Backward Stochastic Difference Equations

• Backward stochastic difference equations (BSΔEs) are discrete analogues of BSDEs that model backward evolution of adapted random processes using martingale differences.
• They guarantee well-posed solutions through Lipschitz conditions, bijectivity in Y, and discrete-time martingale representation, ensuring uniqueness and stability.
• BSΔEs have practical applications in stochastic control, robust risk measurement, and financial pricing, extending naturally to coupled forward-backward systems.

Backward stochastic difference equations (BSΔEs) are discrete-time analogues of backward stochastic differential equations (BSDEs), designed to capture the conditional dynamics of stochastic systems on finite, countable, or lattice state spaces driven by martingale increments. The theory characterizes the evolution of adapted random processes backward in time, given a terminal condition, incorporating nonlinear drivers and often martingale representation principles. BSΔEs possess broad utility in stochastic control, financial mathematics, dynamic risk measurement, and equilibrium theory, extending naturally to fully coupled forward-backward systems, infinite-horizon formulations, ergodic optimization, and multi-agent applications.

1. Core Formulation and Existence Theory

Consider a filtered probability space supporting a discrete process (often a Markov or semi-Markov chain) and martingale differences $(M_{k+1})$. A canonical BSΔE on horizon $0,\ldots,T$ takes the form

$$Y_k = Q + \sum_{u=k}^{T-1} f(u, Y_u, Z_u) - \sum_{u=k}^{T-1} Z_u\Delta M_{u+1},\qquad Y_T = Q,$$

where $Q$ is an $\mathcal{F}_T$-measurable terminal value and $f$ is a driver function, possibly nonlinear in $Y$ and $Z$ (Elliott et al., 20 Dec 2025). Existence and uniqueness of adapted solutions are governed by the following foundational assumptions:

• Measurability and Adaptivity: $f(\omega, k, y, z)$ is $\mathcal{F}_k$-measurable in $\omega$.
• Invariance under $Z$-Equivalence: $f$ is insensitive to changes in $Z$ that produce indistinguishable martingale increments.
• Bijectivity in $Y$: The mapping $y \mapsto y - f(k, y, z)$ is a $\mathbb{R} \to \mathbb{R}$ bijection for each $(k,\omega,z)$.
• Lipschitz Bounds: For constants $L_y, L_z$, $$|f(k, y_1, z_1) - f(k, y_2, z_2)| \leq L_y|y_1 - y_2| + L_z\|z_1 - z_2\|$$.

Backward induction leverages discrete martingale representation: at each step, conditional expectations and orthogonal decomposition yield $Z_k$, followed by unique solvability for $Y_k$ using the bijection. Adaptedness and equivalence classes of $Z$ ensure well-posedness even on spaces supporting finite-state chains or lattices (Fukasawa et al., 2023, Allan et al., 2015).

2. Generalizations and Coupled Systems

BSΔEs serve as the backward components in more general forward-backward stochastic difference equations (FBSΔEs) on finite, lattice, or Markov-chain state spaces. The fully coupled system

$$\begin{aligned} \Delta X_t &= b(t,X_t,Y_t,Z_t) + \sigma(t,X_t,Y_t,Z_t)M_{t+1}, \ \Delta Y_t &= -f(t+1, X_{t+1}, Y_{t+1}, Z_{t+1}) + Z_t M_{t+1}, \end{aligned}$$

admits unique adapted solutions under monotonicity and Lipschitz conditions (Ji et al., 2019, Ma et al., 8 Jan 2025). Solvability in infinite horizon settings is established via domination-monotonicity conditions and exponential weighting, which control solution norms without terminal data and produce global existence and unique continuation properties (Ma et al., 8 Jan 2025). In lattice-based formulations, minimal noise structure allows explicit martingale representation and Feynman-Kac-type recursion for value functions and controls (Fukasawa et al., 2023).

3. Comparison Theorems and Robust Expectation

Comparison principles are a fundamental feature. For two BSΔEs, solutions $(Y^{(1)}, Z^{(1)})$ and $(Y^{(2)}, Z^{(2)})$ corresponding to drivers $f_1$ and $f_2$ and terminals $Q_1$, $Q_2$, monotonicity of data yields order-preserving solutions:

$$Q_1 \le Q_2,\quad f_1(k,Y^{(2)}_k,Z^{(2)}_k) \le f_2(k,Y^{(2)}_k,Z^{(2)}_k) \implies Y^{(1)}_k \le Y^{(2)}_k$$

given sufficiently small $L_z$ for positivity (Elliott et al., 20 Dec 2025, Allan et al., 2015, Bielecki et al., 2014). In lattice or concave driver settings, dual representations yield time-consistent robust nonlinear expectations ("g-expectations") via

$$Y_n = \min_{\dot{P}} \left\{ E^{\dot{P}}[Y_N | \mathcal{F}_n] + c_n(\dot{P}) \right\}$$

where $c_n(\dot{P})$ penalizes deviations from reference measures (Fukasawa et al., 2023, Bielecki et al., 2014).

