Backward Stochastic Evolution Equations

• Backward stochastic evolution equations (BSEEs) are infinite-dimensional analogs of BSDEs defined on Banach or Hilbert spaces using cylindrical Wiener processes.
• They employ mild formulations with semigroup theory and γ-radonifying operators to ensure solution existence and uniqueness under Lipschitz and spectral conditions.
• Applications span stochastic control, SPDE analysis, and modeling complex systems with memory, delay, and singular kernel effects.

Backward stochastic evolution equations (BSEEs) are infinite-dimensional extensions of backward stochastic differential equations (BSDEs), formulated on Banach or Hilbert spaces and driven by stochastic processes, typically cylindrical Wiener processes. BSEEs generalize classical BSDEs by replacing finite-dimensional Itô processes with stochastic evolution equations governed by (possibly unbounded) linear operators, often in the context of stochastic partial differential equations (SPDEs). Central to control theory, filtering, stochastic analysis, and the theory of SPDEs, BSEEs serve as adjoint equations in the stochastic maximum principle, encode sensitivities in optimization, and appear in connections to singular Volterra-type kernels, anticipated equations, and set-valued inclusions.

1. Canonical Formulation and Mild Solutions

A backward stochastic evolution equation on a Banach or Hilbert space $X$ equipped with a strongly continuous semigroup $S(t)=e^{-tA}$, is typically written as

$$\begin{cases} dU(t) + A\,U(t)\,dt = f(t, U(t),V(t))\,dt + V(t)\,dW(t), & t\in[0,T],\ U(T) = u_T, \end{cases}$$

where $A$ is (minus) the generator of a $C_0$-semigroup $\{S(t)\}$, $W$ is a Brownian motion, and $f$ is a general driver (Lü et al., 2018). The mild solution is given by

$$U(t) = S(T-t)u_T + \int_t^T S(s-t)f(s,U(s),V(s))\,ds + \int_t^T S(s-t) V(s)\,dW(s).$$

The extension to multivalued generators yields backward stochastic evolution inclusions (BSEIs): $$dY_t + A Y_t\,dt \in G(t, Y_t, Z_t)\,dt + Z_t\,dW_t,\quad Y_T = \xi,$$ with measurable, closed-valued $S(t)=e^{-tA}$0, where a mild $S(t)=e^{-tA}$1-solution consists of processes $S(t)=e^{-tA}$2 and a selector $S(t)=e^{-tA}$3 satisfying an inclusion version of the above mild identity (Essaky et al., 2022).

The solution framework requires stochastic integration in Banach spaces, notably utilizing $S(t)=e^{-tA}$4-radonifying operators to define stochastic integrals when $S(t)=e^{-tA}$5 is a UMD Banach space (Lü et al., 2018).

2. Function Space Setting and Solvability Conditions

Existence and uniqueness of BSEE solutions hinge on the underlying functional setting. In UMD Banach spaces (where the Hilbert transform extends to vector-valued $S(t)=e^{-tA}$6), the Itô isomorphism for $S(t)=e^{-tA}$7-radonifying-valued processes is crucial (Lü et al., 2018). The following network of conditions typically appears:

• Generator and potential regularity: The driver $S(t)=e^{-tA}$8 is required to be Lipschitz in the unknowns $S(t)=e^{-tA}$9 and jointly measurable (Lü et al., 2018, Essaky et al., 2022).
• Semigroup properties: The family $$\begin{cases} dU(t) + A\,U(t)\,dt = f(t, U(t),V(t))\,dt + V(t)\,dW(t), & t\in[0,T],\ U(T) = u_T, \end{cases}$$0 must be $$\begin{cases} dU(t) + A\,U(t)\,dt = f(t, U(t),V(t))\,dt + V(t)\,dW(t), & t\in[0,T],\ U(T) = u_T, \end{cases}$$1-bounded—the upper contraction property suffices (Lü et al., 2018, Essaky et al., 2022).
• Terminal condition: The data $$\begin{cases} dU(t) + A\,U(t)\,dt = f(t, U(t),V(t))\,dt + V(t)\,dW(t), & t\in[0,T],\ U(T) = u_T, \end{cases}$$2 or $$\begin{cases} dU(t) + A\,U(t)\,dt = f(t, U(t),V(t))\,dt + V(t)\,dW(t), & t\in[0,T],\ U(T) = u_T, \end{cases}$$3 belongs to a suitable $$\begin{cases} dU(t) + A\,U(t)\,dt = f(t, U(t),V(t))\,dt + V(t)\,dW(t), & t\in[0,T],\ U(T) = u_T, \end{cases}$$4 space, possibly equipped with the filtration and the Banach space norm.

