∈ (L2) there is a unique sequence (fn)n∈N0 such that fn ∈ L2(Rn) issymmetric and

F =∞∑

n=0

⟨: ·⊗n :, fn

⟩with convergence in (L2). Furthermore, it holds that

E(F 2) =∞∑

n=0

n!∫

Rn

f2n(s) ds =

∞∑n=0

n!|fn|20.

Proposition 3 allows another representation of the chaos decomposition by use of the multiple Wienerintegrals, which can be found, e.g., in [Nu], Theorem 1.1.2.

Consider the operator A = −d2/dx2 + x2 + 1 already mentioned above. Let Γ (A) be an operatoron F =

∑∞n=0 〈: ·⊗n :, fn〉 ∈ (L2) satisfying

∞∑n=0

n!|A⊗nfn|20 < ∞

and defined by

Γ (A)F =∞∑

n=0

⟨: ·⊗n :, A⊗nfn

⟩.

The operator Γ (A) has similar properties as the operator A, e.g., Γ (A) has a set of eigenfunctionswhich forms an orthonormal basis for (L2), Γ (A)−1 is a bounded operator of (L2) and for any p > 1the operator Γ (A)−p is a Hilbert-Schmidt operator of (L2), see [Ku], Chapter 3.3 . Let the space (S)p

be for p ≥ 0(S)p := {φ ∈ (L2) :‖ Γ (A)pφ ‖0< ∞}.

The space (S)n is a Hilbert space with norm ‖ · ‖n, too. As in the construction of the Schwartz spaceS(R) one can define a space (S) as

(S) := ∩p≥0(S)p.

The space (S) is therefore a nuclear space, (S) ⊂ (L2), and it is called Hida test function space. Thedual space (S)∗ of (S) is called Hida distribution space. With (S)′n = (S)−n, it holds that

(S)∗ = ∪n∈N(S)−n.

Let ‖ · ‖−n denote the norm of (S)−n. The topologies of (S) and of (S)∗ are given by the projectivelimit topology and weak topology, respectively (see [Ku], Chapter 2.2). These spaces form a Gel’fandtriple

(S) ⊂ (L2) ⊂ (S)∗.

The bilinear form 〈〈·, ·〉〉 denotes the inner product of (L2), which is extended to the dual mapping〈〈Φ, ζ〉〉 with Φ ∈ (S)∗ and ζ ∈ (S). So a sequence {Φk} converges in (S)∗ if and only if 〈〈Φk, ζ〉〉converges for all ζ ∈ (S). Similar as above there are continuous inclusion maps

(S) ⊂ (S)n ⊂ (L2) ⊂ (S)−n ⊂ (S)∗.

The symmetric tensor product f⊗g of two given functions f, g is the symmetrization of f ⊗ g.

Remark 4 If η ∈ S(R), then it follows from the construction of (S) that 〈·, η〉 ∈ (S). Furthermore, in[Ku], Theorem 6.5, it is shown, that all functions fn in the chaos decomposition of an element

φ =∞∑

n=0

⟨: ·⊗n :, fn

⟩, (5)

φ ∈ (S), are smooth. Thus it is possible to define a Hida distribution Φ by the use of the deltadistribution δ0 ∈ S ′(R) such that⟨⟨

Φ,

∞∑n=0

⟨: ·⊗n :, fn

⟩⟩⟩= 〈δ0, f1〉 = f1(0). (6)

8

The notation that will be used for Φ is 〈·, δ0〉, which is declared by equation (6). The expression〈·, δ0〉 is therefore a Hida distribution and is a mapping from (S) → C. If φ is given by (5), then〈·, δ0〉(φ) = 〈δ0, f1〉 as before. Thus we have an additional extension of the expression 〈·, λ〉 withλ ∈ S′(R) which is a Hida distribution declared by 〈·, λ〉(φ) = 〈λ, f1〉.

Proposition 5 Suppose φ ∈ (S),

φ =∞∑

k=0

⟨: ·⊗k :, ηk

⟩with ηk ∈ (S(R))⊗k and let Ψ ∈ (S)∗ such that

Ψ =∞∑

k=0

⟨: ·⊗k :, Fk

⟩with Fk ∈ (S ′(R))⊗k. Then it holds that

〈〈Ψ, φ〉〉 =∞∑

k=0

k! 〈Fk, ηk〉 .

The proof is in [Ku] Chapter 5.1.

5 S-transform and Wick product

Due to the construction of the Hida test function space (S) it follows that for any η ∈ S(R)

exp�(〈·, η〉) :=∞∑

k=0

1k!⟨: ·⊗k :, η⊗k

⟩is an element of (S). Therefore it is possible to define any Hida distribution Φ on this random variablesexp�(〈·, η〉) in the bilinear mapping:

Definition 4 The S-transform of a Hida distribution Φ ∈ (S)∗ is defined by

S(Φ)(η) := 〈〈Φ, exp�(〈·, η〉)〉〉 , η ∈ S(R).

Proposition 6 Let Φ, Ψ ∈ (S)∗ and Φ =∑∞

k=0

⟨: ·⊗k :, Fk

⟩with

Fk ∈ (S ′(R))⊗k, then

S(Φ)(η) =∞∑

k=0

⟨Fk, η⊗k

⟩.

If S(Φ)(η) = S(Ψ)(η) for all η ∈ S(R), then Φ = Ψ and the set

{exp�(〈·, η〉) : η ∈ S(R)}

is total in (S).

The proof is presented in [Ku] Chapter 5 in the proofs of several Theorems.

Example 4 The S-transform of the ordinary Brownian motion Bt is

S(Bt)(η) = 〈〈Bt, exp�(〈·, η〉)〉〉=⟨⟨〈·, 1[0,t]〉, exp�(〈·, η〉)

⟩⟩= 〈1[0,t], η〉 =

∫R

1[0,t](u)η(u) du =∫ t

0

η(u) du

9

Example 5 We calculate the S-transform of the fractional Brownian motion BHt .

S(BHt )(η) =

⟨⟨BH

t , exp�(〈·, η〉)⟩⟩

=∫S′(R)

BHt exp�(〈·, η〉) dµ

=∫S′(R)

∫R(MH

− 1(0,t))(s) dBs exp�(〈·, η〉) dµ

=∫ t

0

(MH+ η)(s) ds

where several steps like fractional integration by parts and the fact that the integrals are well-definedare used (see [Be], Theorem 1.6.8).

Example 6 For the short range Brownian motion Bst we get

S(Bst )(η) = 〈〈Bs

t , exp�(〈·, η〉)〉〉=⟨⟨〈·, t2 exp(−(u− t)2)〉, exp�(〈·, η〉)

⟩⟩=∫

Rη(u)t2 exp(−(u− t)2) du.

Definition 5 The Wick product Φ � Ψ of two Hida distributions Φ and Ψ in (S)∗ is the Hida distri-bution in (S)∗ such that S(Φ � Ψ)(η) = S(Φ)(η)S(Ψ)(η) for all η ∈ S(R).

In [Ku] it is proved in Chapter 8 that the Wick product Φ�Ψ of any two Hida distributions Φ and Ψexists and is unique. The following algebraic properties of the Wick product are obvious by regardingtheir S-transform Let Φ, Ψ, Γ ∈ (S)∗, then

Φ � Ψ = Ψ � Φ

Γ � (Φ � Ψ) = (Γ � Φ) � Ψ

Γ � (Φ + Ψ) = Γ � Φ + Γ � Ψ.

For F,G ∈ (L2) it may happen that (F �G) /∈ (L2), see [Be], Corollary 1.5.6. In general it holds that

T � (G · F ) 6= (T �G) · F

with T,G, F ∈ (L2). Let F,G ∈ (L2) with F �G ∈ (L2). Then the S-transform of F with η ≡ 0 is theexpectation value of F , 〈〈F, 1〉〉 = E(F ), hence

E(F �G) = E(F )E(G).

