Stochastic differential equations (SDEs) are mathematical models used to describe the evolution of random phenomena in continuous time and space [1]. Forward-backward stochastic differential equations (FBSDEs) are a particular class of SDEs that involve both forward and backward stochastic processes. They arise naturally in a variety of fields, including finance, physics, and engineering [2]. Solving FBSDEs is a challenging task due to their inherent complexity and the need to handle both the forward and backward processes simultaneously. One approach to solving FBSDEs is the method of monotonicity, which has gained significant attention in recent years due to its ability to handle non-linearities and avoid numerical instability issues. The method of monotonicity was first introduce by Pardoux in his pioneering work [3] to solve stochastic partial differential equations, and later, further advancements were made by Krylov and Rozovskii [4], and Gyöngy [5]. The study of fully coupled FBSDEs under monotone type conditions were conducted in [6][7], and favorable results are obtained. It was further developed in [8][9].

Forward-backward stochastic partial differential equations (FBSPDEs) can be viewed as a natural extension of FBSDEs. In light of the nonlinear Feynman-Kac formula, or the Four Step Scheme [10], it is not hard to imagine that the solution of a backward SPDE could be a crucial device for solving an FBSDE with random coefficients [11][12]. It has been shown that the solvability of a large class of non-Markovian FBSDEs is almost equivalent to the solvability of the corresponding backward stochastic partial differential equations (BSPDEs) [13]. Therefore the solvability of FBSPDEs could be considered as part of the effort for a full understanding of the solvability of general strongly coupled FBSDEs with random coefficients. In addition, the interesting structure of FBSPDEs can be used to describe many natural phenomena. For instance, an application to the reaction-diffusion models is provided in [14]. The solvability of FBSPDEs has been attempted using the method of contraction mapping and method of continuation under various of conditions and assumptions in [15]. In [16], Yosida approximation scheme has been employed to establish the solvability of FBSPDEs even in the absence of the Lipschitz conditions. However, this is accomplished by assuming non-degeneracy and a special compatibility condition to compensate for the lack of Lipschitz continuity. Under the method of Galerkin approximation, a weak form solution of FBSPDEs was obtained in [17].

The present paper is distinguished from [14]-[17] chiefly in the method employed and in the resulting class of couplings covered. In [15] well-posedness is obtained by contraction mapping and continuation under a Lipschitz framework, but with the forward and backward equations coupled through Lipschitz coefficients rather than through the sharper, one-sided monotonicity structure used here; [16] trades the Lipschitz assumption for a non-degeneracy and compatibility condition via Yosida approximation, a route that excludes the degenerate diffusion coefficients allowed under monotonicity; and [17] produces only a weak (Galerkin) solution, without the strong, adapted solution obtained below. Relative to all of [14]-[17], the present contribution is threefold: 1) we adapt the monotonicity method of Pardoux [3] and Krylov-Rozovskii [4] - previously used for single-direction (forward-only or backward-only) SPDEs - to the fully coupled forward-backward setting, which requires the new one-sided monotonicity assumptions (A.2)-(A.3) and (B.3) on the coupling terms themselves, rather than only Lipschitz bounds on their size; 2) this structure yields existence and uniqueness under weaker regularity on the coefficients b , σ , k than the Lipschitz-in-all-variables assumptions of [15], since only σ and the diagonal terms of b , k need be controlled via (A.2), while the full Lipschitz bound (A.3) is only needed for the coupling; and (iii) the abstract formulation (4.1) in Sections 4-5, obtained via a method of continuation on a coupling parameter α ∈ [ 0 , 1 ] , subsumes both the Laplacian-type system (1.2) and the coercive-operator system (3.1) as special cases, broadening the class of admissible second-order (or more general continuous linear) operators beyond what the contraction arguments of [15] allow directly. In this paper, we would like to explore the method of monotonicity to FBSPDEs. Let us first start with the following FBSPDEs.

where a and h are positive constants, and G is a bounded domain in ℝ d with smooth boundary conditions. Let |   ⋅   | be the norm of L 2 ( G ) and ‖   ⋅   ‖ be the norm of H 0 1 ( G ) . They are given as follows:

and

For notational simplicity, the norm |   ⋅   | inside the integral signs is also used to denote the standard norm on ℝ n , n ∈ ℕ . Let H − 1 ( G ) be the dual space of H 0 1 ( G ) , and the normal of H − 1 ( G ) be denoted as ‖   ⋅   ‖ H − 1 . Denote 〈 ⋅ , ⋅ 〉 the inner product of L 2 ( G ) , 〈 ⋅ , ⋅ 〉 H the inner product of H 0 1 ( G ) , and 〈 ⋅ , ⋅ 〉 H − 1 , H the duality pairing between H − 1 ( G ) and H 0 1 ( G ) . For any u ∈ L 2 ( G ) , there exists an u ′ ∈ H − 1 ( G ) , such that 〈 u ′ , v 〉 H − 1 , H = 〈 u , v 〉 for all v ∈ H 0 1 ( G ) . The mapping u → u ′ is linear, injective, compact and continuous, and we can identify u ′ with u . In this sense, we identify ( L 2 ( G ) ) − 1 with L 2 ( G ) . Hence L 2 ( G ) is a dense subset of H − 1 ( G ) and we have evolution triple

Hence ‖ u ‖ H − 1 ≤ c | u | ≤ k ‖ u ‖ for all u ∈ H 0 1 ( G ) and some constants c and k .