When $$\begin{cases} dU(t) + A\,U(t)\,dt = f(t, U(t),V(t))\,dt + V(t)\,dW(t), & t\in[0,T],\ U(T) = u_T, \end{cases}$$5 is a Hilbert space, the integral spaces simplify, and classical $$\begin{cases} dU(t) + A\,U(t)\,dt = f(t, U(t),V(t))\,dt + V(t)\,dW(t), & t\in[0,T],\ U(T) = u_T, \end{cases}$$6-theory can be deployed (Al-Hussein, 2012).

For BSEIs (set-valued case), additional assumptions include the nonemptiness and closedness of the set of selectors, and a global Hausdorff–Lipschitz condition on $$\begin{cases} dU(t) + A\,U(t)\,dt = f(t, U(t),V(t))\,dt + V(t)\,dW(t), & t\in[0,T],\ U(T) = u_T, \end{cases}$$7 (Essaky et al., 2022).

Existence and uniqueness under these hypotheses follow via contraction mapping arguments in the relevant process spaces (such as $$\begin{cases} dU(t) + A\,U(t)\,dt = f(t, U(t),V(t))\,dt + V(t)\,dW(t), & t\in[0,T],\ U(T) = u_T, \end{cases}$$8), often constructed on short backward intervals and patched globally, exploiting the smoothing or contraction properties of the semigroup and stochastic integral (Lü et al., 2018, Essaky et al., 2022).

3. Extensions: Inclusions, Anticipated and Volterra-Type Equations

BSEEs extend beyond single-valued, non-anticipative forms. Significant generalizations include:

• Backward Stochastic Evolution Inclusions (BSEI): With set-valued generator $$\begin{cases} dU(t) + A\,U(t)\,dt = f(t, U(t),V(t))\,dt + V(t)\,dW(t), & t\in[0,T],\ U(T) = u_T, \end{cases}$$9, the problem becomes a mild inclusion in the space of adapted $A$0-radonifying maps. The theory in UMD Banach spaces accounts for measurable selectors and set-valued Hausdorff–Lipschitz continuity (Essaky et al., 2022).
• Anticipated Backward Stochastic Evolution Equations (ABSEE): These arise as adjoints in path-dependent stochastic control, featuring conditional expectations of non-anticipative derivatives that require "anticipated" arguments; the adjoint equation is posed over a time interval potentially extended by delay or memory effects (Liu et al., 26 Apr 2025).
• Volterra and Fractional-Type BSEEs/BSVIEs: In singular backward stochastic Volterra integral equations (BSVIEs), kernels can be singular, e.g., of fractional or rough-Heston type, with the equation driven by two-parameter processes $A$1 (Wang et al., 2023). Fractional BSEEs model systems with long memory, using Caputo-type kernels and fractional resolvent operators (Asadzade et al., 4 Jan 2026).

These generalizations support modeling of systems with memory, delay, and set-valued or nonlocal effects, common in physics (viscoelasticity, heat conduction), finance (rough volatility), and control with state-dependent delays.

4. Duality, Transposition, and Operator-Valued Backward Equations

In infinite-dimensional stochastic control, BSEEs are the adjoint equations in the stochastic maximum principle (SMP), encoding the sensitivity of cost functionals to state perturbations. Vector-valued BSEEs model first-order adjoints, while operator-valued BSEEs describe second-order variations (for nonconvex control or diffusion-control-dependence) (Lü et al., 2012, Lu et al., 2014).

For general filtrations or operator-valued backward equations, the transposition method is adopted: the solution is defined via duality against a family of forward SDEs, not as classically defined processes. For example, the solution $A$2 is realized by verifying, for all test forward solutions $A$3, a duality identity linking the adjoint and forward processes (Lu et al., 2014). In the operator-valued case, relaxed transposition solutions are constructed using limiting arguments to deal with lack of strong topology or only weak sequential compactness in $A$4 (Lü et al., 2012, Lu et al., 2014).

