PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 90(104) (2011), 85–98 DOI: 10.2298/PIM1104085L

CHAOS EXPANSION METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS INVOLVING THE MALLIAVIN DERIVATIVE–PART II

Tijana Levajković and Dora Seleši

Communicated by Stevan Pilipović

Abstract. We solve stochastic differential equations involving the Malliavin derivative and the fractional Malliavin derivative by means of a chaos expansion on a general white noise space (Gaussian, Poissonian, fractional Gaussian and fractional Poissonian white noise space). There exist unitary mappings between the Gaussian and Poissonian white noise spaces, which can be applied in solving SDEs.

1. Introduction

This paper represents the conclusion of the results stated in Part I [11], where we introduced four types of white noise spaces: Gaussian, Poissonian, fractional Gaussian and fractional Poissonian depending on a Hurst parameter H ∈ (0, 1). Generalized stochastic processes, such as the Brownian motion, white noise or Poissonian noise, are given in the form of their chaos expansion in terms of the Fourier–Hermite and Fourier–Charlier polynomials. In [11] we showed that there exist unitary mappings between the Gaussian and Poissonian spaces, as well as between the regular spaces (H = \frac{1}{2}) and their fractional counterparts (H ∈ (0, 1)).

In this paper we focus on some examples of stochastic differential equations involving the Malliavin derivative, the Ornstein–Uhlenbeck operator and their fractional versions. All equations we solve can be interpreted on all four types of white noise spaces. We provide a general method of solving, using the Wiener–Itô chaos decomposition form, also known as the propagator method (see [12, 13, 14, 21]). It is used to set all coefficients in the chaos expansion on the left-hand side of the equation equal to the corresponding coefficients on the right-hand side of the equation, resulting in an infinite system of deterministic equations. Solving these equations and summing up the solutions into a series expansion, one obtains the solution of the initial equation, provided the series converges in some q-weighted space.

2010 Mathematics Subject Classification: 60H40, 60H10, 60H07, 60G20.

Key words and phrases: Gaussian white noise space, Poissonian white noise space, fractional Gaussian white noise space, fractional Poissonian white noise space, series expansion, Malliavin derivative, Skorokhod integral, Ornstein-Uhlenbeck operator, stochastic differential equation.

This paper was supported by the project Functional analysis, ODEs and PDEs with singularities, No. 17024, financed by the Ministry of Science, Republic of Serbia.

The paper is organized in the following manner: In Section 2 we review the notation and some results on chaos expansions and Malliavin derivatives obtained in Part I of the paper. Section 3 is devoted to solving some classes of stochastic differential equations which are driven by the Malliavin derivative and the Ornstein– Uhlenbeck operator.

2. Overview

In this section we provide the most important notation needed for further understanding. For details we refer to [11] and the references cited therein.

Let h_{n}, ξ_{n+1} , n ∈ ℕ_{0} denote the family of Hermite polynomials and Hermite functions respectively. Consider the Schwartz spaces of tempered distributions S′(ℝ) = ⋃ _{l∈ℕ_{0}} S_{−l} (ℝ), where

S_{−l} (ℝ) = { f = ∑_{k=1}^{∞} b_{k} ξ_{k} : ∥f∥_{−l}^{2} = ∑_{k=1}^{∞} b_{k}^{2}(2k)^{−l} < ∞} , l ∈ ℕ_{0},

and the spaces of distributions with exponential growth introduced in [20] expS′(ℝ) = ⋃ _{l∈ℕ_{0}} expS_{−l} (ℝ), where

expS_{−l} (ℝ) = { f = ∑_{k=1}^{∞} b_{k} ξ_{k} : ∥f∥_{exp ,−l}^{2} = ∑_{k=1}^{∞} b_{k}^{2}e^{−2kl} < ∞} , l ∈ ℕ_{0}.

Let I = (ℕ_{0}^{ℕ})_{c} denote the set of sequences of non-negative integers which have finitely many nonzero components α = (α_{1}, α_{2}, …, α_{m}, 0, 0, …), α_{i} ∈ ℕ_{0}, i = 1, 2, …, m, m ∈ ℕ. The k-th unit vector ε^{(k)} = (0, ⋯, 0, 1, 0, ⋯), k ∈ ℕ is the sequence of zeros with the number 1 as the k-th component. The length of a multi-index α ∈ I is defined as |α| = ∑ _{k=1}^{∞} α_{k} and α! = ∏ _{k=1}^{∞} α_{k}!. Let (2ℕ)^{α} = ∏ _{k=1}^{∞} (2k)^{α_{k}} .

The basic probability space is (Ω, F, P) (S′(ℝ), B, P), where S′(ℝ) denotes the space of tempered distributions, B the Borel sigma-algebra generated by the weak topology on S′(ℝ) and P denotes the unique probability measure on (S′(ℝ), B) corresponding to a given characteristic function.

2.1. Chaos expansions on white noise spaces. Suppose that L^{2}(P) = L^{2}(S′(ℝ), B, P) is the Hilbert space of square integrable functions on S′(ℝ) with respect to the measure P and let K_{α}, α ∈ I, denote the orthogonal basis of L^{2}(P).

Theorem 2.1 (Wiener–Itô chaos expansion). Every element F ∈ L^{2}(P) has a unique representation of the form F (ω) = ∑ _{α∈I} c_{α} K_{α} (ω), c_{α} ∈ ℝ, such that

∥F ∥_{L^{2} (P)}^{2} = ∑_{α∈I} c_{α}^{2} ∥K_{α} ∥_{L^{2} (P)}^{2} < ∞.

