On forward and backward SPDEs with non-local boundary conditions

We study linear stochastic partial differential equations of parabolic type with non-local in time or mixed in time boundary conditions. The standard Cauchy condition at the terminal time is replaced by a condition that mixes the random values of the solution at different times, including the terminal time, initial time and continuously distributed times. For the case of backward equations, this setting covers almost surely periodicity. Uniqueness, solvability and regularity results for the solutions are obtained. Some possible applications to portfolio selection are discussed.


Introduction
Stochastic partial differential equations (SPDEs) are well studied in the existing literature for the case of Cauchy boundary conditions at the initial time or at the terminal time. Forward parabolic SPDEs are usually considered with a Cauchy condition at initial time, and backward parabolic SPDEs are usually considered with a Cauchy condition at terminal time. A backward SPDE cannot be transformed into a forward equation by a simple time change. Usually, a backward SPDE is solvable in the sense that there exists a diffusion term being considered as a part of the solution that helps to ensure that the solution is adapted to the driving Brownian motions. In addition, there are results for the pairs of forward and backward equations with separate Cauchy conditions at initial time and the terminal time respectively. The results for solvability of regularity of forward and backward SPDEs can be found in Alós et al (1999), Bally We are given a standard complete probability space (Ω, F, P) and a right-continuous filtration F t of complete σ-algebras of events, t ≥ 0. We are given also a N -dimensional Wiener process w(t) with independent components; it is a Wiener process with respect to F t .
Assume that we are given an open domain D ⊂ R n such that either D = R n or D is bounded with C 2 -smooth boundary ∂D. Let T > 0 be given, and let Q We will study the boundary value problems for forward equations in Q and boundary value problems for backward equations in Q u(x, t, ω) | x∈∂D = 0 u(·, T ) − Γu = ξ.
In these boundary problems, Γ is a linear operator that maps functions defined on Q × Ω to functions defines on D × Ω. The operator A is defined as where b ij , f i , x i are the components of b,f , and x respectively, and measurable with respect to F t for all x ∈ R n , and the function ξ(x, ω) : R n × Ω → R is F 0 -measurable for all x ∈ R n . In fact, we will also consider ϕ and ξ from wider classes. In particular, we will consider generalized functions ϕ.

Spaces and classes of functions
We denote by · X the norm in a linear normed space X, and (·, ·) X denote the scalar product in a Hilbert space X.
We introduce some spaces of real valued functions.
Let G ⊂ R k be an open domain, then W m q (G) denote the Sobolev space of functions that belong to L q (G) together with the distributional derivatives up to the mth order, q ≥ 1.
We denote by | · | the Euclidean norm in R k , andḠ denote the closure of a region G ⊂ R k .
Let H 0 ∆ = L 2 (D), and let H 1 ∆ = 0 W 1 2 (D) be the closure in the W 1 2 (D)-norm of the set of all smooth functions u : D → R such that u| ∂D ≡ 0. Let H 2 = W 2 2 (D) ∩ H 1 be the space equipped with the norm of W 2 2 (D). The spaces H k and W k 2 (D) are called Sobolev spaces, they are Hilbert spaces, and H k is a closed subspace of W k 2 (D), k = 1, 2. Let H −1 be the dual space to H 1 , with the norm Let C 0 (D) be the Banach space of all functions u ∈ C(D) such that u| ∂D ≡ 0 equipped with the norm from C(D).
We shall write (u, v) H 0 for u ∈ H −1 and v ∈ H 1 , meaning the obvious extension of the bilinear form from u ∈ H 0 and v ∈ H 1 .
We denote byl k the Lebesgue measure in R k , and we denote byB k the σ-algebra of Lebesgue sets in R k .
We denote byP the completion (with respect to the measurel 1 × P) of the σ-algebra of subsets of [0, T ] × Ω, generated by functions that are progressively measurable with respect to F t .
We introduce the spaces The spaces X k (s, t) and Z k t (s, t) are Hilbert spaces. In addition, we introduce the spaces For brevity, we shall use the notations ) be such that all ζ k (·, t, ω) are progressively measurable with respect to F t , and let ζ − ζ k X 0 → 0. Let t ∈ [0, T ] and j ∈ {1, . . . , N } be given. Then the sequence of the integrals t 0 ζ k (x, s, ω) dw j (s) converges in Z 0 t as k → ∞, and its limit depends on ζ, but does not depend on {ζ k }.
Proof follows from completeness of X 0 and from the equality where the sequence {ζ k } is such as in Proposition 2.1.