Define the following operator

for any u ∈ L 2 ( G ) . Let { λ k } k = 1 ∞ be a family of nondecreasing unbounded positive numbers such that for each k , λ k is an eigenvalue of the operator ℒ . For every k ∈ ℕ , let e k ∈ H 0 1 ( G ) be a corresponding eigenfunction such that { e k } k = 1 ∞ forms an orthonormal basis of L 2 ( G ) . Let { q i } i = 1 ∞ be a family of positive numbers such that ∑ i = 1 ∞ q i < ∞ . Let our Wiener process W be defined as

where { B i ( t ) } is a sequence of iid Brownian motions in ℝ . Let Q be the operator from L 2 ( G ) to L 2 ( G ) such that Q e i = q i e i for all i . Note that Q is a self-adjoint and positive definite nuclear operator, and W is a L 2 ( G ) -valued Q Wiener process. Let L Q denote the space of all linear operators E such that E Q 1 2 is a Hilbert-Schmidt operator from L 2 ( G ) to L 2 ( G ) , with the inner product

for all F and G ∈ L Q .

Now let us suppress variables in u , v and W , and rewrite the system as follows:

The rest of the paper is organized as follows. The assumptions on monotonicity conditions and Lipschitz continuity are introduced in Section 2, and the well-posedness of system (1.2) are obtained. In Section 3, in the presence of coercivity assumption, similar results can be shown for system (3.1). In this system, second-order terms in (1.2) have been replaced by continuous linear operators. In sections 4 and 5, a more general system of FBSPDEs (4.1) has been introduced, and the existence and uniqueness of solutions of the system have been obtained.

Let us first provide the definition of an adaptive solution to (1.2).

Definition 2.1A triple ( u , v , Z ) is said to be a solution of(1.2)if it satisfies(1.2)P-a.s. and it is inthe space

Let us assume the following assumptions.

(A.1) For fixed u , v ∈ H 0 1 ( G ) and z ∈ L Q , the triple ( b , k , σ ) is in

and g ( u ) is in L ℱ T 2 ( Ω ; L 2 ( G ) ) .

(A.2) There exists a constant c 1 > 0 , such that for every t > 0 , u , u ′ , v , v ′ ∈ H 0 1 ( G ) , and z , z ′ ∈ L Q ,

and

(A.3) There exist constants c 2 , c 1 > 0 , such that for every t > 0 , u , u ′ , v , v ′ ∈ H 0 1 ( G ) and z , z ′ ∈ L Q ,

and

Lemma 2.2.Assume assumptions(A.1)and(A.3). Let ( u , v , z ) be a triple in

(1) If

the forward equation

has a unique adapted solution u in

(2) If

the backward equation

has a unique adapted solution ( v , Z ) in

Proof. We will prove the lemma by an application of the contraction mapping theorem. Let us first prove the existence and uniqueness of a solution of the forward equation. For any u , u ′ ∈ L ℱ 2 ( Ω ; L 2 ( 0 , T ; H 0 1 ( G ) ) ) , let u and u ' be the unique solutions of

and

respectively. Define

Applying the Itô formula to | u ^ | 2 and assumption (A.3) to get

Thus if

the mapping u ↦ u is a contraction, and the existence and uniqueness of a solution is guaranteed.

Now let us study the backward system. For any ( v , z ) and ( v ′ , z ′ ) in

let ( v , Z ) and ( v ′ , Z ′ ) be solutions of

and

respectively. Define

An application of the Itô formula to | v ^ | 2 and assumption (A.3) yield

Clearly that if

the backward admits a unique adapted solution. □

Remark 2.3.Lemma 2.2 is a decoupling device: with ( v , z ) frozen, the forward equation for u and the backward equation for ( v , Z ) are, individually, classical monotone SPDEs of the type first treated by Pardoux [3] and Krylov-Rozovskii [4], and assumptions (A.1) and (A.3) place b , σ , k exactly in the framework for which the well-posedness of such single-direction (forward-only or backward-only) SPDEs is classical. The contraction argument above is the standard way of recovering that well-posedness once the coefficients are frozen in the “other” variable; it plays the same role here that Picard-iteration on the frozen coefficients plays for the linear system in Lemma 4.2 below. The size restrictions on c 2 2 simply say that the Lipschitz constant of b (respectively k ) in u (respectively ( v , z ) ) must not overwhelm the coercivity constant a (respectively h ) of the leading operator ℒ - this is the usual competition, in the monotonicity method, between the dissipation supplied by the principal part and the growth of the lower-order coupling terms.