In [11] we considered two important cases of the measure P: the Gaussian measure μ and the Poissonian measure ν. Their fractional versions, for a fixed Hurst parameter H ∈ (0, 1), are obtained using the mapping M = M^{(H)} : S(ℝ) → L^{2}(ℝ) ∩ C^{∞} (ℝ) defined by M̂f (y) = |y|^{\frac{1}{2} −H} f̂(y), y ∈ ℝ, f ∈ S(ℝ), which has the property M^{−1} = M^{(1−H)} . This map extends by linearity and continuity to S′(ℝ), and thus by setting L^{2}(P_{H}) = L^{2}(P ∘ M^{−1} ) = {G : Ω → ℝ; G ∘ M ∈ L^{2}(P)} one obtains the fractional version of the square integrable random variables on the white noise space L^{2}(P), i.e., L^{2}(μ_{H}) in the fractional Gaussian case and L^{2}(ν_{H}) in the fractional Poissonian case.

The orthogonal basis of the four white noise spaces L^{2}(P) is thus obtained in the following manner:

Table 1.

white noise  classical  fractional
space  Gaussian Poissonian Gaussian Poissonian
measure P  μ  ν  μ_{H}  ν_{H}
basis K_{α}  H_{α}  C_{α}  H̃_{α}  C̃_{α}
basis 𝔢_{k}  ξ_{k}  ξ_{k}  e_{k}  e_{k}

where the family of Fourier–Hermite polynomials is defined by

H_{α} (ω) = ∏_{k=1}^{∞} h_{α_{k}} (⟨ω, ξ_{k}⟩), α ∈ I,

and the family of Fourier–Charlier polynomials is defined by

C_{α} (ω) = C_{|α|} (ω; \underbrace{ξ_{1}, …, ξ_{1}}_{α_{1}}, …, \underbrace{ξ_{m}, …, ξ_{m}}_{α_{m}}), α = (α_{1}, …, α_{m}, 0, 0, …) ∈ I,

where

C_{k}(ω; φ_{1}, …, φ_{k})

= \frac{∂^{k}}{∂u_{1} …∂u_{k}} exp[⟨ ω, log( 1 + ∑_{j=1}^{k} u_{j} φ_{j} ) − ∑_{j=1}^{k} u_{j} ∫_{ℝ} φ_{j} (y) dy⟩ ] |_{u_{1} =···=u_{k} =0} ,

for k ∈ ℕ and φ_{j} ∈ S(ℝ).

In the fractional spaces the bases are given by H̃_{α} (ω) = ∏ _{k=1}^{∞} h_{α_{k}} (⟨ω, e_{k}⟩), α ∈ I and

C̃_{α}(ω) = C_{|α|} (ω; \underbrace{e_{1}, …, e_{1}}_{α_{1}}, …, \underbrace{e_{m}, …, e_{m}}_{α_{m}}), α = (α_{1}, …, α_{m}, 0, 0, …) ∈ I,

where e_{i} = M^{−1} ξ_{i}, i ∈ ℕ.

The spaces of generalized random variables are introduced by imposing weights on the convergence condition in the Wiener–Itô chaos expansion and thus weakening the L^{2}(P)-norm. For a weight sequence q_{α} > 1, α ∈ I, let (Q)_{−1}^{P} = ⋃ _{p∈ℕ_{0}} (Q)_{−1,−p}^{P} be the inductive limit of the spaces

(Q)_{−1,−p}^{P} = { F = ∑_{α∈I} b_{α} K_{α} : ∥F ∥_{(Q)_{−1,−p}^{P}}^{2} = ∑_{α∈I} b_{α}^{2} q_{α}^{−p} < ∞} , p ∈ ℕ_{0}.

Two important special cases are given by weights of the form: q_{α} = (2ℕ)^{α} and q_{α} = e^{(2ℕ)^{α}} . In these cases we denote the q-weighted spaces by (S)_{−1}^{P} and exp(S)_{−1}^{P} respectively.

We consider generalized stochastic processes of type (I) as linear and continuous mappings from a topological vector space X into the space of q-weighted generalized functions (Q)_{−1}^{P} i.e., elements of L(X, (Q)_{−1}^{P} ). If at least one of the spaces X or (Q)_{−1}^{P} is nuclear, then L(X, (Q)_{−1}^{P} ) ≅ X′⊗ (Q)_{−1}^{P} .

Theorem 2.2. Let X = ⋂ _{k=0}^{∞} X_{k} be a nuclear space endowed with a family of seminorms {∥ · ∥_{k}; k ∈ ℕ_{0}} and let X′ = ⋃ _{k=0}^{∞} X_{−k} be its topological dual. Generalized stochastic processes as elements of X′⊗ (Q)_{−1}^{P} have a chaos expansion of the form

(2.1)  u = ∑_{α∈I} f_{α} ⊗ K_{α} , f_{α} ∈ X_{−k} , α ∈ I,

where k ∈ ℕ_{0} does not depend on α ∈ I, and there exists p ∈ ℕ_{0} such that

(2.2)  ∥u∥_{X′⊗(Q)_{−1,−p}^{P}}^{2} = ∑_{α∈I} ∥f_{α} ∥_{−k}^{2} q_{α}^{−p} < ∞.

The expectation of the process u is given by the zeroth order chaos expansion coefficient E(u) = f_{(0,0,0,...)} .