Proposition 7 Let 〈: ·⊗n :, fn〉 and 〈: ·⊗m :, gm〉 be Hida distributions with fn ∈ (S ′(R))⊗n, gm ∈(S ′(R))⊗n, n, m ∈ N0. Then⟨

: ·⊗n :, fn

⟩�⟨: ·⊗m :, gm

⟩=⟨: ·⊗(n+m) :, fn⊗gm

⟩.

See [Ku], Chapter 8, for a proof.

Example 7 Let f ∈ L2(R), then〈·, f〉�2 =

⟨: ·⊗2 :, f⊗2

⟩By use of equation (2) it follows that⟨

: ·⊗2 :, f⊗2⟩

= 〈·, f〉2 − |f |20. (7)

Note that the right hand side is only well-defined if f ∈ L2(R), because | · |0 is not declared for anarbitrary element of S ′(R).

Generalising this example the next lemma follows.

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Lemma 4 Let f ∈ L2(R), then the following equations hold with n ∈ N0

〈·, f〉�n =n/2∑k=0

(n

2k

)(2k − 1)!!(−|f |20)k〈·, f〉n−2k

and

〈·, f〉n =n/2∑k=0

(n

2k

)(2k − 1)!!(|f |20)k〈·, f〉�(n−2k)

and

exp(〈·, f〉 − 1/2|f |20) =∞∑

k=0

〈·, f〉�k

k!= exp�(〈·, f〉).

Proof The proof is obtained by equation (3) and Proposition 7. �

The term exp�(〈·, f〉) is called the Wick exponential of 〈·, f〉.The following result appears as Theorem 8.2 in [Ku].

Theorem 3 (Characterization of Hida distributions ) Let Φ ∈ (S)∗. Then its S-transformF = S(Φ) satisfies the conditions

– For any η1 and η2 in S(R), the function F (η1 + zη2) is an entire function of z ∈ C.– There exists nonnegative constanst K, a and p such that

|F (η)| ≤ K exp(a|η|2p) (8)

for all η ∈ S(R).

Conversely, suppose a function F defined on S(R) satisfies the two conditions above. Then there existsa unique Φ ∈ (S)∗ such that F = S(Φ).

The following example shows that the conditions of Theorem 3 may fail.

Example 8 Let h ∈ L2(R). Then exp�(〈·, h〉�k) /∈ (S)∗ for k > 2. Calculate the S-transform and get

S

( ∞∑n=0

(〈·, h〉)�(kn)

n!

)(η) =

∞∑n=0

(〈h, η〉k)n

n!= exp(〈h, η〉k).

Choose η = h, then condition (8) fails.

Theorem 4 Let {Φn} be a sequence in (S)∗ and let Fn = S(Φn). Then Φn converges in the weaktopology of (S)∗ if and only if the following conditions are satisfied:

– limn→∞ Fn(η) exists for each η ∈ S(R).– There exist nonnegative constants K, a and p, indepedent of n such that

|Fn(η)| ≤ K exp(a|η|2p),

for all n ∈ N and for all η ∈ S(R).

This theorem can be proved by Theorem 3 as shown in [Ku], Theorem 8.6.

11

6 Stochastic integrals driven by Gaussian processes with dependent increments

Definition 6 Let I be an interval in R. A mapping X : I → (S)∗ is called a stochastic distributionprocess. A stochastic distribution process is called differentiable in the (S)∗-sense, if

limh→0

Xt+h −Xt

h

exists in (S)∗.

The next theorem presents a criterion for differentiability (see [Be], Theorem 5.3.9). Let I be asabove. A function F : I → S ′(R) is differentiable, if the limit (F (t + h)− F (t))/h exists in S ′(R) withrespect to the weak topology of S ′(R).

Theorem 5 Let F : I → S ′(R) be differentiable. Then 〈·, F (t)〉 is a differentiable stochastic distribu-tion process and

d

dt〈·, F (t)〉 =

⟨·, dF (t)

dt

⟩.

Proof Let {hn} be a sequence that tends to zero. It is true that⟨F (t + hn)− F (t)

hn, η

⟩=〈F (t + hn), η〉 − 〈F (t), η〉

hn

=S(〈·, F (t + h)〉)(η)− S(〈·, F (t)〉)(η)

hn

for each η ∈ S(R), where the second equality follows from Proposition 6 considering the single termwith k = 1. Letting n →∞ the assertion is proved. �

Theorem 6 (Derivative of Bvt ) The stochastic distribution process Bv

t is differentiable in the (S)∗-sense and satisfies

W vt :=

d

dtBv

t =⟨·, d

dtm(u, t)

⟩;

its derivative W vt is called white noise of Bv

t . The S-transform of W vt equals

S(W vt )(η) =

⟨d

dtm(u, t), η(u)

⟩.

Proof The proof is given by Theorem 5. �

Example 9 (Ordinary Brownian motion) To compute the derivative of the ordinary Brownian motionBt, it will be used that d

dt1[0,t] = δt in S ′(R), hence

d

dt

⟨⟨〈·, 1[0,t]〉, exp(〈·, η〉 − 1/2|η|20)

⟩⟩=

d

dt〈1[0,t](u), η〉

=d

dt

∫ t

0

η(u) du = η(u) = 〈δt, η〉

=⟨⟨〈·, δt〉, exp(〈·, η〉 − 1/2|η|20)

⟩⟩.

Therefore dBt/dt =: Wt = 〈·, δt〉 and S(Wt)(η) = η(t) (see [Ku], Chapter 3.1).

Example 10 (Fractional Brownian motion) For H ∈ (0, 1) the fractional Brownian motion BH· : R →

(S)∗ is differentiable in the (S)∗-sense and

WHt :=

d

dtBH

t = 〈·, δt ◦MH+ 〉

and for all η ∈ S(R)〈δt ◦MH

+ , η〉 = (MH+ η)(t).

12

The S-transform of WHt is given by

S(WHt )(η) =

⟨⟨d

dtBH

t , exp(〈·, η〉 − 1/2|η|20)⟩⟩

=d

dtS(BH

t )(η) = (MH+ η)(t).

This example is presented in [Be], Chapter 5.

Example 11 (Short range Brownian motion) The derivative of Bst is obviously given by⟨

·, d

dtt2 exp(−(t− u)2)

⟩= 〈·, 2t exp(−(t− u)2) + t2(−2(u− t)) exp(−(u− t)2)〉.

Definition and properties of the white noise integral will be sketched only. A more detailled discus-sion can be found in [Ku], Chapter 13. Suppose that Xt is a mapping from R → (S)∗.

Definition 7 The stochastic distribution process Xt is white noise integrable, if there is a Ξ ∈ (S)∗such that, for all η ∈ S(R), (SX·)(η) ∈ L1(R) and

(SΞ)(η) =∫

R(SXt)(η) dt.

This definition makes sense as the S-transform is injective.

Definition 8 Let the stochastic distribution process Xt be such that Xt�W vt is white noise integrable,

then Xt is integrable with respect to Bvt , and

∫R Xt dBv

t is defined by∫R

Xt dBvt :=

∫R

Xt �W vt dt.

This definition coincides in the case of a fractional Brownian motion with the definition of the fractionalIto integral (see Bender ([Be]), Øksendal and Hu [HuOk]). Therefore it is a generalization of the Itointegral, too. This is further discussed in [Ku] Chapter, 13.

