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Pathwise Convergence of a Modular Simulation Scheme for Hybrid SDEs with Memory

Oscar Peralta

Abstract

This work introduces hybrid stochastic differential equations with memory (mH-SDEs), a new class of stochastic systems where transition rates depend on the joint history of both Euclidean and discrete components. This extends existing hybrid stochastic differential equation models that condition transitions only on the Euclidean process history, enabling richer dependencies such as age-based transitions and self-reinforcing dynamics. For mH-SDEs driven by Lévy processes, we develop the Modular-Poisson algorithm, which employs path-dependent uniformization to generate discrete jumps while advancing the Euclidean evolution between jumps using any established SDE solver as a micro-algorithm. The main theoretical contribution establishes pathwise convergence with explicit rates, developing new techniques to control error accumulation across regime changes and bound the probability of process decoupling. The modular design allows practitioners to employ existing, well-studied SDE simulation methods while preserving theoretical convergence properties. This work provides the first rigorous convergence analysis for hybrid systems with joint history dependence under jump-diffusion dynamics.

Pathwise Convergence of a Modular Simulation Scheme for Hybrid SDEs with Memory

Abstract

This work introduces hybrid stochastic differential equations with memory (mH-SDEs), a new class of stochastic systems where transition rates depend on the joint history of both Euclidean and discrete components. This extends existing hybrid stochastic differential equation models that condition transitions only on the Euclidean process history, enabling richer dependencies such as age-based transitions and self-reinforcing dynamics. For mH-SDEs driven by Lévy processes, we develop the Modular-Poisson algorithm, which employs path-dependent uniformization to generate discrete jumps while advancing the Euclidean evolution between jumps using any established SDE solver as a micro-algorithm. The main theoretical contribution establishes pathwise convergence with explicit rates, developing new techniques to control error accumulation across regime changes and bound the probability of process decoupling. The modular design allows practitioners to employ existing, well-studied SDE simulation methods while preserving theoretical convergence properties. This work provides the first rigorous convergence analysis for hybrid systems with joint history dependence under jump-diffusion dynamics.

Paper Structure

This paper contains 19 sections, 6 theorems, 65 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model Definition
  3. Illustrative Applications of the Framework
  4. Regime Transitions Driven by Euclidean Path Functionals
  5. Age-Dependent and Semi-Markov Transitions
  6. Transitions Based on the Euclidean Jump Activity
  7. Self-Referential Dynamics and Path Reinforcement
  8. Higher-Order Memory Without State Space Augmentation
  9. Path Reinforcement and Self-Excitation Mechanisms
  10. The Modular-Poisson Algorithm and Convergence
  11. Algorithmic Structure and Convergence Rate
  12. Proof of Theorem \ref{['theo:rateofconvergence']}
  13. Advantages of the Simulation Approach
  14. Conclusion
  15. Numerical Examples
  16. ...and 4 more sections

Key Result

Theorem 2.1

Under Assumption ass:main_construction, there exists a unique strong solution $Y=(J, X)$ to the system eq:hybrid_J–eq:hybrid_X.

Figures (4)

  • Figure A.1: Insurance reserve simulation over 10 years starting from $X_0 = 0.9$ (below regulatory minimum). Top: reserve levels with regulatory threshold (red line) and breach periods (shaded). Middle: market regime evolution showing 6 transitions. Bottom: risk indicators driving transitions.
  • Figure A.2: Semi-Markov reliability system over 15 years with 10 component replacements. Top: active component transitions between high-reliability (State 0) and standard (State 1) components. Middle: component age evolution showing sawtooth pattern with resets at replacements and key age thresholds. Bottom: instantaneous failure rates displaying bathtub curve behavior for component 0 (purple) and linear wear for component 1 (red), all bounded by 2.0.
  • Figure A.3: Lévy-driven financial model over 10 years showing 8 regime changes. Top: asset price evolution with dramatic discontinuous jumps (vertical red lines) driven by the compound Poisson process. Middle: market regime transitions between bull (State 0) and bear (State 1) markets. Bottom: rolling window counts of catastrophic positive jumps (green) and negative jumps (red) exceeding 15% thresholds that drive regime changes.
  • Figure A.4: Path reinforcement model over 20 time units showing 10 mode changes. Top: production mode evolution between standard (State 0) and optimized (State 1) with clustering of transitions in early periods. Middle: accumulated time in current mode displaying sawtooth pattern with vertex reinforcement effects. Bottom: instantaneous transition rates increasing through edge reinforcement, bounded by $\lambda = 2.0$.

Theorems & Definitions (15)

  • Remark 2.1
  • Theorem 2.1
  • proof
  • Remark 4.1
  • Remark 4.2
  • Theorem 4.1: Pathwise Convergence of the Modular-Poisson Algorithm
  • Definition 4.2
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • ...and 5 more