In [11] we have constructed two unitary mappings: the mapping U acting between the Gaussian and Poissonian spaces, and the mapping M acting between a regular space and its fractional version. The essence of both operators is to establish a mapping between the orthogonal bases of the corresponding spaces, leaving the coefficients of the chaos expansion unaffected. Let U : X ⊗ (Q)_{−1}^{μ} → X ⊗ (Q)_{−1}^{ν} be defined by

(2.3)  U[ ∑_{α∈I} u_{α} ⊗ H_{α} ] = ∑_{α∈I} u_{α} ⊗ C_{α} , u_{α} ∈ X, α ∈ I,

and let M : X ⊗ (Q)_{−1}^{μ_{H}} → X ⊗ (Q)_{−1}^{μ} be defined by

(2.4)  M[ ∑_{α∈I} v_{α} ⊗ H̃_{α} ] = ∑_{α∈I} v_{α} ⊗ H_{α} , v_{α} ∈ X, α ∈ I.

This resulted in the following commutative diagram:

[figure: L^{2}(μ); M^{−1}; U∘M^{−1}; M^{−1} ∘U; U; L^{2}(μ_{H}); U; L^{2}(ν); M^{−1}; L^{2}(ν_{H})]

Diagram 1

Another important operator is obtained by extending M : S′(ℝ) → S′(ℝ) into M = M ⊗ Id : S′(ℝ) ⊗ (Q)_{−1}^{P} → S′(ℝ) ⊗ (Q)_{−1}^{P} given by

M( ∑_{α∈I} a_{α} (t) ⊗ K_{α} (ω)) = ∑_{α∈I} Ma_{α} (t) ⊗ K_{α} (ω).

2.2. The Malliavin derivative and the Ornstein–Uhlenbeck operator. In this subsection P will denote either the Gaussian measure μ or the Poissonian measure ν, and P_{H} will denote its corresponding fractional measure μ_{H} or ν_{H}. The notation (Q)_{−1}^{P} will refer to either (S)_{−1}^{P} or exp(S)_{−1}^{P} , and (Q)_{−1}^{P_{H}} to either (S)_{−1}^{P_{H}} or exp(S)_{−1}^{P_{H}} .

Definition 2.1. The Malliavin derivative of a process u ∈ Dom(𝔻) of the form u = ∑ _{α∈I} f_{α} ⊗ K_{α}, f_{α} ∈ X, α ∈ I is defined by

𝔻u = ∑_{α∈I} ∑_{k∈ℕ} α_{k} f_{α} ⊗ 𝔢_{k} ⊗ K_{α−ε^{(k)}} ,

where Dom(𝔻) is defined as an appropriate subset of X ⊗ (Q)_{−1}^{P} and called the domain of the Malliavin derivative.

Note that 𝔻u is a generalized stochastic process with values in a distribution space i.e., 𝔻u ∈ X ⊗ S′(ℝ) ⊗ (S)_{−1}^{P} or 𝔻u ∈ X ⊗ expS′(ℝ) ⊗ exp(S)_{−1}^{P} . In [11] we gave a characterization of the domain and the codomain in both cases.

Definition 2.2. The Skorokhod integral of a process F ∈ Dom(δ) of the form F = ∑ _{α∈I} f_{α} ⊗ v_{α} ⊗ K_{α}, f_{α} ∈ X, α ∈ I and v_{α} ∈ S′(ℝ) or v_{α} ∈ expS′(ℝ) respectively, is defined by δ(F) = ∑ _{α∈I} ∑ _{k∈ℕ} v_{α,k} f_{α} ⊗K_{α+ε^{(k)}} , where v_{α} = ∑ _{k∈ℕ} v_{α,k} 𝔢_{k}, v_{α,k} ∈ ℝ is the expansion of v_{α} in S′(ℝ) or expS′(ℝ) respectively.

Note that Dom(δ) ⊂ X ⊗ S′(ℝ) ⊗ (S)_{−1}^{P} , resp. Dom(δ) ⊂ X ⊗ expS′(ℝ) ⊗

exp(S)_{−1}^{P} and that δ(F) ∈ X ⊗ (S)_{−1}^{P} , resp. δ(F) ∈ X ⊗ exp(S)_{−1}^{P} . For a detailed characterization of the domain we refer to [11].

Definition 2.3. The composition R = δ ∘ 𝔻 is called the Ornstein–Uhlenbeck operator. For u ∈ Dom(𝔻) its action is given by Ru = ∑ _{α∈I} |α|u_{α} ⊗ K_{α}, but its action can be extended to a larger set, i.e., Dom(R) ⊃ Dom(𝔻) in a general case.

The Hermite i.e., the Charlier polynomials are eigenfunctions of R and the corresponding eigenvalues are |α|, α ∈ I, i.e., R(K_{α}) = |α|K_{α}.

Definition 2.4. The fractional Malliavin derivative of F = ∑ _{α∈I} f_{α} ⊗ K_{α} ∈ X ⊗ (Q)_{−1}^{P} is defined by 𝔻^{(H)} F = ∑ _{α∈I} ∑ _{k∈ℕ} α_{k} f_{α} ⊗ M^{−1} 𝔢_{k} ⊗ K_{α−ε^{(k)}} .

Theorem 2.3. Let 𝔻 and 𝔻^{(H)} denote the Malliavin derivative, respectively the fractional Malliavin derivative on X ⊗(Q)_{−1}^{P} . Let 𝔻̃ denote the Malliavin derivative on X ⊗ (Q)_{−1}^{P_{H}} . Then,

(2.5)  𝔻^{(H)} F = M^{−1} ∘ 𝔻F = M ∘ 𝔻̃ ∘ M^{−1} F,

for all F ∈ Dom(𝔻).

