Forward-Referencing Jumps Task (FRJT)

• FRJT is a framework for modeling stochastic control problems with jumps using coupled forward-backward SDEs and Poisson measures.
• It integrates dynamic programming, viscosity solutions, and maximum principles to formulate optimality conditions under nonconvex and progressive control structures.
• Numerical methods, including asymptotic expansion, linear regression, and machine learning-based schemes, enable scalable solutions in high-dimensional settings.

The Forward-Referencing Jumps Task (FRJT) concerns the rigorous modeling, analysis, and numerical solution of stochastic control and optimization problems where random jumps are essential features of the system dynamics, cost functional, or required solution concept. It encompasses the theory and computation of forward-backward stochastic differential equations (FBSDEs) and related control structures under the presence of jumps, compensated Poisson random measures, and progressively measurable control inputs. FRJT is foundational in fields such as mathematical finance, engineering systems subject to abrupt changes, non-linear filtering, and stochastic games with discontinuities.

1. Mathematical Formulation of State and Cost Dynamics with Jumps

Central to FRJT are coupled forward and backward stochastic differential equations driven by both Brownian motion and compensated Poisson random measures, allowing explicit modeling of jumps in the system’s state and evolution:

• Forward SDE (state process):

$$dX_t = b(t, X_t, u_t) dt + \sigma(t, X_t, u_t) dW_t + \int_{\mathcal{E}} \gamma(t, X_{t-}, u_t, e) \tilde{N}(dt,de)$$

where $X_t \in \mathbb{R}^n$; $u_t$ is the control; $W_t$ is Brownian motion; $\tilde{N}(dt,de)$ is the compensated Poisson random measure over jumps $e$.

• Backward SDE (costate or value process):

$$dY_t = -f(t, X_t, Y_t, Z_t, u_t) dt + Z_t dW_t + \int_{\mathcal{E}} K_t(e) \tilde{N}(dt, de)$$

with $Y_T = g(X_T)$ as the terminal condition.

The system is fully coupled and the control $u_t$ appears in all coefficients, including the jump term, requiring advanced techniques for both analysis and computation (Lin, 2011, Wang et al., 5 Jul 2024).

2. Deterministic Value Function and Dynamic Programming Principle

The optimization objective is typically to minimize a cost functional of the form: $$J(u) = \mathbb{E}\left[ \int_0^T \ell(t, X_t, u_t) dt + g(X_T) \right]$$ with the corresponding value function defined by

$$V(t, x) = \inf_u \mathbb{E}\left[ \int_t^T \ell(s, X_s, u_s) ds + g(X_T) \mid X_t = x \right]$$

A critical property established in FRJT is the deterministic nature of $V(t,x)$, enabling the rigorous application of the dynamic programming principle (DPP): $$V(t,x) = \inf_u \mathbb{E}\left[ \int_t^{t+h} \ell(s, X_s, u_s) ds + V(t+h, X_{t+h}) \mid X_t = x \right]$$ This links the stochastic optimization problem to analytic partial (integro-)differential equations (Lin, 2011).

3. Hamilton–Jacobi–Bellman Equations and Viscosity Solutions

Application of the DPP yields Hamilton–Jacobi–Bellman (HJB) equations with integral–differential operators to accommodate the jump part: $$0 = \sup_{u \in U} \Big\{ -\partial_t V - b \cdot D_x V - \frac{1}{2} \mathrm{Tr}\left[\sigma \sigma^T D_{xx}^2 V\right] - \int_{\mathcal{E}} [V(x+\gamma)-V(x)-D_x V \cdot \gamma] \lambda(de) - \ell \Big\}$$ with $V(T,x) = g(x)$.

Because classical solutions are often ruled out by jumps and nonlinearity, FRJT theory relies on viscosity solutions for well-posedness:

• Uniqueness and stability: Under technical conditions, viscosity solutions are unique and stable.
• Convergence: They justify numerical approximation schemes for HJB and PIDEs with jumps. This framework underpins theoretical and computational work in FRJT (Lin, 2011).

