Multivalued BSDEs with Jumps

• Multivalued backward stochastic differential equations with jumps are defined by incorporating maximal monotone operators to enforce time-dependent constraints.
• They integrate both Brownian motion and jump processes, modeled via Poisson random measures, to capture system uncertainties and moving boundaries.
• Analytical techniques like penalization methods and Itô’s formula with jump components ensure the existence, uniqueness, and stability of solutions.

A multivalued backward stochastic differential equation (MBSDE) with jumps is a backward SDE in which the drift or reflection term is governed by a multivalued maximal monotone operator or, equivalently, a constraint that forces the unknown process to remain within a time-dependent (possibly random) domain. The noise driving the SDE includes both a Brownian motion and a jump source, typically modeled by a Poisson random measure. MBSDEs with jumps frequently arise in stochastic control, constrained optimization, and mathematical finance where solutions must remain feasible with respect to moving constraints or domains.

1. Mathematical Formulation

The general framework for MBSDEs with jumps is as follows. Consider a filtered probability space $(\Omega, \mathcal{F}, \mathbf{P})$ supporting:

• a $d$-dimensional Brownian motion $W$,
• an independent Poisson random measure $p(dt, de)$ on $[0, T] \times U$ (for some $U \subset \mathbb{R}^\ell \setminus \{0\}$),
• and its compensated martingale measure $\tilde{\mu}(dt, de) = p(dt, de) - \lambda(de) dt$.

The unknowns are quadruples $(Y, Z, V, K)$ of processes (whose precise regularity depends on the setting). The MBSDE with jumps in a time-dependent domain has the abstract form

$$\begin{aligned} Y_t &= \xi + \int_t^T f(s, Y_s, Z_s, V_s(\cdot)) ds - \int_t^T Z_s dW_s - \int_t^T\int_U V_s(e) \tilde{\mu}(ds, de) + K_T - K_t, \ Y_t &\in D_t \text{ for all } t, \ \int_0^T \langle Y_{s-} - x, dK_s \rangle &\le 0 \quad \forall\ x: x_s \in D_s, \end{aligned}$$

where $K$ is a finite-variation adapted process enforcing the constraint, and the Skorokhod-type condition ensures that the increments of $K$ are active only when $Y$ is on the boundary of $D_t$ (Fakhouri et al., 2015). In one dimension or with special domain geometry, the multivalued term can be written via maximal monotone operators $k_t(\cdot)$, leading to the generalized BSDE

$$Y_t + \int_t^T U_s ds = \xi + \int_t^T f(s, Y_s, Z_s, \psi_s) ds - \int_t^T Z_s dW_s - \int_t^T\int_{\mathcal{U}} \psi_s(e) \tilde{N}(ds, de),$$

with $U_t \in k_t(Y_t)$ (Elmansouri et al., 26 Nov 2025). Equivalently, the multivalued reflection is encoded by $dK_t \in \partial I_{D_t}(Y_t)\,dt$, where $\partial I_{D_t}$ is the subdifferential of the indicator of $D_t$.

2. Maximal Monotone Operators and Time-Dependent Domains

The mathematical representation of multivaluedness is via maximal monotone operators. In the multidimensional (convex domain) case, the operator corresponds to the inward-normal cone to $D_t$. In the one-dimensional setting relevant to moving boundaries, each time $t$ is associated with an operator $k_t$ induced by an increasing, right-continuous function $k(t, \cdot)$ with domain $$\mathcal{D}_t = \{x: x > a_t\} \cup \{a_t \text{ if } k(t, a_t) > -\infty\}$$, where $a_t$ is a (possibly random) lower boundary. The graph of $k_t$ is

$$\operatorname{Gr}(k_t) = \{ (x, y): x \in \mathcal{D}_t,\, y \in [k_-(t, x),\, k(t, x)] \},$$

with $k_-(t,x)$ being the left limit. Notably, maximal monotone structure guarantees well-posedness and admits penalization approaches in the existence proofs (Elmansouri et al., 26 Nov 2025).

For multidimensional domains, $D = (D_t)_{t \in [0, T]}$ is assumed closed, convex, nonempty-interior, $\mathcal{F}_t$-adapted, and continuous in the Hausdorff metric. This flexibility is essential to encode time-dependent constraints such as stochastic barriers or moving obstacles (Fakhouri et al., 2015).

3. Main Assumptions and Technical Conditions

The typical set of hypotheses for well-posedness consists of:

• Terminal condition: $\xi$ square-integrable, lying a.s. in the terminal domain, e.g., $\xi \in L^2$ with $\xi \in D_T$ (Fakhouri et al., 2015), or $\xi \geq a_T$ in the moving boundary setting (Elmansouri et al., 26 Nov 2025).
• Driver conditions: $f$ is measurable, square-integrable at zero, and Lipschitz in unknowns; for jump components, monotonicity in the jump parameter is required (e.g., for $\psi$, see (A.2)(iv)) and in general

$$\|f(t, y, z, v) - f(t, y', z', v')\| \leq L \bigl( \|y - y'\| + \|z - z'\| + \|v - v'\|_{L^2(U, \lambda)} \bigr)$$

(Fakhouri et al., 2015).