This definition can be extended to more general integrators:

Definition 9 Let Xt be a differentiable stochastic distribution process. If Yt � (dXt/dt) is white noiseintegrable, then Yt is integrable with respect to the process Xt, and∫

RYt dXt :=

∫R

Yt �dXt

dtdt.

The following theorem is inspired by Bender’s theorem for fractional Ito integrals (see [Be], Chap-ter 5).

Theorem 7 Let a, b ∈ R, and let X· : [a, b] → (S)−p be continuous for some p ∈ N. Further, letW v· : R → (S)−q be continuous for some q ∈ N. Then

∫ b

aXt dBv

t exists. Additionally, for any sequence

of tagged partitions τn =(π

(n)k , t

(n)k

)of [a, b] with limn max{|πk − πk−1| : k = 1, . . . , n} = 0 it follows

that

limn→∞

n∑k=0

Xt(n)k−1

�(

Bv

π(n)k

−Bv

π(n)k−1

)=∫ b

a

Xt dBvt

with limit in (S)∗.

Proof Concerning integrability:

S

(∫ b

a

Xs dBvs

)(η) =

∫ b

a

S(Xs �W vs )(η) ds

=∫ b

a

S(Xs)(η)S(W vs )(η) ds

≤ maxs∈[a,b]

{|S(Xs)(η)|}∫ b

a

S(W vs )(η) ds

≤ K exp(a|η|2p)∫ b

a

S(W vs )(η) ds < ∞

13

By continuity of W vt and∣∣∣∣⟨ d

dtm(u, t)− d

dtm(u, t0), η(u)

⟩∣∣∣∣ ≤ |η(u)∣∣q

∣∣∣∣ d

dtm(u, t)− d

dtm(u, t0)

∣∣∣∣−q

=∣∣η(u)

∣∣q

∥∥W vt −W v

t0

∥∥−q

with t, t0 ∈ [a, b] it follows that⟨

ddtm(u, t), η(u)

⟩is a continuous function in t. So the function

〈m(u, t), η(u)〉 is continuously differentiable on the interval [a, b]. It follows by a well-known resultin real analysis that for all η ∈ S(R) the continuous function S(Xt)(η) is Riemann-Stieltjes integrablewith respect to 〈m(u, t), η(u)〉, thus the approximation is proved. �

The S-transform of∫

R Xt dBvt is therefore given by

S

(∫R

Xt dBvt

)(η) =

∫R

S(Xt �W vt )(η) dt =

∫R(SXt)(η)

⟨d

dtm(u, t), η(u)

⟩dt.

Proposition 8 If∫ b

aXs dBv

s exists and∫ b

aXs dBv

s ∈ (L2), then E(∫ b

aXs dBv

s

)= 0.

Proof Let η ≡ 0, so S(Bvt )(0) is the expectation value of Bv

t and

S

(∫ b

a

XsdBvs

)(0)

=∫ b

a

S(Xs)(0)⟨

dm(u, s)ds

, 0⟩

ds

=∫ b

a

S(Xs)(0)d

ds〈m(u, s), 0〉 ds

= 0,

which proves the claim. �

Remark 5 If the Gaussian process Bvt shall serve as a model for stock prices, the zero expectation

property is highly desirable. Since in this case the stochastic part in the related stochastic differentialequations has expectation value zero, too. Otherwise the stochastic part of the stochastic differentialequation has a drift and that might cause an arbitrage opportunity.

Example 12 Calculate

S

(∫ t

0

Bvs dBv

s

)(η) =

∫ t

0

S(Bvs )(η)S(W v

s )(η) ds

=∫ t

0

S(Bvs )(η)

d

dtS(Bv

t )(η) ds

=12

(S(Bvt )(η))2 =

12S((Bv

t )�2)(η)

where (·)�2 is the Wick square. By equation (7) it follows that

12

(Bvt )�2 =

12

(Bvt )2 − 1

2|m(u, t)|20.

Proposition 9 Let Xt and Yt be differentiable stochastic distribution processes. Then the Wick productrule

d(Xt � Yt)dt

=dXt

dt� Yt + Xt �

dYt

dt

holds in the (S)∗-sense.

14

Proof It holds for the S-transform of Xt � Yt that

d(S(Xt � Yt)(η))dt

=d(S(Xt)(η)S(Yt)(η))

dt

=dS(Xt)(η)

dtS(Yt)(η) + S(Xt)(η)

dS(Yt)(η)dt

.

�

From the proposition above one can derive the formula for partial integration∫R

Yt dXt = Yt �Xt − Y0 �X0 +∫

RXt dYt (9)

in the (S)∗- sense.Let a(·, ·), b(·, ·) : R× (S)∗ → (S)∗ and regard the stochastic differential equation on [0, T ] (T > 0)

dXt = a(t, Xt) dt + b(t, Xt) dBvt .

It is defined in terms of the corresponding stochastic integral equation with t ∈ [0, T ]

Xt = X0 +∫ t

0

a(s,Xs) ds +∫ t

0

b(s,Xs) dBvs .

If this equation is interpreted in the (S)∗-sense, it can be transformed to

Xt = X0 +∫ t

0

(a(s,Xs) + b(s,Xs) �W vs ) ds.

Set f(s,Xs) := (a(s,Xs) + b(s,Xs) �W vs ). Then this equation takes the form

Xt = X0 +∫ t

0

f(s,Xs) ds, t ∈ [0, T ] (10)

as a white noise integral equation, where f(·, ·) is a mapping R× (S)∗ → (S)∗ such that the integral isdefined.

A stochastic distribution process Xt is called weakly measurable if S(Xt)(η) is measurable for allη ∈ S(R).

Definition 10 A stochastic distribution process Xt is called a weak solution of the white noise integralequation (10) on [0, T ], if it satisfies the following conditions:

(a) X is weakly measurable,(b) the stochastic distribution process f(t, Xt) is white noise integrable,(c) For each η ∈ S(R) the equality holds for almost all t ∈ [0, T ].

S(Xt)(η) = S(X0)(η) +∫ t

0

S(f(s,Xs))(η) ds.

Example 13 Consider the stochastic differential equation

dXt = µXt dt + σXt dBvt

with constants µ and σ. It is has to be interpreted by the associated stochastic integral equation

Xt =∫ t

0

µXs ds +∫ t

0

σXs dBvs

=∫ t

0

(µXs + σXs �W vs ) ds

=∫ t

0

Xs � (µ + σW vs ) ds.

15

Its solution is given by

Xt = X(0) exp�(µt + σBvt ) = X(0) exp(µt + σBv

t −12σ2|m(u, t)|20)

= X(0) exp(µt + σBvt −

12σ2v(t, t)).

The following theorem on existence and uniqueness of solutions of white noise integral equations istaken from [Ku], Theorem 13.43.

Theorem 8 Suppose the function f : [0, T ]× (S)∗ → (S)∗ satisfies the following conditions:

(a) (Measurability condition) The function f(s,Xs), s ∈ [0, T ], is weakly measurable for any weaklymeasurable stochastic distribution process X : [0, T ] → (S)∗.

(b) (Lipschitz condition) For almost all t ∈ [0, T ],

|S(f(t, Φ))(η)− S(f(t, Ψ))(η)| ≤ L(t, η)|S(Φ)(η)− S(Ψ)(η)|,

for all η ∈ S(R) and Ψ,Φ ∈ (S)∗, where L is nonnegative and∫ T

0

L(t, η) dt ≤ K(1 + |η|2p)

for some K, p ≥ 0.(c) (Growth condition) For almost all t ∈ [0, T ],

|S(f(t, Φ))(η)| ≤ ρ(t, η)(1 + |S(Φ)(η)|),

for all η ∈ S(R) and Φ ∈ (S)∗, where ρ is nonnegative and∫ T

0

ρ(t, η) dt ≤ K exp(c|η|2p

),

where K, p are the same as above and c ≥ 0.