4. Maximum Principles for Progressive and Nonconvex Controls

Beyond the DPP/HJB approach, FRJT employs stochastic maximum principles (SMP) for deriving first-order necessary optimality conditions. Notable advances include:

• Maximum Principle for Doubly Stochastic FBSDEs with Jumps: Establishes sufficient conditions for optimality via the Hamiltonian maximization and adjoint processes, extending classical Pontryagin-type results (Al-Hussein et al., 2013).
• General Maximum Principle in Progressive Framework: Utilizes spike variation techniques to treat nonconvex control domains and progressive measurability, resulting in optimality conditions where the backward solution component $Z$ depends explicitly on the jump variable $e$:

$$H(t, X_t, Y_t, Z(t,\cdot), u^*, p_t, q(t,\cdot)) \geq H(t, X_t, Y_t, Z(t,\cdot), u, p_t, q(t,\cdot)), \quad \forall u$$

This provides a robust variational approach even when the control affects all coefficients, including jumps, and is essential for systems where full adaptation to jump events is required (Wang et al., 5 Jul 2024).

5. Progressive Structure, Jump-Time Optimality, and Impulse Controls

Classical predictable control frameworks cannot react at the jump instants, as controls are freeze just before jumps. FRJT leverages progressive control structures, enabling systems to respond precisely at jump times:

• Three-part SMP decomposition:

1. Continuous-time optimality condition (holds $\nu$-a.s.)
2. Jump-time optimality condition (holds $\mu$-a.s.)
3. Impulse control optimality condition (discrete sum over impulses) Such decompositions allow accurate characterization and adjustment of optimal control at both regular and jump/impulse times, delivering strictly better cost performance than predictable controls—explicitly demonstrated in linear-quadratic examples (Chen et al., 2023).

6. Numerical Methods and Machine Learning-Based Approaches

Addressing numerical solution of high-dimensional FRJT poses significant technical challenges, especially with jumps. Several converged and scalable schemes have been established:

• Asymptotic Expansion (Semi-analytic Approximation):
  • Expands FBSDE solutions as polynomial series around small-variance limit.
  • Higher-order coefficients computed via linear ODE systems.
  • Rigorous $S^p$ error bounds; explicitly accommodates state-dependent (non-Poissonian) jumps via change of measure (Fujii et al., 2015).
• Linear Regression Methods:
  • Discretize time and project conditional expectations onto basis functions via least-squares regression.
  • Demonstrated convergence and robustness to jump effects in extensive numerical testing (Ye et al., 2018).
• Machine Learning-Enabled Forward Schemes:
  • Decouples jump terms, recasts the PIDE as a nonlinear PDE, and applies recursive Picard iteration to linearize.
  • Implements conditional expectations either via Least Squares Monte Carlo or neural networks (affine/ReLU layers, Adam optimization, batch normalization).
  • Achieves exponential convergence in iteration index $m$:

  $$\sup_{t,x} |u(t,x) - w_m(t,x)| \leq C 2^{-m}(1 \wedge \eta^m)$$

  with fast error decay for lower jump intensities; scaling demonstrated up to 100 dimensions (Kawai et al., 12 Oct 2024).

Methodology	Handles Jumps	Dimensionality	Error Bound / Convergence
Asymptotic Expansion (Fujii et al., 2015)	Yes	High	$O(\varepsilon^{N+1})$
Linear Regression (Ye et al., 2018)	Yes	Moderate	Convergent via regression
ML-based Forward Scheme (Kawai et al., 12 Oct 2024)	Yes	Very High	$C 2^{-m}(1 \wedge \eta^m)$

7. Extensions: Environment-Dependent Jumps and Regime Switching

FRJT theory is naturally extended to random environments and regime-switching models:

• FBSDEs in Random Environments:

  • Jump intensity modulated by exogenous measure-valued processes (environmental signals).
  • Existence and uniqueness established under general monotonicity conditions, robust to rare or state-dependent jump arrivals.
  • Directly incorporates Cox and Hawkes processes, allowing modeling of self-exciting and environmental jump effects (Hernández-Hernández et al., 2023).
• Regime-switching Conditional McKean-Vlasov Equations:
  • Captures joint state and regime dynamics, regime transitions via jump events, offering methodologies for multi-agent control, systemic risk, and mean-field games.

8. Applications and Impact

FRJT provides the analytic and computational foundation for:

• Optimal control and risk management for discontinuous systems and markets
• Stochastic games and mean-field control with abrupt transitions
• High-frequency finance (option pricing, counterparty risk, mean-variance hedging)
• Engineering systems with fault detection and rapid adaptation
• Probabilistic representations for non-linear (integro-)PDEs and stochastic partial differential equations related to jumps

The synergy between dynamic programming, maximum principles (including progressive formulations), viscosity theory, and modern computational techniques marks FRJT as a unifying framework for contemporary stochastic optimization in systems subject to unpredictable sharp changes.