Under certain regularity conditions, we are able to prove the following result.

Theorem 2.4.Suppose assumptions(A.1),(A.2)and(A.3)hold. In addition, suppose that

The system (1.2) admits a unique adapted solution in the space

Proof. We are going to prove the theorem using the contraction mapping theorem. Let ( v , z ) and ( v ′ , z ′ ) be any two pairs from

Suppose that u and u ′ be the solutions of the forward equations

and

respectively. Then by solving backward equations

and

one obtains solutions ( v , Z ) and ( v ′ , Z ′ ) . Let

The remaining of the theorem is to show that

for some q < 1 . An application of the Itô formula yields

Thus assumptions (A.2) and (A.3) imply that

Then utilizing the Itô formula, assumptions (A.2) and (A.3), one gets

Substituting (2.2) into (2.3), and applying (A.3) to obtain

The two parts of condition (2.1) now play complementary roles: the requirement a h ≤ c 2 2 / ( 4 c 3 2 ) forces the coefficient c 3 2 − c 2 2 / ( 4 a h ) of E | u ^ ( T , x ) | 2 to be non-positive, so that term may be dropped, while the requirement c 2 4 ( 1 + 2 c 1 ) / ( 8 c 1 2 ) < a h makes the remaining coefficient c 2 4 ( 1 + 2 c 1 ) / ( 8 a h c 1 2 ) strictly less than 1. Hence

for some q ∈ ( 0 , 1 ) , and the mapping ( v , z ) ↦ ( v , Z ) is a contraction on L ℱ 2 ( Ω ; L 2 ( 0 , T ; H 0 1 ( G ) ) ) × L ℱ 2 ( Ω ; L 2 ( 0 , T ; L Q ) ) . Thus by (2.1) and the contraction mapping theorem, system (1.2) admits a unique adapted solution in the desired space. □

Remark 2.5. Condition (2.1) should be read as a window for the product a h of the two diffusion constants, sandwiched between a lower bound coming from the forward-to-backward coupling (through b , σ ) and an upper bound coming from the terminal coupling g : a h must be large enough that the iteration ( v , z ) ↦ ( v , Z ) in the proof of Lemma 2.2 contracts, yet not so large that the terminal condition v ( T , x ) = g ( u ( T , x ) ) destroys that contraction by feeding too much of the forward solution’s fluctuation back into v . This mirrors the well-known tension in fully coupled FBSDEs between the length of the time horizon and the size of the coupling: here the roles of “short horizon” and “weak coupling” are played by the sizes of a , h relative to c 2 and c 3 .

Let A and B be two continuous linear operators from H 0 1 ( G ) to H − 1 ( G ) with domains dense in L 2 ( G ) , such that the following coercivity assumption is held.

(A.4) For every u ∈ H 0 1 ( G ) , there exist constants α > 0 , β > 0 and γ , such that

Let us modify system (1.2) as follows to make it more general:

The following result is quite straightforward, and it concludes this section.

Theorem 3.1.Suppose assumptions(A.1),(A.2),(A.3)and(A.4)hold. In addition, suppose that

The system (3.1) admits a unique adapted solution in the space

Proof. Simply we apply the Itô formula to e − γ t | u ^ | 2 and e − γ t | v ^ | 2 . The rest of the proof is very similar to the proof of Theorem 2.4. □

Remark 3.2. The exponential weights e − γ t absorb the lower-order (non-coercive) part γ | u | 2 that (A.4) allows A and B to carry; this is the usual device for passing from a strictly coercive operator such as − a ℒ in (1.2) to a merely coercive-up-to-a-shift operator A . Theorem 3.1 therefore covers, in addition to second-order elliptic operators, e.g., non-symmetric or lower-order perturbations of ℒ , at the price of only bookkeeping the extra constant γ .

In the next two sections we are going to consider a more general system

with the following assumptions.

(B.1) For fixed u , v ∈ H 0 1 ( G ) and z ∈ L Q , the triple ( S , T , σ ) is in

and g ( u ) is in L ℱ T 2 ( Ω ; L 2 ( G ) ) .

(B.2) There exists a constant c 1 > 0 , such that for every t > 0 , u , u ′ , v , v ′ ∈ H 0 1 ( G ) and z , z ′ ∈ L Q ,

and

(B.3) There exists a constant c 2 > 0 , such that for every t > 0 , u , u ′ , v , v ′ ∈ H 0 1 ( G ) , and z , z ′ ∈ L Q ,

and

Let us prove some preliminary results.

Lemma 4.1.If a nonnegative sequence { a i } i = 0 ∞ satisfies

for all i ≥ 1 , then there exists a constant c , such that