• Domain conditions: The family $D_t$ (or $a_t$ for 1D) is adapted, with regularity to ensure that projections $\Pi_{D_t}$ and Hausdorff-continuity arguments apply.
• Operator integrability: For moving boundaries, local-in-time $L^1$ and $L^2$ conditions for $k(t, y)$ on compact intervals above the lower boundary (see (B.1)-(B.2) in (Elmansouri et al., 26 Nov 2025)), supporting uniform estimates in penalized equations.
• Interior point condition: Existence of a suitable interior point process $A_t$ strictly inside $D_t$ for all $t$, with a nonzero minimum distance to the boundary, as in the multidimensional framework (Fakhouri et al., 2015).

4. Existence, Uniqueness, and Penalization Methods

The existence and uniqueness of solutions rely on approximating the multivalued problem with a sequence of standard BSDEs with single-valued penalized drivers, and then passing to the limit.

Penalization for Multidimensional RBSDEs:

Penalized equations take the form

$$\begin{cases} Y^n_t = \xi + \int_t^T f(s, Y^n_s, Z^n_s, V^n_s) ds - \int_t^T Z^n_s dW_s - \int_t^T\int_U V^n_s(e) \tilde{\mu}(ds, de) + (K^n_T - K^n_t), \ K^n_t = -n \int_0^t (Y^n_s - \Pi_{D_s}(Y^n_s)) ds, \end{cases}$$

where $\Pi_{D_s}$ is the Euclidean projection onto $D_s$ (Fakhouri et al., 2015). Existence and unique solvability for each $n$ is standard; convergence as $n \to \infty$ is established using uniform estimates, stability, and monotonicity.

Penalization for 1D Moving Boundary:

A monotone sequence of Lipschitz approximations $k_n$ to $k$ is constructed so that

$$Y^n_t = \xi + \int_t^T (f(s, Y^n_s, Z^n_s, \psi^n_s) - k_n(s, Y^n_s)) ds - \int_t^T Z^n_s dW_s - \int_t^T\int_U \psi^n_s(e) \tilde{N}(ds, de),$$

and the associated finite variation process is $K^n_t = -\int_0^t k_n(s, Y^n_s) ds$ (Elmansouri et al., 26 Nov 2025). The monotonicity ensures $Y^n$ increases to the solution, and that $Z^n, \psi^n, K^n$ converge in the appropriate norms.

Both frameworks utilize Itô’s formula (including jump terms and local times) to derive the required a priori estimates and establish convergence.

5. Core Estimates and Skorokhod-Type Conditions

A priori estimates are critical, both to ensure tightness in solution spaces and to pass to the multivalued limit. These typically take the form: $\E\left[ \sup_{0 \leq t \leq T} |Y_t|^2 + \int_0^T |Z_s|^2 ds + \int_0^T \int_U |V_s(e)|^2 \lambda(de) ds + |K_T|^2 \right] \leq C \cdot \text{(data norm)},$ where constants depend on the data and domain geometry (Fakhouri et al., 2015).

Skorokhod conditions generalize normal reflection to the multivalued/jump setup. For domains, the constraint is that

$$\int_0^T \langle Y_{s-} - x, dK_s \rangle \leq 0, \quad \forall\, x \text{ adapted with } x_s \in D_s,$$

ensuring that $K$ pushes minimally to confine $Y$ to $D_t$. In the maximal monotone formalism, this becomes the condition

$$\int_0^T (Y_s - \alpha_s)\, (dK_s + \beta_s ds) \leq 0 \quad \forall\, (\alpha_s, \beta_s) \in \text{Graph}(k_s),$$

encoding both reflection on the boundary and non-intrusive evolution in the interior (Elmansouri et al., 26 Nov 2025).

6. Extensions: Unbounded Domains and Local Theory

The theory accommodates unbounded time-varying domains provided growth-control conditions for $k(t, x)$ are imposed (cf. assumption (C) in (Elmansouri et al., 26 Nov 2025)), ensuring that penalization remains well-posed and solutions do not explode. The existence–uniqueness methodology is local-in-time on stopping intervals and then patched globally via monotonicity and a comparison principle.

Local results for RBSDEs with jumps on subintervals are also established in the multidimensional theory, allowing for patching of local solutions when the domain is only piecewise constant in time (Fakhouri et al., 2015).

7. Functional Settings and Notation

Common function spaces for well-posedness are as follows:

• $S^2(\mathbb{R}^m)$: càdlàg adapted processes $Y$ with $\E [\sup_{t \leq T} |Y_t|^2]<\infty$,
• $M^2(\mathbb{R}^{m \times d})$: predictable processes $Z$ with $\E \int_0^T \|Z_s\|^2 ds<\infty$,
• $L^2(\tilde{\mu}; \mathbb{R}^m)$: predictable $V(s,e)$ with $\E \int_0^T\int_U |V_s(e)|^2 \lambda(de) ds < \infty$,
• $A^2(\mathbb{R}^m)$: adapted, nondecreasing, vanishing at zero, $\E|K_T|^2<\infty$,
• Hausdorff metric for moving convex sets, and analogous spaces ($$\mathcal{S}^2, \mathcal{H}^2, \mathcal{H}^2_\pi, \mathcal{A}^2$$) in the 1D setting (Fakhouri et al., 2015, Elmansouri et al., 26 Nov 2025).

These frameworks allow uniform estimates to be derived and ensure the requisite compactness for passing to the limit in penalizations.

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For precise statements, assumptions, and proofs, see "Reflected backward stochastic differential equations with jumps in time-dependent random convex domains" (Fakhouri et al., 2015) and "Multivalued backward stochastic differential equations with jumps and moving boundary" (Elmansouri et al., 26 Nov 2025).