Conditions for the coefficients
To proceed further, we assume that Conditions 2.1-2.3 remain in force throughout this paper.
Condition 2.1 The matrix b = b ⊤ is symmetric and bounded. In addition, there exists a constant δ > 0 such that

Condition 2.3 There exists an integer
, and linear continuous operatorsΓ : L 2 (Q) → H 0 ,Γ i : H 0 → H 0 , i = 1, .., N , such that the operatorsΓ : By Condition 2.3, the mapping Γ : Y 1 → Z 0 T is linear and continuous. This condition covers cases whenΓ where k 0 (·) ∈ L 2 (0, T ) and k i ∈ R. It covers also Γ such that where k i (·) are some regular enough kernels.
We introduce the set of parameters Sometimes we shall omit ω.

The definition of solution
We will study the following boundary value problem in Q We do not exclude an important special case when the functions b, f , λ, ϕ, and ξ, are deterministic, and h i ≡ 0, B i ≡ 0 (∀i). In this case, equation (3.1) is deterministic.
Definition 3.1 Let u ∈ Y 1 , ϕ ∈ X −1 , and h i ∈ X 0 . We say that equations (3.1)-(3.2) are satisfied if for all r, t such that 0 ≤ r < t ≤ T , and this equality is satisfied as an equality in Z −1 T .
Note that the condition on ∂D is satisfied in the sense that u(·, t, ω) ∈ H 1 for a.e. t, ω. Further, u ∈ Y 1 , and the value of u(·, t, ω) is uniquely defined in Z 0 T given t, by the definitions of the corresponding spaces. The integrals with dw i in (3.4) are defined as elements of Z 0 T . The integral with ds in (3.4) is defined as an element of Z −1 T . In fact, Definition 3.1 requires for (3.1) that this integral must be equal to an element of Z 0 T in the sense of equality in Z −1 T .

Existence and regularity results
Theorem 3.1 There exist a number κ = κ(P) > 0 such that problem (3.1)-(3.3) has an unique solution in the class Y 1 , for any ϕ ∈ X −1 , h i ∈ X 0 , ξ ∈ Z 0 0 , and any Γ such that Γ ≤ κ, where Γ is the norms of the operator Γ : where C = C(κ, P) > 0 is a constant that depends only on κ and P.
Starting from now and up to the end of this section, we assume that Condition 3.1 holds.
, ω, and the corresponding derivatives are bounded.
It follows from this condition that there exist modifications of β i such that the functions β i (x, t, ω) are continuous in x for a.e. t, ω. We assume that β i are such functions.
Theorem 3.2 Let F 0 be the P-augmentation of the set {∅, Ω}. Assume that at least one of the following conditions is satisfied: where C > 0 is a constant that does not depend on ϕ, h i , and ξ.
Theorem 3.3 Let the functions b, f and λ be non-random and such that the operator A can be represented as where λ(x, t) ≤ 0, and where f i are bounded functions. Further, let Then problem (3.1)-(3.3) has a unique solution u in the class Y 1 for any ϕ ∈ X −1 , h i ∈ X 0 , and ξ ∈ H 0 . In addition, (3.6) holds with a constant C > 0 that does not depend on ϕ, h i , and ξ.
The following corollary is a special case of Theorem 3.3 for deterministic parabolic equation with the boundary condition that covers the condition of periodicity.
Corollary 3.1 Under the assumptions of Theorem 3.3, for any k ∈ [−1, 1], the deterministic boundary value problem where C > 0 is a constant that does not depend on Φ and ϕ.
The classical result about well-posedness of the Cauchy condition at initial time corresponds to the special case of k = 0.