Then for any X0 ∈ (S)∗ the equation

Xt = X0 +∫ t

0

f(s,Xs) ds

has a unique weak solution X such that for all η ∈ S(R)

ess supt∈[0,T ]

|SXt(η)| < ∞.

This theorem will be used to prove the following theorem on existence and uniqueness of solutionsof stochastic integral equations driven by Gaussian processes with dependent increments.

Theorem 9 (Existence and Uniqueness Theorem for bilinear SDE’s) Let σ, µ ∈ C([0, T ]) andX0 ∈ (S)∗ and t ∈ [0, T ]. Further, let m(u, t) such that there are K, p > 0 satisfying∫ T

0

∣∣∣∣⟨ d

dtm(u, t), η(u)

⟩∣∣∣∣ dt ≤ K(1 + |Apη|20)

for all η ∈ S(R). Then there exists a unique weak solution of

Xt = X0 +∫ t

0

(µ(s)Xs + σ(s)Xs �W vs ) ds, t ∈ [0, T ],

which is given by

Xt = X0 exp�(∫ t

0

µ(s) ds +∫ t

0

σ(s) dBvs ).

16

Proof We show that the conditions of Theorem 8 are valid. The measurability condition is met dueto continuity of the Wick product and measurability of dm(u, t)/dt. To prove the Lipschitz conditioncheck that for almost all t ∈ [0, T ], for all Φ, Ψ ∈ (S)∗, and for all η ∈ S(R)∣∣∣∣S(µ(t)Φ)(η) + S(σ(t)Φ)(η)

⟨d

dtm(u, t), η(u)

⟩−S(µ(t)Ψ)(η)− S(σ(t)Ψ)(η)

⟨d

dtm(u, t), η(u)

⟩∣∣∣∣≤ (|µ(t)|+ |σ(t)

⟨d

dtm(u, t), η(u)

⟩|)|S(Φ)(η)− S(Ψ)(η)|.

Therefore choose

L(t, η) := (|µ(t)|+ |σ(t)⟨

d

dtm(u, t), η(u)

⟩|)

and observe that an upper bound for∫ T

0L(t, η) dt is given by∫ T

0

(|µ(t)|+ |σ(t)⟨

d

dtm(u, t), η(u)

⟩|) dt

≤ max{|µ(t)|}T + max{|σ(t)|}∫ T

0

|〈 d

dtm(u, t), η(u)〉| dt.

This proves the Lipschitz condition. In order to check the growth condition proceed in a similar manner∣∣∣∣S(µ(t)Φ)(η) + S(σ(t)Φ)(η)⟨

d

dtm(u, t), η(u)

⟩∣∣∣∣≤(

maxt{|µ(t)|}+ max

t{|σt|}

∣∣∣∣⟨ d

dtm(u, t), η(u)

⟩∣∣∣∣) |S(Φ)(η)|

≤(

maxt{|µ(t)|}+ max

t{|σt|}

∣∣∣∣⟨ d

dtm(u, t), η(u)

⟩∣∣∣∣) (1 + |S(Φ)(η)|).

Take ρ(t, η) := maxt∈[0,T ]{|µ(t)|}+ maxt∈[0,T ]{|σt|}|⟨

ddtm(u, t), η(u)

⟩| and the condition for ρ(t, η) is

obviously satisfied. The solution turns out to be

Xt =∫ t

0

(µ(s) + σ(s)W vs ) �Xs ds

= exp�(∫ t

0

µ(s) ds +∫ t

0

σ(s) dBvs ).

�

Remark 6 A more general type of stochastic differential equations is discussed in Section 9.

Now the ordinary Ornstein-Uhlenbeck process in the (S)∗-sense and a generalized Ornstein-Uhlenbeckprocess are presented.

Example 14 (Ordinary Ornstein-Uhlenbeck process) The centered Ornstein-Uhlenbeck process BOUt is

a stationary centered Gaussian process with covariance function v(s, t) = (σ2/2α) exp(−α|t− s|) withpositiv constants α, σ, if the initial vaule has a normal BOU

0 distribution with mean zero and variance(σ2/2α) (see Karatzas and Shreve [KaSh], Chapter 5.6). Therefore the Ornstein-Uhlenbeck process inthe (S)∗-sense is given by

BOUt := BOU

0 exp(−αt) + 〈·, σ1[0,t] exp(−α(t− u))〉.

Its derivative in the (S)∗-sense is

d

dtBOU

t = BOU0 (−α) exp(−αt) +

⟨·, σ d

dt(1[0,t] exp(−α(t− u)))

⟩.

17

The stochastic integral and differential equations with respect to the Ornstein-Uhlenbeck processcan be defined similarly as before.

Example 15 (Generalized Ornstein-Uhlenbeck process) Let α, σ > 0. The generalized Ornstein-Uhlenbeckprocess is the solution of the stochastic differential equation

dXt = −αXt dt + σ dBvt ,

which is identical todXt

dt= −αXt + σW v

t (11)

in the (S)∗-sense. Note that by Theorem 9 on existence and uniqueness one can show with slightmodifications that the solution is unique. Solve the corresponding non-disturbed differential equationdXt = −αXt dt and get the solution Xt = C exp(−αt). Assume that C is a stochastic distributionprocess. By the product rule, given in Proposition 9, the derivative is computed to

d(Ct exp(−αt))dt

=dCt

dtexp(−αt)− αCt exp(−αt).

Comparing this with the stochastic differential equation (11) one deduces

dCt

dtexp(−αt) = σW v

t .

Hence Ct = C0 +∫ t

0exp(αs)σW v

s ds = C0 + σ∫ t

0exp(αs) dBv

s . Thus the solution of (11) is

BGOUt := Xt = C0 exp(−αt) + σ

∫ t

0

exp(−α(t− s)) dBvs ,

where C0 is the initial random variable of the process BGOUt . By Proposition 8 the expectation value

of BGOUt is S(BGOU )(0) = E(C0) exp(−αt), and its covariance function is equal to

E((BGOU

t − E(C0) exp(−αt))(BGOUs − E(C0) exp(−αs))

)= exp (−α(t + s))

(E(C2

0 )− E(C0)2)

+ E

(C0 exp(−αt)σ

∫ s

0

exp(−α(s− u)) dBvs

)+ E

(C0 exp(−αs)σ

∫ t

0

exp(−α(t− u)) dBvs

)+ E

(σ2

∫ t

0

exp(−α(t− u)) dBvs

∫ s

0

exp(−α(s− u)) dBvs

).

Finally a process that can be considered as an extension of the Brownian Bridge is presented.

Example 16 (Brownian bridge with Bvt ) For fixed 0 < a < b an ordinary Brownian bridge is given by

Ba→bt := a

(1− t

T

)+ b

t

T+(

Bt −t

TBT

), t ∈ [0, T ].

This definition motivates the extension to the generalized Brownian Bridge

Bv,a→bt := a

(1− t

T

)+ b

t

T+(

Bvt −

t

TBv

T

).

The expectation value is E(Bv,a→bt ) = E(Ba→b

t ) = a + (b− a)t/T . The covariance function of Bv,a→bt

is equal to

E

((Bv

s −s

TBv

T

)(Bv

t −t

TBv

T

))= v(s, t)− t

Tv(s, T )− s

Tv(t, T ) +

st

Tv(T, T ).

