Infinite-Dimensional FBSDE Analysis

• Infinite-dimensional FBSDEs are systems coupling forward processes in Banach/Hilbert spaces with backward stochastic equations, providing a probabilistic framework for complex PDEs.
• They employ generalized Dirichlet forms and infinite-dimensional stochastic calculus to ensure unique mild solutions and robust martingale representations.
• Applications include stochastic control, SPDE analysis, and delay/jump systems, illustrating their central role in modern infinite-dimensional stochastic analysis.

An infinite-dimensional forward-backward stochastic differential equation (FBSDE) is a system coupling a forward stochastic process, typically evolving in a Banach or Hilbert space, with a backward stochastic equation driven by possibly infinite-dimensional martingale additive functionals. Infinite-dimensional FBSDEs occur naturally in the probabilistic representation of nonlinear partial differential equations (PDEs) in infinite-dimensional state spaces, in stochastic optimal control, and in the analysis of stochastic partial differential equations (SPDEs). Such systems arise as the Markov process representation of generalized Dirichlet forms, in delay systems leading to Banach-valued states, and as functional analytic frameworks for SPDEs such as the Kardar-Parisi-Zhang equation.

1. Infinite-Dimensional Forward Process

The forward dynamics are formulated on a real, separable Banach space $E$, typically equipped with a densely embedded real separable Hilbert space $H \subset E$. The dynamics are characterized by a generalized Dirichlet form, which emerges from a formal operator $L$ of the type

$$L u(z) = \tfrac12 \mathrm{Tr}[A(z)D^2u(z)] + (A(z)b(z), Vu(z))_H,$$

where $A: E \rightarrow L_{\text{sym}}(H)$ is a measurable, nonnegative-definite, possibly degenerate operator and $b: E \rightarrow H$ is Borel-measurable. The associated non-symmetric bilinear form has the structure

$$ℰ(u,v) = \int_E (A(z)V u(z), V v(z))_H dp(z) + \int_E (A(z)b(z),V u(z))_H v(z) dp(z),$$

with $V$ the $H$-gradient and $FC_b^\infty$ the algebra of smooth, finitely-based cylinder functions. Under sector and positivity (A2–A3) conditions, the closure $H \subset E$0 is a generalized Dirichlet form yielding a contraction semigroup $H \subset E$1 on $H \subset E$2, whose generator is $H \subset E$3. The process is realized as a Hunt process $H \subset E$4, with transition function $H \subset E$5 for integrable $H \subset E$6 (Zhu, 2012).

2. Backward SDE in Infinite Dimension

Given a terminal condition $H \subset E$7 and driver $H \subset E$8, the backward component is a BSDE evolving along the trajectory of the forward process:

$H \subset E$9

where $L$0 is the $L$1-valued martingale additive functional (MAF) obtained by Fukushima’s decomposition of the process and $L$2 is a predictable process taking values in $L$3. The $L$4 term generalizes the Brownian integral to the infinite-dimensional martingale context. The existence of such a representation exploits the closure of coordinate forms and the domain $L$5 in the corresponding Dirichlet form setting (Zhu, 2012).

3. Analytic and Structural Hypotheses

The well-posedness of the infinite-dimensional FBSDE system requires:

• Lipschitz Continuity in $L$6: For all $L$7, $L$8.
• Monotonicity in $L$9: There exists $$L u(z) = \tfrac12 \mathrm{Tr}[A(z)D^2u(z)] + (A(z)b(z), Vu(z))_H,$$0 such that $$L u(z) = \tfrac12 \mathrm{Tr}[A(z)D^2u(z)] + (A(z)b(z), Vu(z))_H,$$1.
• Terminal Data: $$L u(z) = \tfrac12 \mathrm{Tr}[A(z)D^2u(z)] + (A(z)b(z), Vu(z))_H,$$2.
• Growth and Regularity: $$L u(z) = \tfrac12 \mathrm{Tr}[A(z)D^2u(z)] + (A(z)b(z), Vu(z))_H,$$3 is continuous in $$L u(z) = \tfrac12 \mathrm{Tr}[A(z)D^2u(z)] + (A(z)b(z), Vu(z))_H,$$4, measurable in $$L u(z) = \tfrac12 \mathrm{Tr}[A(z)D^2u(z)] + (A(z)b(z), Vu(z))_H,$$5, and grows at most polynomially on bounded $$L u(z) = \tfrac12 \mathrm{Tr}[A(z)D^2u(z)] + (A(z)b(z), Vu(z))_H,$$6-balls.
• Dirichlet Form Closability: The forms $$L u(z) = \tfrac12 \mathrm{Tr}[A(z)D^2u(z)] + (A(z)b(z), Vu(z))_H,$$7 must be closable in $$L u(z) = \tfrac12 \mathrm{Tr}[A(z)D^2u(z)] + (A(z)b(z), Vu(z))_H,$$8 (Zhu, 2012).