The definition of solution
For backward SPDEs, we will study the following boundary value problem in Q D × Ω. For instance, the case where Γu = u(·, 0) is not excluded; this case corresponds to the periodic type boundary condition u(·, T ) − u(·, 0) = ξ.
.., N , and ϕ ∈ X −1 . We say that equations for all r, t such that 0 ≤ r < t ≤ T , and this equality is satisfied as an equality in Z −1 T .
Note that the condition on ∂D is satisfied in the sense that u(·, t, ω) ∈ H 1 for a.e. t, ω. Further, u ∈ Y 1 , and the value of u(·, t, ω) is uniquely defined in Z 0 T given t, by the definitions of the corresponding spaces. The integrals with dw i in (4.4) are defined as elements of Z 0 T . The integral with ds in (4.4) is defined as an element of Z −1 T . In fact, Definition 4.1 requires for (4.1) that this integral must be equal to an element of Z 0 T in the sense of equality in Z −1 T .

Existence and regularity results
Starting from now and up to the end of this section, we assume that Condition 4.1 holds.
In particular, it follows from this condition that there exist modifications of β i such that the functions β i (x, t, ω) are continuous in x for a.e. t, ω. We assume that β i are such functions.
Up to the end of this section, we assume that the following condition is satisfied.
(ii) The functions b, f and λ are such that the operator A can be represented as where the functions f (x, t, ω), λ(x, t, ω), and β i (x, t, ω) are bounded and are differentiable in x for a.e. t, ω, and the corresponding derivatives are bounded. Then problem (4.1)-(4.2),(4.5) has a unique solution (u, χ 1 , ..., χ N ) in the class Y 1 × (X 0 ) N for any ϕ ∈ X −1 and ξ ∈ Z 0 T . In addition, (4.4) holds with C > 0 that does not depend on ϕ and ξ.

Some applications: portfolio selection problems
Theorem 4.2 can be applied to portfolio selection for continuous time diffusion market model, where the price dynamic is described by Ito stochastic differential equations. Examples of these models can be found in, e.g., Karatzas and Shreve (1998).
We consider the following model of a securities market consisting of a risk free bond or bank account with the price B(t), t ≥ 0, and a risky stock with the price S(t), t ≥ 0. The prices of the stocks evolve as where (w(t), w(t)) is a Wiener process, a(t) is a appreciation rate, (σ(t), s(t)) is a vector of volatility coefficients. The initial price S(0) > 0 is a given deterministic constant. The price of the bond evolves as where B(0) is a given constant, r ≥ 0 is a short rate. For simplicity, we assume that r = 0 and We assume that (w(·), w(t)) is a standard Wiener process on a given standard probability where Ω is a set of elementary events, F is a complete σ-algebra of events, and P is a probability measure.
Let F t be the filtration generated by w(t), and let F t be the filtration generated by (w(t), w(t)).
In particular, we assume that F 0 and F 0 are trivial σ-algebras, i.e., they are the P-augmentations of the set {∅, Ω}.

Strategies for bond-stock-options market
The rules for the operations of the agents on the market define the class of admissible strategies where the optimization problems have to be solved.
Let X(0) > 0 be the initial wealth at time t = 0 and let X(t) be the wealth at time t > 0.
We assume that the wealth X(t) at time t ∈ [0, T ] is Here β(t) is the quantity of the bond portfolio, γ(t) is the quantity of the stock portfolio, t ≥ 0.
The pair (β(·), γ(·)) describes the state of the bond-stocks securities portfolio at time t. Each of these pairs is called a strategy.
A pair (β(·), γ(·)) is said to be an admissible strategy if the processes β(t) and γ(t) are progressively measurable with respect to the filtration F t .
In particular, the agents are not supposed to know the future (i.e., the strategies have to be adapted to the flow of current market information).
Let P * be an equivalent probability measure such that S(t) is a martingale under P * . By the assumptions on (a, σ, s), this measure exists and is unique.