18

7 Stochastic integrals driven by stochastic processes in (S)∗

In this section the definition of the stochastic integral driven by a stochastic distribution in (S)∗ willbe further discussed. According to the chaos decomposition theorem a stochastic distribution processX(t) can be represented as

X(t) =∞∑

k=0

⟨: ·⊗k :, dk(u, t)

⟩, (12)

where dk(u, ·) : R → (S ′(R))⊗k for each u ∈ Rk and k ∈ N, and d0(t) a function from R → C. As inTheorem 5 one can prove that for a differentiable Fk(·) : R → (S ′(R))⊗k it holds that

d⟨: ·⊗k :, Fk(t)

⟩dt

=⟨

: ·⊗k :,dFk(t)

dt

⟩.

This will be used in the proof of the following theorem.

Theorem 10 Let b > a and dk(u, ·) : [a, b] → (S ′(R))⊗k be continuously differentiable for k ∈ N, suchthat there exists a, p > 0 and a positive bounded function L(t) on [a, b] such that∣∣∣∣⟨d(dk(u, t))

dt, η⊗k(u)

⟩∣∣∣∣ ≤ L(t)ak|η|2kp

k!(13)

for all η ∈ S(R), for all t ∈ [0, T ] and for all k ∈ N. Further let d0(·) be continuously differentiable on[a, b] in the classical sense. Then

X(t) =∞∑

k=0

⟨: ·⊗k :, dk(u, t)

⟩is a differentiable stochastic distribution process and

dX(t)dt

=d(d0(t))

dt+

∞∑k=1

⟨: ·⊗k :,

d(dk(u, t))dt

⟩. (14)

Proof Define

gnη (t) := d0(t) +

n∑k=1

⟨dk(u, t), (η(u))⊗k

⟩.

The function gnη (t) is continuously differentiable and

dgnη (t)dt

:=d(d0(t))

dt+

n∑k=1

⟨d(dk(u, t))

dt, (η(u))⊗k

⟩

=d

dt

(d0(t) +

n∑k=1

⟨dk(u, t), (η(u))⊗k

⟩).

Letting n →∞ one obtains

d(d0(t))dt

+∞∑

k=1

⟨d(dk(u, t))

dt, (η(u))⊗k

⟩

=d

dt

(d0(t) +

∞∑k=1

⟨dk(u, t), (η(u))⊗k

⟩).

Due to equation (13) the left hand side converges and the order of differentation and summation can bechanged. Furthermore, this expression is an element of (S)∗ as it satisfies the conditions of Theorem 3.The right hand side is the derivative of S(X(t))(η) with respect to t. So X(t) is a continuouslydifferentiable stochastic distribution process. �

19

Remember that the definition of the integral∫

R Yt dXt is given by Definiton 9.

Example 17 It holds that ∫ b

a

X(t) dX(t) =12(X(b))�2 − 1

2(X(a))�2

which is obvious by regarding the S-transforms

∫ b

a

S(X(t))(η)S(

dX(t)dt

)(η) dt =

∫ b

a

S(X(t))(η)dS(X(t))(η)

dtdt

=12(S(X(b))(η))2 − 1

2(S(X(a))(η))2.

Example 18 (Associated Ornstein-Uhlenbeck process) Consider the stochastic differential equation

dY (t) = −αY (t) dt + σ dX(t) (15)

with constants α, σ ∈ R. The solution can be calculated similar to Example 15. The solution of theundisturbed differential equation is C exp(−αt). Supposing C is a stochastic distribution process itfollows by Proposition 9 that

d(exp(−αt)C(t)) = −α exp(−αt)C(t) dt + exp(−αt) dC(t).

Comparing this equation with (15) it follows that σ dX(t) = exp(−αt) dC(t). Therefore the solutionof (15) equals

Y (t) = exp(−αt)C0 + σ

∫ t

0

exp(−α(t− s)) dX(s) (16)

with a stochastic distribution C0. For fixed η the S-transform of (15) is an ordinary differential equationin t which has a unique solution. The latter is given by the S-transform of equation (16) evaluated atthe same η. This shows that equation (15) has a unique solution.

Example 19 (The failing bilinear stochastic differential equation driven by X(t)) It is natural to ask ifthe bilinear stochastic differential equation with constants µ, σ ∈ R

dY (t) = µY (t) dt + σY (t) dX(t)

can be considered and solved in (S)∗. Assume that X(t) is already in the form of (12). It will be shownthat this stochastic differential equation has no solution in (S)∗, if there is a k ∈ N such that for allt ∈ R the function dk(·, t) isn’t the null element L2(Rk). To prove this evaluate the S-transform ofY (t) at some arbitrary η

dS(Y (t))(η)dt

= µS(Y (t))(η) + σS(Y (t))(η)dS(X(t))(η)

dt.

This ordinary differential equation in t has a unique solution of the form

S(Y (t))(η) = S(Y (0))(η) exp(µt + σS(X(t))(η)). (17)

If there exists a dk(u, t) 6≡ 0 for some k ∈ N an η can be found such that the growth condition is notsatisfied. Hence there is no stochastic distribution process Y (t) with values in (S)∗ satisfying equation(17).

20

8 A chain formula for stochastic distribution processes

In this section a chain formula for a certain class of stochastic distribution processes is given. It includesIto’s rule in the special case of Brownian motion.

Theorem 11 Let f(·, ·) : R× (S)∗ → (S)∗, such that

f(t, X) =n∑

k=0

ak(t)X�k

with continuously differentiable functions ak(t). Then the chain formula

f(b, Bvb )− f(a,Bv

a) =∫ b

a

(∂

∂tf(t, Bv

t ) +∂

∂xf(t, Bv

t ) �W vt

)dt

=∫ b

a

∂

∂tf(t, Bv

t ) dt +∫ b

a

∂

∂xf(t, Bv

t ) dBvt

holds.

Proof The S-transform of f(t, Bvt ) is given by Ft(η) :=

∑nk=0 ak(t)(S(Bv

t )(η))k. By use of Theorems 5and 6 one can show that the derivative of the function Ft(η) equals

dFt(η)dt

=n∑

k=0

(dak(t)

dt(S(Bv

t )(η))k + ak(t)k(S(Bvt )(η))k−1S(W v

t )(η))

.

Integrating both sides with respect to t the S-transform of the chain formula under considerationfollows. �

In the next step the chain rule for limn→∞∑n

k=0 a(t)(Bvt )�k will be considered. The idea is to

approximate functions f as considered in Theorem 11 by these Wick polynomials and to define aclass of functions to which the chain rule applies. One of these functions is the already known Wickexponential exp�(〈·, g〉) =

∑∞k=0(〈·, g〉)�k/k! with g ∈ S ′(R).

Definition 11 Let D be a subset of (S)∗. A function f : R×D → (S)∗ admits a Wick representationwith respect to D, if there exists a sequence of Wick monomials {ak(t)X�k : k ∈ N0} with continuouslydifferentiable ak(t), such that for all X ∈ D

f(t, X) = limn→∞

n∑k=0

ak(t)X�k

with convergence in (S)∗.

The following proposition contains a sufficient condition for a function f to admit a Wick repre-sentation.

Proposition 10 Suppose that h : R → R has a power series representation with h(x) =∑∞

k=0 hkxk

for all x ∈ R, and that there exist positive constants C, p and a such that for X ∈ (S)∗ and for allη ∈ S(R)

|h(S(X)(η))| ≤ C exp(a|η|2p

).

Then

h�(X) :=

( ∞∑k=0

hkX�k

)∈ (S)∗.

Proof We have to check the first condition of Theorem 3. Convergence of Fn(η) :=∑n

k=0 hk(S(X)(η))k

follows from convergence of∑n

k=0 hkxk for all x ∈ R. �

21

Theorem 12 Let D := {〈·, g〉 : g ∈ S ′(R)}. Suppose that f : R×D → (S)∗ admits a Wick represen-tation with respect to D. Then the following chain rule for f holds with b > a

f(b, Bvb )− f(a,Bv

a) =∫ b

a

(∂f(t, Bv

t )∂t

+∂f(t, Bv

t )∂x

�W vt

)dt.

