Fractional Backward Stochastic Evolution Equations

• FBSEEs are backward stochastic evolution equations that incorporate fractional derivatives to model infinite-dimensional systems with hereditary and memory-dependent behaviors.
• They employ Caputo derivatives and fractional resolvent operators on Hilbert spaces to ensure well-posedness, existence, and uniqueness of mild solutions.
• FBSEEs are applied in optimal control, physics, and finance to capture long-range dependencies and complex dynamics in systems with delays.

Fractional Backward Stochastic Evolution Equations (FBSEEs) are a class of stochastic evolution equations posed backward in time and driven by non-Markovian (memory-dependent) fractional dynamics. They generalize backward stochastic differential equations (BSDEs) and backward stochastic evolution equations (BSEEs) by incorporating fractional (nonlocal-in-time) derivatives, thus enabling the modeling of infinite-dimensional systems with hereditary and long-range dependent behaviors. FBSEEs are formulated on separable Hilbert spaces and play a central role in the optimal control of systems, especially those encountered in mathematical physics and finance with memory effects.

1. Mathematical Formulation and Fractional Analytic Structure

The canonical FBSEE, in a separable Hilbert space $H$, is governed by a backward in time Caputo fractional derivative of order $\alpha \in (0,1)$: $${^{C}_t}D_T^\alpha Y(t) = A Y(t) + f\bigl(t, Y(t), Z(t), u(t)\bigr) + Z(t) \frac{dW(t)}{dt}, \quad Y(T) = \xi,$$ where:

• $A$ generates a strongly continuous ($C_0$) semigroup $\{S(t)\}_{t\ge0}$,
• $W(\cdot)$ is a $Q$-Wiener process on $H$,
• $f\colon [0,T]\times H\times L_2^0\times U \to H$ is the coefficient,
• $u(\cdot)$ is a control process,
• $Z(t)$ is an $L_2^0$-valued adapted process,
• $\xi$ is the terminal condition in $L^2(\Omega, \F_T; H)$.

The Caputo derivative is expressed as: $${^{C}_t}D_T^\alpha Y(t) = -\frac{1}{\Gamma(1-\alpha)} \int_t^T (s-t)^{-\alpha} Y'(s)\,ds$$ for right-sided derivatives. The mild solution is represented by fractional resolvent operators: $$Y(t) = S_{\alpha}(T-t)\xi + \int_t^T (s-t)^{\alpha-1} P_{\alpha}(s-t) f(s, Y(s), Z(s), u(s))\,ds - \int_t^T (s-t)^{\alpha-1}P_{\alpha}(s-t) Z(s)\,dW(s),$$ where $S_{\alpha}, P_{\alpha}$ are obtained via the Wright function convolution with $S(t)$ and satisfy uniform bounds in operator norm (Asadzade et al., 4 Jan 2026).

2. Functional Analytic Hypotheses and Solution Theory

Well-posedness of the FBSEE requires:

• $A:\Dom(A)\subset H\to H$ generates a $C_0$-semigroup ($S(t)$),
• $f$ is $\F_t$-measurable in $t$, $C^1$ in $(y, z, u)$ with bounded derivatives,
• Control set $U\subset\mathbb{R}^k$ is convex and closed,
• Terminal data $\xi\in L^2(\Omega,\F_T;H)$.

Existence and uniqueness of mild solutions leverage the boundedness of fractional resolvent operators and the contraction mapping principle in weighted Hilbert space norms. For $\alpha>\frac12$, moment estimates for the linearized FBSEE yield control over second and fourth moments of variations, necessary for stability and perturbation analysis (Asadzade et al., 4 Jan 2026, Mahmudov et al., 2021).

FBSEEs also fit as a special case of singular backward stochastic Volterra integral equations (BSVIEs) with singular kernels $k(t,s) = (t-s)^{-\alpha}$, provided the kernel satisfies global integrability and local smallness on subintervals. The general BSVIE framework results in unique adapted M-solutions under appropriate Lipschitz and integrability conditions on the coefficients and kernel (Wang et al., 2023).

3. Connection to Singular BSVIEs and the Volterra Structure

FBSEEs are embedded within the broader class of backward stochastic Volterra integral equations with singular kernels. Specifically, for given $A(s, y, z)$ and kernel $k(t,s) = (t-s)^{-\alpha}$, the FBSEE is written as: $$Y(t) = \xi + \int_t^T (t-s)^{-\alpha}A(s, Y(s), Z(t,s))\,ds - \int_t^T Z(t,s)\,dW(s), \quad 0 < \alpha < 1.$$ Adapted M-solutions satisfy a martingale representation form, which is crucial for uniqueness and probabilistic interpretation. Localized contraction arguments and patching over partitions of $[0,T]$ provide well-posedness in infinite-dimensional frameworks (Wang et al., 2023).

This Volterra-type structure allows the final cost functional or adjoint process to depend on the entire past state trajectory, capturing memory effects crucial in hereditary systems such as viscoelasticity or stochastic optimal control with delay (Wang et al., 2023).