Definition 2.5. Let δ : X ⊗S′(ℝ)⊗(Q)_{−1}^{P} → X ⊗(Q)_{−1}^{P} denote the Skorokhod

integral. The fractional Skorokhod integral δ^{(H)} : X ⊗ S′(ℝ)⊗ (Q)_{−1}^{P} → X ⊗ (Q)_{−1}^{P} is defined for every F ∈ Dom(δ) by δ^{(H)} F = δ ∘ MF.

For the Ornstein–Uhlenbeck operator we note that its fractional version coincides with the regular one, i.e., R^{(H)} = δ^{(H)} ∘ 𝔻^{(H)} = δ ∘ M ∘ M^{−1} ∘ 𝔻 = δ ∘ 𝔻 = R.

For further information on fractional white noise spaces and Malliavin derivatives we refer to the basic literature [1]–[9], [16]–[19], [22], [23].

3. Stochastic differential equations

In this section we investigate the existence of solutions for stochastic differential equations involving the Malliavin derivative and the Ornstein–Uhlenbeck operator. The method of Wiener–Itô chaos expansions, used to set all coefficients in the chaos expansion on the left-hand side of the equation equal to the corresponding coefficients on the right-hand side of the equation, is a general and useful tool, also known as the propagator method. With this method we reduce a problem to an infinite system of deterministic equations. Summing up all coefficients of the expansion and proving convergence in an appropriate weight space, one obtains the solution of the initial equation. Other types of equations investigated by the same method can be found in several papers: [10], [12]–[15], [21], [24]–[26].

All stochastic equations solved in this section can be interpreted, by the use of the isometric transformations U and M defined in (2.3) and (2.4), in all four white noise spaces we have considered so far. Also, due to Theorem 2.3 the Malliavin derivative and the Skorokhod integral can be interpreted as their fractional counterparts in the corresponding fractional white noise space. With this argumentation we state the equations and solve them in a white noise space of general type.

3.1. Equations with the Malliavin derivative. Denote by r = r(α) = min{k ∈ ℕ : α_{k} ≠ 0}, for nonzero α ∈ I. Then the first nonzero component of α is the rth component α_{r}, i.e., α = (0, 0, …, 0, α_{r}, …, α_{m}, 0, 0, …). Denote by α_{ε^{(r)}} the multi-index with all components equal to the corresponding components of α, except the r-th, which is α_{r} − 1. We call α_{ε^{(r)}} the representative of α and write

(3.1)  α = α_{ε^{(r)}} + ε^{(r)} , α ∈ I, |α| > 0.

For example, the first nonzero component of α = (0, 0, 2, 1, 0, 5, 0, 0, …) is its third component. It follows that r = 3, α_{r} = 2 and the representative of α is α_{ε^{(r)}} = α − ε^{(3)} = (0, 0, 1, 1, 0, 5, 0, 0, …).

The set K_{α} = {β ∈ I : α = β + ε^{(j)} , for some j ∈ ℕ}, α ∈ I, |α| > 0 is nonempty, because α_{ε^{(r)}} ∈ K_{α}. Moreover, if α = nε^{(r)} , n ∈ ℕ, then Card(K_{α}) = 1 and in all other cases Card(K_{α}) > 1. For example if α = (0, 1, 3, 0, 0, 5, 0, …), then the set K_{α} has three elements

K_{α} = {α_{ε^{(2)}} = (0, 0, 3, 0, 0, 5, 0, …), (0, 1, 2, 0, 0, 5, 0, …), (0, 1, 3, 0, 0, 4, 0, …)}.

3.2. A first order equation. Let us consider an equation of the form

(3.2)  \{ \begin{matrix}𝔻u = h, & h ∈ X ⊗ S′(ℝ) ⊗ (S)_{−1} \\ Eu = ũ_{0}, & ũ_{0} ∈ X \end{matrix} .

Theorem 3.1. Let h = ∑ _{α∈I} ∑ _{k∈ℕ} h_{α,k} ⊗ 𝔢_{k} ⊗ K_{α} ∈ X ⊗ S′(ℝ) ⊗ (S)_{−1} , with coefficients h_{α,k} ∈ X such that

(3.3)  \frac{1}{α_{r}} h_{α_{ε^{(r)}} ,r} = \frac{1}{α_{j}} h_{β, j} ,

for the representative α_{ε^{(r)}} of α ∈ I, |α| > 0 and all β ∈ K_{α}, such that α = β +ε^{(j)} , for j ⩾ r, r ∈ ℕ. Then, equation (3.2) has a unique solution in X ⊗ (S)_{−1} . The chaos expansion of the generalized stochastic process, which represents the unique solution of equation (3.2) is given by

(3.4)  u = ũ_{0} + ∑_{α=α_{ε^{(r)}} +ε^{(r)} ∈I}  \frac{1}{α_{r}} h_{α_{ε^{(r)}} ,r} ⊗ K_{α} .