Proof Observe that by Proposition 10 f(t, Bvt ) = limn→∞

∑nk=0 ak(t)(Bv

t )�k and, by Theorem 11,

n∑k=0

(ak(b)(Bv

b )�k − ak(a)(Bva)�k

)=∫ b

a

n∑k=0

(dak(u)

du(Bv

u)�k + ak(u)k(Bvu)�(k−1) �W v

u

)du

Taking limits on both sides for n →∞ proves the assertion. �

Theorem 13 Let D and f be as in Theorem 12 and let

f(t, 〈·, g〉) = limn→∞

n∑k=0

ak(t)(〈·, g〉)k

with convergence in (L2) for g ∈ L2(R). If Bvt is the ordinary Brownian motion Bt, then the chain

rule as given in Theorem 12 and Ito’s rule coincide.

Proof Due to the Wick formulae given in Lemma 4 the function f(t, Bt) has two representations, onewith Wick product, the other one with ordinary product. The chain rule is declared with respect tothe representation with the Wick product and Ito’s rule ist declared with respect to the representationwith the ordinary product. Both will be computed. By comparing their summands it will be shownthat they are equal. As the chain rule and Ito’s rule are linear mappings it is sufficient to check theassertion for a(t)(Bt)n with some continuously differentiable a(t) and some n ∈ N, and then to proceedto infinite sums. First regard the representation with the Wick product of a(t)(Bt)n by use of the Wickcalculus (Lemma 4) and the chain rule as given in Theorem 12

a(t)(Bt)n

= a(t)n/2∑k=0

(n

2k

)(2k − 1)!! tk(Bt)�(n−2k)

=∫ t

0

da(u)du

n/2∑k=0

(n

2k

)(2k − 1)!!uk(Bu)�(n−2k) du

+∫ t

0

a(u)n/2∑k=0

(n

2k

)(2k − 1)!! kuk−1(Bu)�(n−2k) du

+∫ t

0

a(u)n/2∑k=0

(n

2k

)(2k − 1)!!uk(n− 2k)(Bu)�(n−2k−1) dBu

=: I1 + I2 + I3

Applying Ito’s rule to a(t)(Bt)n and then using the Wick representation it follows

a(t)(Bt)n

=∫ t

0

da(u)du

(Bu)n du +∫ t

0

a(u)n(Bu)n−1 dBu +12

∫ t

0

a(u)n(n− 1)(Bu)n−2 du

=∫ t

0

da(u)du

n/2∑k=0

(n

2k

)(2k − 1)!!uk(Bu)�(n−2k) du

22

+∫ t

0

a(u)n(n−1)/2∑

k=0

(n− 12k

)(2k − 1)!!uk(Bu)�(n−2k−1) dBu

+12

∫ t

0

a(u)n(n− 1)(n−2)/2∑

k=0

(n− 22k

)(2k − 1)!!uk(Bu)�(n−2k−2) du

=: I4 + I5 + I6.

Term I1 is equal to I4. Integral I2 is equal to I6, because with k + 1 =: m in I6 one gets

I6

=12

∫ t

0

a(u)n(n− 1)(n−2)/2∑

k=0

(n− 22k

)(2k − 1)!!uk(Bu)�(n−2k−2) du

=∫ t

0

a(u)n/2∑m=1

(n− 2)!n (n− 1) 2m (2m− 1)2(n− 2m)!(2m)!

(2m− 3)!!um−1(Bu)�(n−2m) du

=∫ t

0

a(u)n/2∑m=1

(n

2m

)(2m− 1)!!mum−1(Bu)�(n−2m) du

= I2.

Integral I3 is equal to I5. Note that both sums have the same largest k, if n is odd. If n is even, thenthe last term in I3 vanishes because of the factor (n− 2k), hence both sums end up again at the samek. In order to show that both integrals are equal, we calculate

I5 =∫ t

0

a(u)(n−1)/2∑

k=0

n!(n− 1− 2k)!(2k)!

(2k − 1)!!uk(Bu)�(n−2k−1) dBu

=∫ t

0

a(u)n/2∑k=0

(n

2k

)(2k − 1)!!(n− 2k) uk(Bu)�(n−2k−1) dBu

= I3.

So Ito’s rule and the chain formula are identical in the case of the Brownian motion for finite polyno-mials

∑nk=0 ak(t)(Bt)n. For n →∞ this sum converges in (L2). The theorem is proved. �

Replacing the Wick product in ak(t)(Bvt )�k by the usual product, the chain rule appears as follows.

Proposition 11 Assume that fo : R×D → (S)∗ admits a Wick representation with respect to D :={〈·, g〉 : g ∈ L2(R)}, and assume that fo(t, Bv

t ) =∑∞

k=0 fok (t)(Bv

t )k converges in (L2). Then the chainrule has the additional representation

dfo(t, Bvt ) = fo

t (t, Bvt )dt + fo

x(t, Bvt )dBv

t +12

d|m(u, t)|20dt

foxx(t, Bv

t )dt. (18)

Proof As the chain rule is a linear mapping, it is sufficient to prove it for f(t, Bvt ) = an(t)(Bv

t )n only.By Lemma 4

d (an(t)(Bvt )n)

= d

an(t)n/2∑k=0

(n

2k

)(2k − 1)!!(|m(u, t)|20)k(Bv

t )�(n−2k)

=

dan(t)dt

n/2∑k=0

(n

2k

)(2k − 1)!!(|m(u, t)|20)k(Bv

t )�(n−2k))

23

+ an(t)n/2∑k=0

(n

2k

)(2k − 1)!!k

d|m(u, t)|20dt

(|m(u, t)|20)k−1(Bvt )�(n−2k))

+ an(t)n/2∑k=0

(n

2k

)(2k − 1)!!(|m(u, t)|20)k(n− 2k − 1)(Bv

t )�(n−2k))dBvt

= fot (t, Bv

t )dt +12

d|m(u, t)|20dt

fxx(t, Bvt )dt + fo

x(t, Bvt )dBv

t

arguing as in the proof of Theorem 13. �

This representation is already known for the fractional Brownian motion (see [Be] Chapter 5).As in the classical Ito setting it is helpful to have a chain rule which applies to functions of solutions

of stochastic differential equations, too.

Theorem 14 Suppose f : R×D → (S)∗ admits a Wick representation, and let the stochastic distri-bution process X· : R → D be given by

X(t)−X(0) =∫ t

0

(µ(u) + σ(u) �W vu ) du

with two stochastic distribution processes µ and σ such that the integral exists. Then the chain rule forf(t, Xt) holds with b > a

f(b, Xb)− f(a,Xa) =∫ b

a

(∂f(u, Xu)

∂t+

∂f(u, Xu)∂x

� (µ(u) + σ(u) �W vu ))

du.

Proof As in the proof of Theorem 11 take the S-transform of both sides in the statement, and use theWick representation of f and the integral representation of Xt. Fubini’s theorem is applied to changethe order of integration and S-transform. �

Theorem 15 (Chain rule) Suppose f : R × D → (S)∗ admits a Wick representation and let X· :R → D be a continuously differentiable stochastic distribution process. Then for b > a it holds

f(b, Xb)− f(a,Xa) =∫ b

a

∂f(u, Xu)∂t

+∂f(u, Xu)

∂x� dXu

dudu.