4. Stochastic Maximum Principle and Adjoint Fractional Dynamics

The stochastic maximum principle (SMP) for FBSEEs is formulated via:

• Construction of a Hamiltonian that includes the fractional time-weight,
• Use of spike variation in the control to generate variational equations,
• Linearization yielding a variational FBSEE with zero terminal condition,
• Moment bounds on the variations, typically requiring $\alpha>\frac12$.

The adjoint equation is a forward FBSEE of fractional type: $$\psi(t) = S_{\alpha}(t) h_y(Y^0(0)) + \int_0^t (t-s)^{\alpha-1} P_{\alpha}(t-s)\{f_y^*[s]\,\psi(s) + l_y[s]\}\,ds + \cdots$$ The necessary optimality condition requires the Hamiltonian to achieve a pointwise maximum over admissible control values. In the case of open control sets $U$, this reduces to stationarity of the Hamiltonian gradient with respect to control (Asadzade et al., 4 Jan 2026).

For stochastic control problems where the state equation is a forward Caputo-fractional SEE, the natural adjoint equation is a backward singular BSVIE, confirming the duality and completeness of the SMP for fractional stochastic systems (Wang et al., 2023).

5. Linear-Quadratic (LQ) Control: Explicit Solutions and Fractional Riccati Equations

For LQ FBSEEs, where the dynamics are governed by linear operators and the cost is a quadratic form weighted with fractional kernels, optimal control is constructed explicitly via the adjoint process. The state-adjoint system is coupled: $$\begin{cases} {^{C}_t}D_T^\alpha Y(t) = [A - B R^{-1} B^* P(t)]\,Y(t) + C Z(t) + Z(t) \frac{dW(t)}{dt},\ {^{C}_t}D_T^\alpha \psi(t) = (A^* - P(t) B R^{-1} B^*) \psi(t) + Q Y(t), \end{cases}$$ with

$u^*(t) = - R^{-1} B^* P(t)\,Y(t).$

The feedback gain $P(\cdot)$ is a self-adjoint operator-valued solution to the fractional Riccati equation: $${^{C}_t}D_T^\alpha P(t) + A^* P(t) + P(t) A + Q - P(t) B R^{-1} B^* P(t) = 0, \quad P(T) = G,$$ with well-posedness established by fixed-point arguments in operator space, provided $R$ is coercive and the relevant operators are bounded (Asadzade et al., 4 Jan 2026).

6. Weighted Norms, Volterra Integral Equations, and Infinite-Dimensional Extensions

The analysis of FBSEEs utilizes weighted $L^2$ norms encoding the singular fractional kernels: $$\|Y\|^2_{2,\alpha,0} = \sup_{0\le \tau \le T} \mathbb{E}\left[ \int_\tau^T (s-\tau)^{\alpha-1} |Y(s)|^2\,ds \right].$$ These norms are essential in proving contraction for the mapping that produces the unique mild solution either directly in the case of Caputo-type equations or via equivalent stochastic Volterra integral formulations (Mahmudov et al., 2021).

The equivalence between mild solutions of FBSEEs and solutions to backward stochastic Volterra integral equations extends the theory to a broad class of memory-dependent problems, where classical comparison and maximum principles often fail (Mahmudov et al., 2021, Wang et al., 2023). Infinite-dimensional FBSEEs employ Hilbert-Schmidt norms for stochastic integrals when the noise is cylindrical over separable Hilbert spaces.

7. Relevance to Applied Fields and Open Problems

FBSEEs model infinite-dimensional systems with hereditary features such as viscoelastic materials, heat conduction with memory, and optimal control of distributed parameter systems affected by anomalous diffusion or long memory processes (Wang et al., 2023). In stochastic calculus of variations and financial mathematics, FBSEEs provide the rigorous adjoint process for stochastic maximum principles in non-Markovian and path-dependent frameworks.

Current research directions include:

• Development of maximum principles for broader non-linear and quadratic-growth generators,
• Extension to multi-dimensional and Banach-space settings,
• Handling of backward problems driven by general fractional kernels beyond power-law,
• Analysis of singular BSVIEs and their role in delay and hereditary optimization (Wang et al., 2023, Mahmudov et al., 2021).

The absence of comparison theorems for general fractional BSDEs remains a significant technical challenge, indicating an active frontier for both theoretical research and applications in stochastic control (Mahmudov et al., 2021).

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References:

• (Asadzade et al., 4 Jan 2026): Asadzade, J. A., Mahmudov, N. I., "Stochastic Maximum Principles and Linear-Quadratic Optimal Control Problems for Fractional Backward Stochastic Evolution Equations in Hilbert Spaces," 2026.
• (Wang et al., 2023): Wang, G., Zheng, H., "Singular backward stochastic Volterra integral equations in infinite dimensional spaces," 2023.
• (Mahmudov et al., 2021): Mahmudov, N. I., Ahmadova, G., "Some Results on Backward Stochastic Differential Equations of Fractional Order," 2021.