Proof. We seek the solution in the form u = ũ_{0} + ∑ \begin{matrix}α∈I \\ |α|>0 \end{matrix} u_{α} ⊗ K_{α}. Thus,

𝔻( ũ_{0} + ∑_{\begin{matrix}α∈I \\ |α|>0 \end{matrix}} u_{α} ⊗ K_{α} ) = ∑_{α∈I} ∑_{k∈ℕ} h_{α,k} ⊗ 𝔢_{k} ⊗ K_{α}

∑_{\begin{matrix}α∈I \\ |α|>0 \end{matrix}} ( ∑_{k∈ℕ} α_{k}u_{α} ⊗ 𝔢_{k}) ⊗ K_{α−ε^{(k)}} = ∑_{α∈I} ( ∑_{k∈ℕ} h_{α,k} ⊗ 𝔢_{k}) ⊗ K_{α}

∑_{α∈I} ( ∑_{k∈ℕ} (α_{k} + 1)u_{α+ε^{(k)}} ⊗ 𝔢_{k}) ⊗ K_{α} = ∑_{α∈I} ( ∑_{k∈ℕ} h_{α,k} ⊗ 𝔢_{k}) ⊗ K_{α}

Due to uniqueness of the Wiener-Itô chaos expansion it follows that, for all α ∈ I

∑_{k∈ℕ} (α_{k} + 1)u_{α+ε^{(k)}} ⊗ 𝔢_{k} = ∑_{k∈ℕ} h_{α,k} ⊗ 𝔢_{k}.

Due to the uniqueness of the series expansion in S′(ℝ) we obtain a family of deterministic equations

(3.5)  u_{α+ε^{(k)}} = \frac{1}{α_{k} + 1} h_{α,k} , for all α ∈ I, k ∈ ℕ,

from which we can calculate u_{α}, by induction on the length of α. For α = (0, 0, 0, …), the equations in (3.5) reduce to u_{ε^{(k)}} = h_{α,k} , α ∈ I, k ∈ ℕ, i.e.,

u_{(1,0,0,... )} = h_{(0,0,0,... ),1}
u_{(0,1,0,... )} = h_{(0,0,0,... ),2}
u_{(0,0,1,0... )} = h_{(0,0,0,... ),3}
⋮

and we obtain the coefficients u_{α} for α of length one. Note, u_{α} are obtained in the terms of h_{α_{ε^{(r)}} ,r} = h_{(0,0,0,...),r} , r ∈ ℕ.

For |α| = 1 the multi-indices are of the form α = ε^{(j)} , j ∈ ℕ, so several cases occur. For j = 1, α = ε^{(1)} = (1, 0, 0, …), we have

(3.6)  u_{(2,0,0,... )} = \frac{1}{2}h_{(1,0,0,... ),1} ; u_{(1,1,0,... )} = h_{(1,0,0,... ),2} ; u_{(1,0,1,0... )} = h_{(1,0,0,... ),3} ; u_{(1,0,0,1,0... )} = h_{(1,0,0,... ),4} ; ⋮

Continuing, for j = 2, α = ε^{(2)} = (0, 1, 0, …) the equations in (3.5) reduce to

(3.7)  u_{(1,1,0,0,... )} = h_{(0,1,0,0,... ),1} ; u_{(0,2,0,... )} = \frac{1}{2}h_{(0,1,0,0,... ),2} ; u_{(0,1,1,0... )} = h_{(0,1,0,0,... ),3} ; u_{(0,1,0,1,0... )} = h_{(0,1,0,0,... ),4} ; ⋮

and then, for α = ε^{(3)} = (0, 0, 1, 0, …) we obtain

(3.8)  u_{(1,0,1,0,... )} = h_{(0,0,1,0,... ),1} ; u_{(0,1,1,0,... )} = h_{(0,0,1,0,... ),2} ; u_{(0,0,2,0... )} = \frac{1}{2}h_{(0,0,1,0,... ),3} ; u_{(0,0,1,1,0... )} = h_{(0,0,1,0,... ),4} ; ⋮

The coefficient u_{(1,1,0,0,... )} appears in systems (3.6) and (3.7) and thus the additional condition h_{(1,0,0,... ),2} = h_{(0,1,0,0,... ),1} has to hold in order to have a solvable system. Also, from expressions for u_{(0,1,1,0,... )} and u_{(0,1,0,1,... )} in (3.7) and (3.8) we obtain conditions h_{(0,1,0,... ),3} = h_{(0,0,1,0,... ),2} and h_{(0,0,0,1,0,... ),2} = h_{(0,1,0,0,... ),4} respectively, which need to be satisfied, in order to have a unique u_{α}. In the same manner we obtain all coefficients u_{α}, for α of the length two, expressed as a function of h_{α_{ε^{(r)}} ,r} .

Let now |α| = 2. Then different combinations for the multi-indices occur: if we choose α = (1, 1, 0, 0, …) then (3.5) transforms into the system

(3.9)  u_{(2,1,0,0,... )} = \frac{1}{2}h_{(1,1,0,0,... ),1} ; u_{(1,2,0,... )} = \frac{1}{2}h_{(1,1,0,0,... ),2} ; u_{(1,1,1,0... )} = h_{(1,1,0,0,... ),3} ; u_{(1,1,0,1,0... )} = h_{(1,1,0,0,... ),4} ; ⋮

and if we choose α = (1, 0, 1, 0, 0, …), then the equations in (3.5) transform into

(3.10)  u_{(2,0,1,0,... )} = \frac{1}{2}h_{(1,0,1,0,0,... ),1} ; u_{(1,1,1,0,... )} = h_{(1,0,1,0,0,... ),2} ; u_{(1,0,2,0... )} = \frac{1}{2}h_{(1,0,1,0,0,... ),3} ; u_{(1,0,1,1,0... )} = h_{(1,0,1,0,0,... ),4} ; ⋮