4. Applications in Stochastic Control and Finance

BSΔEs play a pivotal role in discrete-time stochastic optimal control, furnishing adjoint equations and Hamiltonian verification principles. In control problems for processes driven by semi-Markov chains or fractional noise, the value function (performance index) coincides with the solution to a BSΔE with generator built as the supremum over Hamiltonians:

$$f(k,y,z) := \text{ess}\,\sup_{u \in U} f^u(k, y, z)$$

yielding the highest value function $Y_0 = \sup_u J(u)$, with any measurable $u^*$ attaining the supremum delivering optimal controls (Elliott et al., 20 Dec 2025, Han et al., 22 Dec 2024). Linear-quadratic (LQ) problems admit closed-loop feedback laws derived from explicit backward induction formulas and matrix inversion, ensuring explicit computation of optimal strategies (Han et al., 22 Dec 2024, Ma et al., 8 Jan 2025, Ji et al., 2019).

In financial mathematics, BSΔEs underpin nonlinear pricing under market incompleteness, robust risk measurement, and model-free pricing. They underlie dynamic convex risk measures and bid–ask pricing via conic finance, with properties such as monotonicity, cash translation invariance, and time-consistency ensured by the nonlinear g-expectation induced by a convex driver (Bielecki et al., 2014). Reflected BSΔEs (RBSΔEs) generalize to include optimal stopping and American contingent claim pricing via a "barrier" process, with solutions corresponding to nonlinear Snell envelopes (An et al., 2013).

5. Ergodic and Infinite Horizon Theory

Infinite-horizon BSΔEs extend the theory to ergodic cost optimization, often on uniform-ergodic Markov chains. The ergodic BSΔE is formulated:

$$Y_t = Y_T + \sum_{s=t}^{T-1} [ f(s, Y_{s+1}, Z_s) - \lambda ] - \sum_{s=t}^{T-1} Z_s^* M_{s+1}$$

for $t < T < \infty$, with $\lambda$ the ergodic constant (Allan et al., 2015). Existence and uniqueness follow from ergodicity estimates (e.g., Nummelin splitting) and contraction arguments. In control, application of ergodic BSΔEs yields long-run averages and optimal feedback in Markovian regimes.

Infinite-horizon fully coupled FBSΔEs, solved under domination-monotonicity and exponential weights, yield unique solutions and provide class-characterizations for LQ control schemes without terminal data (Ma et al., 8 Jan 2025). A priori estimates quantify solution norms, and continuation methods ensure solvability across parameter regimes.

6. Extensions: Fractional Noise, Set-Valued Risk, and Multivariate Systems

BSΔE theory accommodates non-classical noises, such as fractional Brownian increments. Fractional noise, orthogonalized via lower-triangular matrix decompositions (Cholesky-type), interacts with constructed martingale increments to yield BSΔEs with hybrid noise inputs. Recursive explicit formulas for both $Y_n$ and $Z_n$ apply; optimal control in LQ settings is recovered, with feedback and equilibrium characterized precisely (Han et al., 22 Dec 2024).

Set-valued risk measures in discrete time are represented via backward stochastic difference inclusions (BSDIs) and set-valued BSΔEs, extending scalar BSΔEs to multivariate systems and portfolios. BSDIs encode risk on selectors of capital allocations, while SV-BSΔEs propagate full risk sets backward, linking to dynamic programming and time consistency for multi-agent or regulatory regimes (Ararat et al., 2019).

7. Structural Properties, Computation, and Further Directions

BSΔEs exhibit time-consistency, convexity, and locality, and yield filtration-consistent nonlinear expectations and risk measures. Explicit backward induction recursions, martingale representation, and robust dual formulations provide computational frameworks, with applications ranging from dynamic risk measurement, superhedging, robust pricing, and multi-agent market equilibrium (Fukasawa et al., 2023, Bielecki et al., 2014).

The discrete-time setting provides mathematical simplification over continuous-time BSDEs, as conditional expectations, algebraic decompositions, and equivalence classes of $Z$ admit explicit solutions on sufficiently regular filtrations. Extensions to continuous time, higher-order couplings, and general noise structures remain active research areas, with technical challenges including predictable representation and measurable selection for infinite horizon and high-dimensional drivers (Ararat et al., 2019).

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References:

• (Elliott et al., 20 Dec 2025, Han et al., 22 Dec 2024, An et al., 2013, Fukasawa et al., 2023, Ji et al., 2019, Ji et al., 2018, Allan et al., 2015, Bielecki et al., 2014, Ararat et al., 2019, Ma et al., 8 Jan 2025).