In the context of coupled systems with jumps or doubly stochastic drivers, monotonicity and coercivity conditions adapt to the product space and all involved operator coefficients; for example, see the monotonicity and terminal conditions (A1)–(A4) for the existence/uniqueness of FBDSDEJs in Hilbert spaces with Poisson noise (Al-Hussein, 2024).

4. Existence, Uniqueness, and Nonlinear Feynman-Kac Formula

Under the stated analytic and monotonicity conditions, one obtains both:

• Unique Solutions of the Nonlinear PDE: The quasi-linear backward evolution equation

$$L u(z) = \tfrac12 \mathrm{Tr}[A(z)D^2u(z)] + (A(z)b(z), Vu(z))_H,$$9

admits a unique generalized (mild) solution $A: E \rightarrow L_{\text{sym}}(H)$0, given by the variation-of-constants formula

$A: E \rightarrow L_{\text{sym}}(H)$1

• Unique Solutions of the Coupled BSDE: For every $A: E \rightarrow L_{\text{sym}}(H)$2 outside a polar set, the pair $A: E \rightarrow L_{\text{sym}}(H)$3 solving the BSDE exists uniquely in $A: E \rightarrow L_{\text{sym}}(H)$4 and realizes

$A: E \rightarrow L_{\text{sym}}(H)$5

This establishes the nonlinear infinite-dimensional Feynman-Kac correspondence, linking analytic PDE solutions to probabilistic BSDE representations (Zhu, 2012).

5. Martingale Representation and Infinite-Dimensional Stochastic Calculus

A central tool is the infinite-dimensional generalization of the Brownian martingale representation theorem:

• Any bounded $A: E \rightarrow L_{\text{sym}}(H)$6-measurable random variable $A: E \rightarrow L_{\text{sym}}(H)$7 admits a representation

$A: E \rightarrow L_{\text{sym}}(H)$8

where $A: E \rightarrow L_{\text{sym}}(H)$9 are coordinate martingale additive functionals associated to an orthonormal basis $b: E \rightarrow H$0 of $b: E \rightarrow H$1, and $b: E \rightarrow H$2 are predictable, square-integrable processes.

• Stochastic integration is with respect to these martingale components, replacing the Itô–Brownian integral in the infinite-dimensional setting.
• Fukushima's decomposition for $b: E \rightarrow H$3, with energy and covariation formulas, underpins the construction of the martingale and finite-variation parts of additive functionals (Zhu, 2012).

This structure permits the identification of the $b: E \rightarrow H$4-process in the BSDE as a functional of the gradient of the PDE solution.

6. Extensions: Delay, Jumps, and Nonlinear/Path-Dependent Generators

Infinite-dimensional FBSDE theory extends to:

• Delayed Systems: Where the state $b: E \rightarrow H$5 depends on its history, the FBSDE is recast in the path space $b: E \rightarrow H$6. Existence and explicit solutions for delayed, linear D-FBSDEs are constructed via discretization, backward induction, and passage to continuous infinite-dimensional Riccati–Volterra equations (Ma et al., 2020).
• Jumps and Nonlinearities: FBSDEs with jump components driven by Poisson measures, in both the forward and backward equations, admit existence and uniqueness under local monotonicity and Lipschitz conditions, with the solution in Hilbert product spaces (Al-Hussein, 2024, Bandini et al., 2018).
• Highly Nonlinear/SPDE Context: Non-Lipschitz drivers such as quadratic growth in the gradient, as in infinite-dimensional FBSDE representations of the KPZ equation, require renormalization, functional transforms (Cole–Hopf), and intricate martingale representation arguments. These frameworks yield rigorous probabilistic representations for SPDEs outside the reach of classical Lipschitz theory (Monter et al., 2012).

7. Stochastic Control and Viscosity Solutions in Infinite Dimensions

Infinite-dimensional FBSDEs form the foundation for BSDE-based representations of solutions to Hamilton–Jacobi–Bellman (HJB) equations in stochastic control of SPDEs:

• The value function of the stochastic control problem is represented as the solution $b: E \rightarrow H$7 to a suitably constructed infinite-dimensional BSDE, possibly with jump or path-dependent features (Bandini et al., 2018).
• In the Markovian case, the FBSDE solution coincides with the value of the associated nonlinear Feynman–Kac formula, and the BSDE structure guarantees that the value function is a viscosity solution (in the sense of Swiech–Zabczyk) of the fully nonlinear HJB–integro–PDE in the Hilbert space (Bandini et al., 2018).
• The randomized dynamic programming principle follows, providing a probabilistic foundation for infinite-dimensional dynamic programming and stochastic optimal control.

This broad theoretical scope, encompassing existence, uniqueness, representation, and applications to SPDEs and control, is emblematic of the central role of infinite-dimensional FBSDEs in modern stochastic analysis.