Tables can summarize solution types:

Setting	Solution Concept	Key Reference
Hilbert/UMD space	Mild, classical	(Lü et al., 2018)
Set-valued generator	Adapted selector, mild inc.	(Essaky et al., 2022)
General filtration	Transposition (dual)	(Lü et al., 2012, Lu et al., 2014)
Operator-valued BSEE	(Relaxed) transposition	(Lü et al., 2012, Lu et al., 2014)
Singular kernel	M-solution (BSVIE)	(Wang et al., 2023)

5. Applications: Optimal Control, Observability, and Memory Systems

BSEEs are integral in both stochastic control theory and analysis of SPDEs:

• Stochastic Maximum Principle (SMP): In infinite dimensions, the first- and second-order adjoints for control problems are given by vector- and operator-valued BSEEs, respectively. For convex/regular cases, a vector BSEE suffices; nonconvexity or control in the diffusion mandates the operator-valued backward equation in the transposition sense (Lü et al., 2012, Lu et al., 2014, Meng et al., 2016). SMPs for fractional and anticipated equations rely similarly on forward–backward systems involving fractional- or delay-type BSEEs (Liu et al., 26 Apr 2025, Asadzade et al., 4 Jan 2026, Wang et al., 2023).
• Observability and Null Controllability: Observability inequalities for BSEEs, derived via spectral methods and telescoping arguments, yield duality-based control results for forward SPDEs. Sharp observability estimates in backward equations with degenerate, fourth-order, or heat operators underpin null controllability of corresponding forward equations with minimal controls (Liu et al., 2023).
• Systems with Memory and Delay: BSEEs with fractional or Volterra kernels model systems with hereditary properties. The memory effect appears naturally in the structure of the BSEE's solution, typically via convolution with singular kernels, encoding long-range temporal dependence (Wang et al., 2023, Asadzade et al., 4 Jan 2026).

6. Technical Frameworks and Proof Techniques

The resolution of BSEEs relies on various advanced analytic and probabilistic tools:

• $A$5-Radonifying Operator Techniques: Crucial for defining stochastic integrals in UMD spaces and for transferring Hilbert space martingale inequalities to wider Banach-space contexts (Lü et al., 2018, Essaky et al., 2022).
• Contraction Mappings and Local-Global Arguments: Existence by Picard iteration or contraction on suitable spaces, with time-localized patching (Lü et al., 2018, Essaky et al., 2022, Hu et al., 2017).
• Duality and Riesz Representation: The operator- and vector-valued transposition methods hinge on the representation theorems in vector-valued Bochner spaces to derive dual process identities (Lü et al., 2012, Lu et al., 2014).
• Banach–Alaoglu and Weak Compactness: Relaxed or generalized solutions are constructed via weak convergence and diagonalization in spaces of operators (Lü et al., 2012).
• Weighted a priori estimates: Singular kernels, fractional equations, and space-time white noise require delicate norm estimates with time-dependent weights, Gronwall-type lemmas, and decay bounds (Hu et al., 2017, Asadzade et al., 4 Jan 2026, Wang et al., 2023).

7. Open Directions and Further Extensions

Recent literature points to multiple active research directions:

• Extension to Non-UMD and Non-Hilbert Spaces: Full well-posedness theory for operator-valued BSEEs in Banach spaces remains open; analysis is currently limited to Hilbert (or UMD-lattice) settings (Lü et al., 2012, Lu et al., 2014).
• Weakening of Spectral and Growth Assumptions: Extensions to non-Lipschitz or monotone drivers, non-smoothing semigroups, and degenerate generators are partly open (Hu et al., 2017).
• Stochastic Systems with Path-Dependence: ABSEEs and BSVIEs represent an expanding class of backward objects adapted for delay- and memory-dependent stochastic systems, with applications to infinite-dimensional control and finance (Liu et al., 26 Apr 2025, Wang et al., 2023).

BSEEs thus represent a central and flexible tool, both as structural ingredients in infinite-dimensional stochastic optimization and as primary objects of analysis in the study of SPDEs with backward, inclusion, anticipated, or singular-memory features.