Proof The proof is given by calculating the S-transform as above. �

9 Existence and uniqueness of solutions of stochastic sifferential equations

Let a(t, X) :=∑n

k=0 ak(t)X�k and b(t, X) :=∑n

k=0 bk(t)X�k for all X ∈ (S)∗ and with continuouslydifferentiable functions ak, bk : R → R for k ∈ {0, . . . , n}.Lemma 5 Let Ψ ∈ (S)∗ and Ψ =

∑∞k=0

⟨: ·⊗k :, Fk(u)

⟩, then it holds that

Ψ�k =∞∑

l=0

∑j1+...+jk=l

⟨: ·⊗l :, Fj1(u)⊗ . . . ⊗Fjk

(u)⟩.

Proof The proof is obtained by induction. The assertion is satisfied for k = 2 since

Ψ�2 =∞∑

l=0

∑j1+j2=l

⟨: ·⊗l :, Fj1(u)⊗Fj2(u)

⟩.

The convergence in (S)∗ is granted by the existence of the Wick product. If the claim is satisfied fork then

Ψ�(k+1) =

∞∑l=0

∑j1+...+jk=l

⟨: ·⊗l :, Fj1(u)⊗ . . . ⊗Fjk

(u)⟩ � (

∞∑k=0

⟨: ·⊗k :, Fk(u)

⟩)

24

=∞∑

l=0

∑j1+...+jk+1=l

⟨: ·⊗l :, Fj1(u)⊗ . . . ⊗Fjk+1(u)

⟩.

Again convergence is attained in (S)∗ by the existence of the Wick product. �

Consider the stochastic differential equation

dX(t) = a(t, X(t)) dt + b(t, X(t)) dBvt

= (a(t, X(t)) + b(t, X(t)) �W vt ) dt. (19)

If the solution of this equation exists, then it has the form

X(t) =∞∑

k=0

⟨: ·⊗k :, dk(u, t)

⟩= d0(t) +

∞∑k=1

⟨: ·⊗k :, dk(u, t)

⟩due to the chaos decomposition theorem. Suppose that dk(u, t) satisfies the conditions of Theorem 10.If this representation of X(t) is applied to the stochastic differential equation (19), then one gets withTheorem 10

d∑∞

k=0

⟨: ·⊗k :, dk(u, t)

⟩dt

= a

(t,

∞∑k=0

⟨: ·⊗k :, dk(u, t)

⟩)+ b

(t,

∞∑k=0

⟨: ·⊗k :, dk(u, t)

⟩)�W v

t

=n∑

k=0

(ak(t)∞∑

l=0

∑j1+...+jk=l

⟨: ·⊗l :, dj1(u, t)⊗ . . . ⊗djk

(u, t)⟩

+ bk(t)∞∑

l=0

∑j1+...+jk=l−1

⟨: ·⊗l :, dj1(u, t)⊗ . . . ⊗djk

(u, t)⊗dm(u, t)dt

⟩.

The summands with different l are linearly independent. Furthermore, they are orthogonal to eachother if both belong to (L2). Thus for each l

d⟨: ·⊗l :, dk(u, t)

⟩dt

=n∑

k=0

ak(t)∑

j1+...+jk=l

⟨: ·⊗l :, dj1(u, t)⊗ . . . ⊗djk

(u, t)⟩

+ bk(t)∑

j1+...+jk=l−1

⟨: ·⊗l :, dj1(u, t)⊗ . . . ⊗djk

(u, t)⊗dm(u, t)dt

⟩.

This equation is an inhomogenous linear stochastic differential equation for 〈: ·⊗k :, dk(u, t)〉. Thus ittakes the form

d⟨: ·⊗k :, dk(u, t)

⟩dt

= α(t)⟨: ·⊗k :, dk(u, t)

⟩+⟨: ·⊗k :, g(t)

⟩with a real-valued function α and a function g : R → (S ′(R))⊗k.

Lemma 6 The ordinary differential equation

d⟨: ·⊗k :, dk(u, t)

⟩dt

= α(t)⟨: ·⊗k :, dk(u, t)

⟩+⟨: ·⊗k :, g(t)

⟩, (20)

t ∈ [0, T ], with a continuously differentiable function α and a function g : R → (S ′(R))⊗k has a uniquesolution in (S)∗ for each

⟨: ·⊗k :, dk(u, 0)

⟩.

25

Proof Apply the S-transfrom on both sides of equation (20)

d⟨dk(u, t), (η(u))⊗k

⟩dt

= α(t)⟨dk(u, t), (η(u))⊗k

⟩+⟨g(t), (η(u))⊗k

⟩,

this is a deterministic inhomogen linear differential equation, which has a unique solution Ft(η) de-pending on η ∈ S(R). The solution Ft(η) is multilinear in η and thus it is analytic in η. Further thereexists nonnegative constants K, p and a > such that

|Ft(η)| ≤ K exp(a|η|2p).for all t ∈ [0, T ] So the function Ft(η) satisfy the conditions of Theorem 3 and therefore it is theS-transform of a stochastic distribution process, which solves equation (20).

�

Theorem 16 Let a(t, X) and b(t, X) be finite Wick polynomials as above. Then the stochastic differ-ential equation

dX(t) = a(t, X(t))dt + b(t, X(t)) dBvt = (a(t, X(t)) + b(t, X(t)) �W v

t ) dt (21)

has a unique solution in (S)∗ for all X(0) ∈ (S)∗.

Proof By the considerations above, one can split the stochastic differential equation into equations oftype (20). Due to Lemma 6 they have unique solutions. Adding up these solutions one obtains thesolution of (21). �

Definition 12 Let Pk(·) : (S)∗ → (S)∗ be an operator defined by

Pk

( ∞∑n=0

⟨: ·⊗n :, fn

⟩):=⟨: ·⊗k :, fk

⟩.

It obviously satisfies (Pk)2 = Pk and PnPm = 0 if n 6= m. Pk can be used to modify a stochastic pro-cess X(t) within single subspaces of its chaos decomposition. For instance, let X(t) =

∑3k=0

⟨: ·⊗k :, fk(u, t)

⟩.

Then consider

X(t) :=X(t) + P2(X(t)) �Bvt

=3∑

k=0

⟨: ·⊗k :, fk(u, t)

⟩+⟨: ·⊗3 :, f2(u, t)⊗m(u, t)

⟩.

The process Bvt is plugged in without changing

∑2k=0

⟨: ·⊗k :, fk(u, t)

⟩. Such an effect cannot be realised

by X(t) �Bvt . The covariance function of X(t) is

E((

X(t)− f0(t))(

X(s)− f0(s)))

=2∑

k=0

〈fk(u, t), fk(u, s)〉

+ 〈f3(u, t) + f2(u, t)⊗m(u, t), f3(u, s) + f2(u, s)⊗m(u, s)〉

=3∑

k=0

〈fk(u, t), fk(u, s)〉+ 〈f3(u, t), f2(u, s)⊗m(u, s)〉

+ 〈f2(u, t)⊗m(u, t), f3(u, s)〉+ 〈f2(u, t), f2(u, s)〉〈m(u, t),m(u, s)〉.Suppose one tries to extend Theorem 16 to functions a and b being infinite Wick polynomials. Then

one has to face the problem that infinite Wick polynomials are not necessarily defined on the wholespace (S)∗ as can be seen in Example 8. One can use the operator P1 to get a wide class of functionsa, b : R × D → (S)∗ for which the stochastic differential equation (21) with D := {〈·, f〉, f ∈ S ′(R)}is declared. Suppose that a and b admit a Wick representation in D, then the stochastic differentialequation on [0, T ]

dX(t) = a(t, P1(X(t))) dt + b(t, P1(X(t))) dBvt

is declared for all X(0) ∈ (S)∗.