A portfolio selection problem
In portfolio theory, a typical problem is constructing a portfolio strategy with certain desirable properties. It will be demonstrated below that Theorem 4.2 can be applied to this problem.
Let us consider the following example.
Theorem 5.1 Let where x ∈ D, t ∈ [0, T ] and where S(t) is defined by (5.1) given that S(0) = x. Then E * X(T, x) 2 < +∞, and the process X(t, x) represents the wealth generated by some self-financing strategy given that S(0) = x. In addition, The portfolio described in Theorem 5.1 has the following attractive feature: with a positive ξ, it ensures a systematic gain when τ > T for the case of stagnated marked prices. The event τ < T can be considered as an extreme event if s L is sufficiently small and s U is sufficiently large.
Note that the assumption that the process (a(t), σ(t)) is F t -adapted was used to ensure existence of u. A more general model where this process is F t -adapted leads to a degenerate SPDE in bounded domain where Condition 2.1 is not satisfied. This case is not covered by Theorem 4.2. An example of portfolio selection based on a degenerate backward SPDE in entire space was considered in Ma and Yong (1997).

Introduce operators
where u is the solution in Y 1 of problem (6.1) with s = T , ϕ ∈ X −1 , and Φ ∈ Z 0 T . It is easy to see that these operators are linear and continuous.
For brevity, we denote u(·, t) = u(x, t, ω). Clearly, u ∈ Y 1 is the solution of problem (4.1)- Clearly, Q ≤ Γ L T , where Q , Γ , and L T , are the norms of the operators Q : T is continuous for small enough Q , i.e. for a small enough κ > 0. Hence Starting from now, we assume that Condition 4.1 is satisfied, in addition to Conditions 2.1-2.3.
The following lemma represents an analog of the so-called "the second energy inequality", or "the second fundamental inequality" known for the deterministic parabolic equations (see, e.g., inequality (4.56) from Ladyzhenskaya (1985), Chapter III).
Lemma 6.2 Problem (6.1) has a unique solution (u, χ 1 , ..., χ N ) in the class Y 2 × (X 1 ) N for any ϕ ∈ X 0 , Φ ∈ Z 1 T , and where C > 0 does not depend on ϕ and Φ; it depends on P an on the supremums of the derivatives listed in Condition 4.1(ii).
The lemma above represents a reformulation of Theorem 3.1. from Du and Tang (2012) Dokuchaev (2011Dokuchaev ( ,2012a are still valid.
Lemma 6.3 The operator Q : Z 0 T → Z 0 T is compact.
Proof of Lemma 6.3. Let u = L 0 Φ, where Φ ∈ Z 0 T . By the semi-group property of backward SPDEs from Theorem 6.1 from Dokuchaev (2010), we obtain that u| t∈[0,s] = L s u(·, s) for all s ∈ (0, T ]. By Lemmas 6.1 and 6.2, we have for τ ∈ {t 1 , ..., t m } that for constants C i > 0 which do not depend on Φ. Hence the operator Q : Z 0 T → W 1 2 (D) is continuous. Since the embedding of W 1 2 (D) to H 0 and in Z 0 T is a compact operator, the proof of Lemma 6.3 follows.