We continue with α = (0, 1, 1, 0, 0, …) and α = (2, 0, 0, …) and obtain the systems

u_{(1,1,1,0,... )} = h_{(0,1,1,0,0,... ),1} ; u_{(0,2,1,0,... )} = \frac{1}{2}h_{(0,1,1,0,0,... ),2} ; u_{(0,1,2,0... )} = \frac{1}{2}h_{(0,1,1,0,0,... ),3} ; u_{(0,1,1,1,0... )} = h_{(0,1,1,0,0,... ),4} ; ⋮ (3.11)

and

u_{(3,0,0,... )} = \frac{1}{3}h_{(2,0,0,... ),1} ; u_{(2,1,0,... )} = h_{(2,0,0,... ),2} ; u_{(2,0,1,0... )} = h_{(2,0,0,... ),3} ; u_{(2,0,0,1,0... )} = h_{(2,0,0,... ),4} ; ⋮ (3.12)

respectively. For α = (0, 2, 0, 0, …) the system (3.5) transforms into

(3.13)  u_{(1,2,0,0,... )} = h_{(0,2,0,0,... ),1} ; u_{(0,3,0,... )} = \frac{1}{3}h_{(0,2,0,0,... ),2} ; u_{(0,2,1,0... )} = h_{(0,2,0,0,... ),3} ; u_{(0,2,0,1,0... )} = h_{(0,2,0,0,... ),4} ; ⋮

Combining with the previous results, we obtain u_{α} for |α| = 3. Two different representations of u_{(2,1,0,0,... )} are given in systems (3.9) and (3.12), so an additional condition \frac{1}{2}h_{(1,1,0,0,... ),1} = h_{(2,0,0,0,... ),2} follows. We express u_{(2,1,0,0,... )} = \frac{1}{2}h_{(1,1,0,0,... ),1} in form of the representative of the multi-index α = (2, 1, 0, 0, …). Since the coefficient u_{(1,2,0,... )} appears both in (3.9) and (3.13), we receive another condition \frac{1}{2}h_{(1,1,0,0,... ),2} = h_{(0,2,0,0,... ),1} , and express u_{(1,2,0,... )} = h_{(0,2,0,0,... ),1} by its representative. From (3.9), (3.10) and (3.11) we obtain u_{(1,1,1,0,... )} = h_{(0,1,1,0,0,... ),1} and the condition h_{(1,1,0,0,... ),3} = h_{(1,0,1,0,... ),2} = h_{(0,1,1,0,0,... ),1} . Then \frac{1}{2}h_{(0,1,1,0,... ),2} = h_{(0,2,0,... ),3} follows from (3.11) and (3.13), and u_{(0,2,1,0,... )} = \frac{1}{2}h_{(0,1,1,0,... ),2} .

We proceed by the same procedure for all multi-index lengths to obtain u_{α}. If the set K_{α}, α ∈ I, has at least one more element besides the representative

α_{ε^{(r)}} of α, then the condition for the process h is given in the form (3.3). We obtain the coefficients u_{α} of the solution as functions of the representative α_{ε^{(r)}}

u_{α} = \frac{1}{α_{r}} h_{α_{ε^{(r)}} , r} , for |α| ≠ 0, α = α_{ε^{(r)}} + ε^{(r)} ,

and the form of the solution (3.4).

It remains to prove convergence of the solution (3.4) in X ⊗ (S)_{−1} . Let h ∈ X ⊗ S_{−p} (ℝ) ⊗ (S)_{−1,−p} . Then, there exists p > 0 such that

∥h∥_{X⊗S_{−p} (ℝ)⊗(S)_{−1,−p}}^{2} = ∑_{α∈I} ∑_{k∈ℕ} ∥h_{α,k} ∥_{X}^{2} (2k)^{−p} (2ℕ)^{−pα} < ∞.

Note that for ũ_{0} ∈ X we have ∥ũ_{0}∥_{X} = ∥ũ_{0}∥_{X⊗(S)_{−1,−q}} for all q > 0. Then, the convergence follows from

∥u∥_{X⊗(S)_{−1,−2p}}^{2} = ∥ũ_{0}∥_{X⊗(S)_{−1,−2p}}^{2} + ∑_{\begin{matrix}α∈I,|α|>0, \\ α=α_{ε^{(r)}} +ε^{(r)} \end{matrix}}  \frac{1}{α_{r}^{2}} ∥h_{α_{ε^{(r)}} ,r} ∥_{X}^{2} (2ℕ)^{−2p(α_{ε^{(r)}} +ε^{(r)} )}

⩽ ∥ũ_{0}∥_{X⊗(S)_{−1,−2p}}^{2} + ∑_{α=α_{ε^{(r)}} +ε^{(r)}} ∥h_{α_{ε^{(r)}} ,r} ∥_{X}^{2} (2r)^{−p} (2ℕ)^{−pα}

⩽ ∥ũ_{0}∥_{X⊗(S)_{−1,−2p}}^{2} + ∑_{α∈I} ∑_{r∈ℕ} ∥h_{α,r} ∥_{X}^{2} (2r)^{−p} (2ℕ)^{−pα} < ∞,

where we used the fact that (2ℕ)^{pε^{(r)}} (2ℕ)^{−pα} ⩽ 1 for all α ∈ I, r ∈ ℕ. □

3.2.1. Special cases. Assume that the process h is expressed as a product h = c ⊗ g, c ∈ S′(ℝ) and g ∈ X ⊗ (S)_{−1} .

Theorem 3.2. Let c = ∑ _{k∈ℕ} c_{k} 𝔢_{k} ∈ S′(ℝ) and g = ∑ _{α∈I} g_{α} ⊗ K_{α} ∈ X ⊗

(S)_{−1} with coefficients g_{α} ∈ X such that

(3.14)  \frac{1}{α_{r}} g_{α_{ε^{(r)}}} c_{r} = \frac{1}{α_{j}} g_{β}c_{j} ,

holds for all β ∈ K_{α}, j ⩾ r, r ∈ ℕ, and their representative α_{ε^{(r)}} . Then

(3.15)  𝔻u = c ⊗ g, Eu = ũ_{0}, ũ_{0} ∈ X,

has a unique solution in X ⊗ (S)_{−1} given by

(3.16)  u = ũ_{0} + ∑_{α=α_{ε^{(r)}} +ε^{(r)} ∈I}  \frac{1}{α_{r}} g_{α_{ε^{(r)}}} c_{r} ⊗ K_{α} .

