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A recursive representation for decoupling time-state dependent jumps from jump-diffusion processes

Qinjing Qiu, Reiichiro Kawai

TL;DR

This work develops a recursive framework to decouple time-state dependent jumps from multivariate jump-diffusion processes, enabling weak approximation via iterates that are provably convergent at an exponential rate. By formulating two main schemes and a jump-suppressed representation, the authors obtain both convergent approximations and hard, computable bounds at each iteration, with optional Poisson thinning to ease computation under a uniformly bounded jump rate. The approach yields a Picard-type iteration that connects to PDE representations under suitable smoothness and ellipticity conditions, and the numerical illustrations demonstrate robust convergence and bounding behavior in both pure-jump and jump-diffusion settings. The framework is dimension-agnostic and holds potential for high-dimensional applications and integration with deep learning-based solvers, offering a principled route to decouple and analyze complex time-state dependent jump structures in stochastic systems.

Abstract

We establish a recursive representation that fully decouples jumps from a large class of multivariate inhomogeneous stochastic differential equations with jumps of general time-state dependent unbounded intensity, not of Lévy-driven type that essentially benefits a lot from independent and stationary increments. The recursive representation, along with a few related ones, are derived by making use of a jump time of the underlying dynamics as an information relay point in passing the past on to a previous iteration step to fill in the missing information on the unobserved trajectory ahead. We prove that the proposed recursive representations are convergent exponentially fast in the limit, and can be represented in a similar form to Picard iterates under the probability measure with its jump component suppressed. On the basis of each iterate, we construct upper and lower bounding functions that are also convergent towards the true solution as the iterations proceed. We provide numerical results to justify our theoretical findings.

A recursive representation for decoupling time-state dependent jumps from jump-diffusion processes

TL;DR

This work develops a recursive framework to decouple time-state dependent jumps from multivariate jump-diffusion processes, enabling weak approximation via iterates that are provably convergent at an exponential rate. By formulating two main schemes and a jump-suppressed representation, the authors obtain both convergent approximations and hard, computable bounds at each iteration, with optional Poisson thinning to ease computation under a uniformly bounded jump rate. The approach yields a Picard-type iteration that connects to PDE representations under suitable smoothness and ellipticity conditions, and the numerical illustrations demonstrate robust convergence and bounding behavior in both pure-jump and jump-diffusion settings. The framework is dimension-agnostic and holds potential for high-dimensional applications and integration with deep learning-based solvers, offering a principled route to decouple and analyze complex time-state dependent jump structures in stochastic systems.

Abstract

We establish a recursive representation that fully decouples jumps from a large class of multivariate inhomogeneous stochastic differential equations with jumps of general time-state dependent unbounded intensity, not of Lévy-driven type that essentially benefits a lot from independent and stationary increments. The recursive representation, along with a few related ones, are derived by making use of a jump time of the underlying dynamics as an information relay point in passing the past on to a previous iteration step to fill in the missing information on the unobserved trajectory ahead. We prove that the proposed recursive representations are convergent exponentially fast in the limit, and can be represented in a similar form to Picard iterates under the probability measure with its jump component suppressed. On the basis of each iterate, we construct upper and lower bounding functions that are also convergent towards the true solution as the iterations proceed. We provide numerical results to justify our theoretical findings.

Paper Structure

This paper contains 12 sections, 10 theorems, 123 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Main results
  4. Recursive representations by paths until the first few jumps
  5. Iterative approximation along two routes with monotonicity
  6. Recursive representations by decoupling jumps
  7. Hard bounding functions
  8. Uniformly bounded jump rate
  9. Numerical illustrations
  10. Pure-jump processes
  11. Survival probability with time-state dependent jump rate
  12. Proofs

Key Result

Theorem 3.1

It holds, as $m\to +\infty$, that $w_m(t,{\bf x})\to u(t,{\bf x})$ for all $(t,{\bf x})\in [0,T]\times \overline{D}$.

Figures (3)

  • Figure 1: Plots of the approximate solutions $\{w_m\}_{m\in\{1,2,3,4,5\}}$ at $t\in \{0,\,0.5\}$ and the associated upper bounding functions, according to \ref{['UL example 1']}, at the third, fourth and fifth iterations, with $m=1$ (orange dash-dot), $m=2$ (green dash-dot), $m=3$ (blue dash), $m=4$ (purple dash) and $m=5$ (red solid). The six black unfilled circles in each figure indicate (very accurate Monte Carlo estimates of) the true values $u(\cdot,x)$ at $x=\{0,1,2,3,4,5\}$.
  • Figure 2: Plots of the approximate solutions $\{w_m(0,x)\}_{m\in\{1,2,3,4,5\}}$, with $m=1$ (orange dash-dot), $m=2$ (green dash-dot), $m=3$ (blue dash), $m=4$ (purple dash) and $m=5$ (red solid). The six black unfilled circles in each figure indicate (very accurate Monte Carlo estimates of) the true values $u(0,x)$ at six states.
  • Figure 3: The approximate solutions $\{w_m(t,x)\}_{m\in\{0,1,2,3\}}$ and the associated upper and lower bounding functions at two timepoints $t\in \{0,\,0.5\}$ for $x\in [x_L,x_U]$ with $m=0$ (pink dash-dot), $m=1$ (orange dash-dot), $m=2$ (green dash) and $m=3$ (blue dash). The unfilled circles in (a) and (c) and the vertical whisker plots in (b) and (d) are, respectively, Monte Carlo estimates and 99% confidence intervals, constructed based on $10^5$ iid sample paths by the exact simulation method for jump-diffusion processes casellaroberts2011.

Theorems & Definitions (20)

  • Remark 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Proposition 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Theorem 3.7
  • Theorem 3.8
  • Theorem 3.9
  • ...and 10 more