Let us introduce operators
Here b ij , x i , β ik are the components of b, β i , and x.
Let ρ ∈ Z 0 s , and let p = p(x, t, ω) be the solution of the problem By Theorem 3.4.8 from Rozovskii (1990), this boundary value problem has an unique solution p ∈ Y 1 (s, T ). Introduce an operator M s : Z 0 s → Y 1 (s, T ) such that p = M s ρ, where p ∈ Y 1 (s, T ) is the solution of this boundary value problem.
Proof of Theorem 4.2. By Theorem 3.1 from Dokuchaev (2005), problem (6.1) has an unique solution p ∈ Y 2 for any ρ ∈ Z 1 s , and where C > 0 does not depend on ρ (Dokuchaev (2005)). This C depends on P and on the supremums of the derivatives in Condition 4.1.
. Introduce an operator Q : H 0 → H 0 such that κp(·, T ) = Qρ. By (6.6), the properties of Φ lead to the equality It suffices to show that the set {ρ − κp(·, T )} ρ∈H 0 is dense in H 0 . For this, it suffices to show that the equation ρ − Qρ = z is solvable in H 0 for any z ∈ H 0 .
Let us show that the operator Q : H 0 → H 0 is compact. Let p be the solution of (6.5). This means that κEp(·, T ) = Qρ. By Lemma 6.2, it follows that where C * > 0 is a constant that does not depend on p, s, and τ .
We have that p| t∈[s,T ] = M s p(·, s) for all s ∈ [0, T ], and, for τ > 0, for constants C i > 0 that do not depend on Φ. Hence the operator Q : H 0 → H 1 is continuous.
The embedding of H 1 into H 0 is a compact operator (see, e.g., Theorem 7.3 from Ladyzhenskaia (1985), Chapter I).
Without a loss of generality, we assume that there exist functionsβ i : andβ i has the similar properties as β i . (Note that, by Condition 2.1, 2b ≥ N i=1 β i β ⊤ i ). Letw(t) = (w 1 (t), . . . ,w M (t)) be a new Wiener process independent on w(t). Let a ∈ L 2 (Ω, F, P; R n ) be a vector such that a ∈ D. We assume also that a is independent from (w(t) − w(t 1 ), w(t) − w(t 1 )) for all t > t 1 > s. Let s ∈ [0, T ) be given. Consider the following Ito equation (6.10) Let y(t) = y x,s (t) be the solution of (6.10), and let τ x,s ∆ = inf{t ≥ s : y a,s (t) / ∈ D}. For Lemma 6.4 Let Conditions 4.1-4.2 hold. Let Φ ∈ L ∞ (Ω, F T , P, C 0 (D)), and let u ∈ Y 1 be solution of (6.1) with ϕ = 0. Then the process γ x, Proof of Lemma 6.4. For the case where u ∈ X 2 c and χ j ∈ X 1 c , this lemma follows from the proof of Lemma 4.1 from Dokuchaev (2011) (see Remark 6.1).
Let us consider the general case. Let ρ ∈ Z 0 s be such that ρ ≥ 0 a.e. and D ρ(x)dx = 1 a.s. Let a ∈ L 2 (Ω, F, P; R n ) be such that a ∈ D a.s. and it has the conditional probability density function ρ given F s . We assume that a is independent from (w(t 1 ) − w(t 0 ), w(t 1 ) − w(t 0 )}, s < t 0 < t 1 . Let p = M s ρ, and let y a,s (t) be the solution of Ito equation (6.10) with the initial condition y(s) = a.
To prove the theorem, it suffices to show that for any t. For this, it suffices to prove that for any ρ ∈ Z 0 s such as described above. By Theorem 6.1 from Dokuchaev (2011) and Remark 6.1, we have that This means that E(E t q(a, s, t)) = E(E T q(a, s, T )), where q(a, s, t) = γ a,s (t ∧ τ a,s )u(y a,s (t ∧ τ a,s ), t ∧ τ a,s ).
Therefore, E{u(S(T ∧ τ ), T ∧ τ )|F t } is the wealth for the self-financing strategy such that replicates ζ.
Hence condition (5.7) is satisfied. This completes the proof of Theorem 5.1.