Proof. Providing an analogue procedure as in the previous theorem, we reduce equation (3.15) to a family of deterministic equations

(3.17)  u_{α+ε^{(k)}} = \frac{1}{α_{k} + 1} g_{α} c_{k}, for all α ∈ I, k ∈ ℕ,

from which, by induction on |α|, we obtain the coefficients u_{α} of the solution u, as functions of the representative α_{ε^{(r)}} . Let α ∈ I, |α| > 0 be given by (3.1). Condition (3.14) implies u_{α} = \frac{1}{α_{r}} g_{α_{ε^{(r)}}} c_{r}. The proof of convergence of the solution (3.16) in X ⊗ (S)_{−1} follows in the same way as in the previous theorem. □

Especially, if we choose c = 𝔢_{i}, for fixed i ∈ ℕ, then equation (3.15) transforms into

(3.18)  \begin{matrix}𝔻u = 𝔢_{i} ⊗ g, & g ∈ X ⊗ (S)_{−1} \\ Eu = ũ_{0}, & ũ_{0} ∈ X \end{matrix}

Theorem 3.3. Let g ∈ X ⊗ (S)_{−1} . Then (3.18) has a unique solution in X ⊗ (S)_{−1} of the form

(3.19)  u = ũ_{0} + ∑_{n∈ℕ} \frac{1}{n} g_{(n−1)ε^{(i)}} ⊗ K_{nε^{(i)}} ,

if and only if g is of the form

(3.20)  g = ∑_{n=0}^{∞} g_{nε^{(i)}} ⊗ K_{nε^{(i)}} = ∑_{n=0}^{∞} g_{nε^{(i)}} (I(𝔢_{i}))^{⋄ n} ,

where I(·) represents the Itô integral.

Proof. Let u ∈ X ⊗(S)_{−1} be a process of the form (3.19). Then, u ∈ Dom(𝔻) and from

𝔻u = ∑_{n=1}^{∞} \frac{1}{n} g_{(n−1)ε^{(i)}} ⊗ n K_{(n−1)ε^{(i)}} ⊗ 𝔢_{n} = ∑_{n=1}^{∞} g_{nε^{(i)}} ⊗ K_{nε^{(i)}} ⊗ 𝔢_{n}

follows that it is a solution to (3.18).

Conversely, let a process g ∈ X ⊗ (S)_{−1} be of the form (3.20). Then, following the notation of Theorem 3.2, c = 𝔢_{i} has the expansion c = ∑ _{k=1}^{∞} c_{k} 𝔢_{k}, where c_{k} = 1 for k = i and c_{k} = 0 for k ≠ i, k ∈ ℕ. The family of equations (3.17) transforms to the family of deterministic equations

(3.21)  \begin{matrix}(α_{i} + 1) u_{α+ε^{(i)}} = g_{α} , & g_{α} ∈ X \\ u_{α+ε^{(k)}} = 0, & k = 1, 2, 3 …, k ≠ i \end{matrix} , α ∈ I.

If (3.20) holds, then for fixed i ∈ ℕ, g_{α} = 0, for all α ≠ nε^{(i)} , and from (3.21) similarly as in Theorem 3.2 the coefficients are obtained by induction on |α|,

u_{α} = \{ \begin{matrix}\frac{1}{n} g_{(n−1)ε^{(i)}} , & α = nε^{(i)} \\ 0, & α ≠ nε^{(i)} \end{matrix} , n ∈ ℕ.

The chaos expansion of the solution is

u = ũ_{0} + ∑_{n∈ℕ} \frac{1}{n} g_{(n−1)ε^{(i)}} ⊗ K_{nε^{(i)}} = ũ_{0} + ∑_{n∈ℕ} \frac{1}{n} g_{(n−1)ε^{(i)}} ⊗ (I(𝔢_{k}))^{⋄n} .

Convergence in X ⊗ (S)_{−1} can be proven by the same method as in Theorem 3.2. Clearly, there exists p ∈ ℕ, such that

∥u∥_{X⊗(S)_{−1,−p}}^{2} = ∥ũ_{0}∥_{X}^{2} + ∑_{n=1}^{∞} \frac{1}{n^{2}} ∥g_{(n−1)ε^{(i)}} ∥_{X}^{2} (2ℕ)^{−pnε^{(i)}}

⩽ ∥ũ_{0}∥_{X}^{2} + ∑_{n=1}^{∞} ∥g_{(n−1)ε^{(i)}} ∥_{X}^{2} (2ℕ)^{−p(n−1)ε^{(i)}}

= ∥ũ_{0}∥_{X}^{2} + ∑_{n=0}^{∞} ∥g_{nε^{(i)}} ∥_{X}^{2} (2ℕ)^{−pnε^{(i)}} < ∞.  □

3.3. An eigenvalue problem. Using the same method as in the previous cases, in [12] we solved an eigenvalue problem of the form

(3.22)  \{ \begin{matrix}𝔻u = C ⊗ u, & C ∈ S′(ℝ) \\ Eu = ũ_{0}, & ũ_{0} ∈ X. \end{matrix}

In the special case, for C = 𝔢_{i}, i ∈ ℕ fixed, we obtained that a unique solution u of the equation (3.22) belongs to X ⊗ (S)_{−1} and is of the form u = ũ_{0} ⊗ exp{I(𝔢_{i})}.