26

Theorem 17 Let

a(t, 〈·, f〉) =∞∑

k=0

ak(t)〈·, f〉�k

b(t, 〈·, f〉) =∞∑

k=0

bk(t)〈·, f〉�k

with f ∈ S ′(R), this a, b admitting a Wick representation in D := {〈·, f〉, f ∈ S ′(R)}. Then the equation

dX(t) = a(t, P1(X(t))) dt + b(t, P1(X(t))) dBvt (22)

has a unique solution X : [0, T ] → (S)∗ for all X(0) ∈ (S)∗.

Proof Equation (22) can be written as

∞∑k=0

d⟨: ·⊗k :, dk(u, t)

⟩dt

=∞∑

k=0

ak(t)⟨: ·⊗k :, d1(u, t)⊗k

⟩+ bk(t)

⟨: ·⊗k :, d1(u, t)⊗k⊗dm(u, t)

dt

⟩,

seperating this equation into a sequence of equations indexed by k we obtain

d⟨: ·⊗k :, dk(u, t)

⟩dt

= ak(t)⟨: ·⊗k :, d1(u, t)⊗k

⟩+ bk−1(t)

⟨: ·⊗k :, d1(u, t)⊗(k−1)⊗dm(u, t)

dt

⟩applying the S-transform on both sides one gets a sequence of deterministic equations. These haveunique solutions F k

t (η). As F kt (η) is multilinear in η it is analytic in η and satisfies the boundary

condition of Theorem 3. It is the S-transform of a stochastic distribution process. Thus a uniquesolution of (22) exists whose S-transform coincides with the sum

∑∞k=0 F k

t (η). �

Example 20 Let µ, σ ∈ R. Consider the equation

dX(t)dt

= µP1(X(t)) dt + σP1(X(t)) �W vt dt,

Thus the equations ⟨·, dd1(u, t)

dt

⟩= µ 〈·, d1(u, t)〉 ,⟨

: ·⊗2 :,dd2(u, t)

dt

⟩=⟨

: ·⊗2 :, d1(u, t)⊗dm(u, t)dt

⟩,

have unique solutions. Therefore

X(t) = exp(µt)〈·, h(u)〉+ 〈·, h(u)〉 �(∫ t

0

exp(µt) dBvt

).

Example 21 Let

sin�(〈·, f〉) :=∞∑

k=0

(−1)k+1

(2k + 1)!(〈·, f〉)�(2k+1)

and

cos�(〈·, f〉) :=∞∑

k=0

(−1)k

(2k)!(〈·, f〉)�(2k)

27

with f ∈ S ′(R). Then the stochastic differential equation

dX(t)dt

= sin�(P1(X(t))) + cos�(P1(X(t))) �W vt

is declared and the solution can be calculated by regarding the corresponding linear stochastic differ-ential equations as in the proof above. Let X(t) =

∑∞k=0

⟨: ·⊗k :, dk(u, t)

⟩. Then for each k ∈ N

d⟨: ·⊗(2k+1) :, d2k+1(u, t)

⟩dt

=(−1)k+1

(2k + 1)!(⟨: ·⊗(2k+1) :, d1(u, t)⊗(2k+1)

⟩)

+(−1)k

(2k)!

(⟨: ·⊗(2k+1) :, (d1(u, t))⊗(2k)⊗dm(u, t)

dt

⟩)and

d⟨: ·⊗(2k) :, d2k(u, t)

⟩dt

= 0.

The equation for 2k + 1 = 1 ⟨·, dd1(u, t)

dt

⟩= 〈·, d1(u, t)〉+

⟨·, dm(u, t)

dt

⟩is the stochastic differential equation solved by the Ornstein-Uhlenbeck process

〈·, d1(u, t)〉 = C0 exp(t) +∫ t

0

exp(t− s) dBvt . (23)

The remaining the solutions for 2k + 1 > 1 are⟨: ·⊗(2k+1) :, d2k+1(u, t)

⟩=∫ t

0

(−1)k+1

(2k + 1)!〈: ·⊗(2k+1) :, d1(u, t)⊗(2k+1)〉

+(−1)k

(2k)!

⟨: ·⊗(2k+1) :, (d1(u, t))⊗(2k)⊗dm(u, t)

dt

⟩dt,

the solutions for 2k are 〈: ·⊗(2k) :, f2k〉 with f2k ∈ (S ′(R)⊗(2k).

Example 22 Many other types of stochastic differential equations may be treated in a similar manneras above. As a last example, the SDE contains the projection P1(X(t)) and the process X(t) itself:

dX(t) = (sin�(P1(X(t))) �W vt + X(t)) dt.

This problem can be treated by solving the following sequence of differential equations⟨: ·⊗(2k+1) :,

d d2k+1(u, t)dt

⟩=⟨: ·⊗(2k+1) :, d2k+1(u, t)

⟩⟨

: ·⊗(2k) :,d d2k(u, t)

dt

⟩=

(−1)k+1

(2k − 1)!

⟨: ·⊗(2k) :, d2k−1(u, t)⊗dm(u, t)

dt

⟩+⟨: ·⊗(2k) :, d2k(u, t)

⟩Their solutions are given by⟨

: ·⊗(2k+1) :, d2k+1(u, t)⟩

= exp(t)⟨: ·⊗(2k+1) :, f2k+1(u)

⟩

28

and ⟨: ·⊗(2k) :, d2k(u, t)

⟩= C(0) exp(t)

⟨: ·⊗(2k) :, f2k(u)

⟩+∫ t

0

exp(t− s)(−1)k+1

(2k − 1)!

⟨: ·⊗(2k) :, d2k−1(u, s)⊗dm(u, s)

ds

⟩ds.

with fi(u) ∈ (S ′(R))⊗i.

Remark 7 For the classical existence and uniqueness theorem for stochastic differential equations drivenby Brownian motion it is required that the coefficient functions are bounded by a quadratic function.This not necessary here.

10 Notes

Remark 8 The theory of semimartingales does not apply to the class of Gaussian processes investigatedin this paper. As an example we mention fractional Brownian motion which may serve as an integratoras described above. but which clearly isn’t a semimartingale.

Remark 9 We conjecture that the class of functions which can be derivated by any chain rule presentedhere can be extended to continuously twice differentiable functions by approximating these functionsby infinite Wick polynomials.

Remark 10 The stochastic integral calculus developped in this paper can be generalized without essen-tial efforts to vector-valued stochastic processes. This fact opens access to stochastic partial differentialequations. S-transform allows to move from a stochastic partial differential equation to a determin-istic partial differential equation. Then the stochastic process appears only in form of its covariancefunction.

Remark 11 In [BjHu] there are several critical comments on the use of stochastic calculi based on theWick product in mathematical finance. However, the authors of the present paper discuss in [DiSc]how to deal with these issues.

Remark 12 In [NuTa] a Wick-Ito rule for Gaussian processes is proved. Their proof uses some propertiesof the variation of the Gaussian process. This approach differs from the approach in the present paper.Furthermore, their Wick-Ito rule is applicable to fractional Brownian motion BH

t with H ∈ (1/4, 1)only, whereas the approach in the present paper allows H ∈ (0, 1). The calculus in [NuTa] is based onthe Malliavin calculus as developped in [Nu].

References

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[BjHu] Bjork T., and Hult H. A note on Wick products and the fractional Black-Scholes model, FinanceStochast., 9(2005), 197–209

[DiSc] Dippon J. and Schiemert D. Option pricing in a Black-Scholes market with memory. Submitted. Seealso http://www.isa.uni-stuttgart.de/LstStoch/Dippon/Papers/OptionPricing.pdf

[HiKuPoSt] Hida T., Kuo H.-H., Potthoff J., and Streit L. White Noise: An Infinite Dimensional Calculus,1993, Dordrecht: Kluwer Academic Publishers

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