3.4. An equation involving the exponential of the Ornstein–Uhlenbeck operator. Consider now a stochastic differential equation of the form

(3.23)  e^{cR} u = h,

where e^{cR} = ∑ _{k=0}^{∞} \frac{c^{k} R^{k}}{k!} , c ∈ ℝ and h ∈ X ⊗exp(S)_{−1,−p} is a generalized stochastic process.

Theorem 3.4. Let h ∈ X ⊗ exp(S)_{−1,−p} , for some p > 0. Then, there exists q > 0 such that equation (3.23) has a unique generalized solution in X⊗exp(S)_{−1,−q} given by the form

(3.24)  u = ∑_{α∈I} e^{−c|α|} h_{α} ⊗ K_{α} .

Proof. Assume u ∈ X ⊗exp(S)_{−1,−p} is a generalized stochastic process of the form (2.1), satisfying condition (2.2) with q_{α}^{−p} = e^{−p(2ℕ)^{α}} . Note that the differential operator e^{cR} satisfies the identity

e^{cR} K_{α} = ∑_{k=0}^{∞} \frac{c^{k}R^{k}K_{α}}{k!} = ∑_{k=0}^{∞} \frac{c^{k}|α|^{k}}{k!} K_{α} = e^{c|α|} K_{α} , α ∈ I.

Then

(3.25)  e^{cR} u = ∑_{α∈I} e^{c|α|} u_{α} ⊗ H_{α} , u_{α} ∈ X.

For c > 0 the operator e^{cR} is a continuous and bounded mapping from X ⊗ exp(S)_{−1,−p} into X ⊗ exp(S)_{−1,−q} , for some q > p + 2c. From e^{|α|} ⩽ e^{(2ℕ)^{α}} , α ∈ I it follows

∥e^{cR} u∥_{X⊗exp (S)_{−1,−q}}^{2} = ∑_{α∈I} e^{2c|α|} ∥u_{α} ∥_{X}^{2} e^{−q(2ℕ)^{α}}

⩽ ∑_{α∈I} e^{2c|α|} e^{−p(2ℕ)^{α}} ∥u_{α} ∥_{X}^{2} e^{−(q−p)(2ℕ)^{α}}

⩽ ( ∑_{α∈I} e^{2c|α|} e^{−(q−p)(2ℕ)^{α}} ) ( ∑_{α∈I} ∥u_{α} ∥_{X}^{2} e^{−p(2ℕ)^{α}} )

⩽ ( ∑_{α∈I} e^{−(q−p−2c)(2ℕ)^{α}} ) ∥u∥_{X⊗exp (S)_{−1,−p}}^{2} < ∞,

for q > p + 2c.

If c ⩽ 0, then the operator e^{cR} is a continuous and bounded mapping from X ⊗ exp(S)_{−1,−p} into X ⊗ exp(S)_{−1,−q} , for q > p:

∥e^{cR} u∥_{X⊗exp (S)_{−1,−q}}^{2} = ∑_{α∈I} e^{2c|α|} ∥u_{α} ∥_{X}^{2} e^{−q(2ℕ)^{α}}

⩽ ( ∑_{α∈I} e^{−(q−p)(2ℕ)^{α}} ) ∥u∥_{X⊗exp (S)_{−1,−p}}^{2} < ∞.

Let h ∈ X ⊗ exp(S)_{−1,−p} be of the form h = ∑ _{α∈I} h_{α} ⊗ H_{α} such that h_{α} ∈ X and

(3.26)  ∑_{α∈I} ∥h_{α} ∥_{X}^{2} e^{−p(2ℕ)^{α}} < ∞.

We are looking for the solution u of (3.23) in the form (2.1), where u_{α} ∈ X are the coefficients to be determined. We apply (3.25) to transform equation (3.23) into the system of deterministic equations e^{c|α|} u_{α} = h_{α}, α ∈ I. Thus u_{α} = e^{−c|α|} h_{α} and we obtain a unique solution of equation (3.23) in the form (3.24).

Finally, the convergence of the solution in X ⊗exp(S)_{−1,−p} , in the case of c > 0, follows directly from (3.26). But, in the case of c ⩽ 0 the solution converges in the space X ⊗ exp(S)_{−1,−q} , for some q > p − 2c, i.e.,

∥u∥_{X⊗exp (S)_{−1,−q}}^{2} = ∑_{α∈I} e^{−2c|α|} ∥h_{α} ∥_{X}^{2} e^{−q(2ℕ)^{α}}

⩽ ( ∑_{α∈I} e^{−2c|α|} e^{−(q−p)(2ℕ)^{α}} ) ( ∑_{α∈I} ∥h_{α} ∥_{X}^{2} e^{−p(2ℕ)^{α}} )

⩽ M∥h∥_{X⊗exp (S)_{−1,−p}}^{2} < ∞,

where M = ∑ _{α∈I} e^{−(q−p+2c)(2ℕ)^{α}} < ∞ for q > p − 2c. □

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Department of Mathematics and Informatics (Received 10 01 2011) Faculty of Traffic and Transport Engineering University of Belgrade Serbia

 t.levajkovic@sf.bg.ac.rs

Department of Mathematics and Informatics Faculty of Science University of Novi Sad Serbia

 dora@dmi.uns.ac.rs