Regime-Switching Jump Diffusions

• Regime-switching jump diffusions are stochastic processes that merge continuous diffusions, jump elements, and discrete regime shifts governed by Markov or semi-Markov transitions.
• The methodology integrates stochastic differential equations with Brownian motion and jump terms, employing Feynman–Kac representations and infinitesimal generators to derive associated (integro-)differential equations.
• These models are pivotal in applications such as option pricing, risk management, and optimal control, utilizing techniques like Lyapunov analysis and Fourier methods for stability assessments and numerical solutions.

A regime-switching jump diffusion is a stochastic process in which a diffusion with jumps is modulated by a discrete regime (or environment) variable, itself typically a Markov or semi-Markov process taking values in a finite or countably infinite set. The continuous component evolves according to an SDE with both Brownian and jump terms, while the regime variable changes state at random times, possibly with state-dependent transition rates and possibly influenced by the continuous state. These models naturally generalize both classical jump diffusions and regime-switching diffusions, allowing for highly nontrivial statistical and dynamical properties, analytic tractability via infinitesimal generators, and direct application to finance, control, physics, and biology.

1. Stochastic Process Formulation

A canonical formulation consists of a continuous state component $X(t)\in\mathbb{R}^n$ and a discrete regime component $\alpha(t)\in\mathcal M=\{1,\dots,m\}$ (or $S=\mathbb{N}$ if countably infinite). The model is defined by:

$$\begin{aligned} dX(t) &= b\bigl(X(t),\alpha(t)\bigr)\,dt + \sigma\bigl(X(t),\alpha(t)\bigr)\,dW(t) \ &\hspace{1em} + \int_{\mathbb{R}^n\setminus\{0\}} \gamma\bigl(X(t^-),\alpha(t^-),z\bigr)\,\widetilde N(dt,dz), \end{aligned}$$

where $W(t)$ is a standard Brownian motion, $\widetilde N(dt,dz)$ is a compensated Poisson measure with Lévy measure $\nu(dz)$, and $b$, $\sigma$, $\gamma$ are regime-dependent drift, diffusion, and jump amplitude functions. The regime process $\alpha(t)\in\mathcal M=\{1,\dots,m\}$0 is a continuous-time Markov or semi-Markov chain, specified by transition intensities $\alpha(t)\in\mathcal M=\{1,\dots,m\}$1, possibly depending on $\alpha(t)\in\mathcal M=\{1,\dots,m\}$2, yielding dynamics

$\alpha(t)\in\mathcal M=\{1,\dots,m\}$3

Such systems admit unique strong solutions under broad Lipschitz, nondegeneracy, and growth conditions, even for non-Lipschitz coefficients and infinite regime spaces (Xi et al., 2018, 2002.01422, Xi et al., 2017).

2. Infinitesimal Generator and Feynman-Kac Representation

The infinitesimal generator $\alpha(t)\in\mathcal M=\{1,\dots,m\}$4 for a test function $\alpha(t)\in\mathcal M=\{1,\dots,m\}$5 is: $\alpha(t)\in\mathcal M=\{1,\dots,m\}$6 This operator underlies strong Markov properties and the derivation of partial (integro-)differential equations (PDE, IPDE) for expectations of functionals of $\alpha(t)\in\mathcal M=\{1,\dots,m\}$7. The Feynman–Kac formula provides, under suitable regularity, a stochastic representation of classical solutions to initial, terminal, or boundary value problems: $\alpha(t)\in\mathcal M=\{1,\dots,m\}$8 This extends to backward (terminal) and Dirichlet (boundary) problems, linking the SDE dynamics to systems of nonlocal parabolic equations (Zhu et al., 2017).

3. Stability and Ergodicity Criteria

Stability of the origin, ergodicity, and recurrence are central qualitative features of regime-switching jump diffusions. Various modes are formalized:

• Almost sure exponential stability: $\alpha(t)\in\mathcal M=\{1,\dots,m\}$9 almost surely, for some $S=\mathbb{N}$0.
• $S=\mathbb{N}$1th-moment exponential stability: $S=\mathbb{N}$2.
• Recurrence/positive recurrence: Returns to compact domains in finite or arbitrary expected time (Chen et al., 2018).

Lyapunov function methods yield tractable sufficient and sometimes necessary criteria for stability and ergodicity, reducing to linear/bilinear matrix inequalities in linear systems. For one-dimensional linear systems, necessary and sufficient stability is characterized by explicit Lyapunov exponents involving drift, diffusion, and jumps, averaged over the regime stationary distribution (Chao et al., 2017, Yang et al., 2014). For countably infinite regimes, stability and Feller properties require additional control on explosion via Lyapunov–type bounds on switching rates (Xi et al., 2017).

4. Regularity: Feller, Strong Feller, and Irreducibility

The Feller property (preservation of continuity under the Markov semigroup) and strong Feller property (regularization of bounded measurable functions) are established via coupling, resolvent expansions, or Radon–Nikodym derivative arguments (Xi et al., 2018, 2002.01422, Xi et al., 2017). Strong Feller, combined with irreducibility (the possibility of reaching any regime in positive time), guarantees uniqueness of invariant measures and ergodicity. For models with countably infinite regimes, these properties require fine continuity and growth bounds on SDE coefficients and switching rates.

5. Analytical and Numerical Methods

Fourier and matrix-exponential techniques enable efficient computation of transition characteristic functions and probability densities, crucial for numerics—e.g., option pricing and risk management. Closed-form solutions for European and forward-starting options are available via FFT inversion of matrix-exponential–type characteristic functions (Ramponi, 2011, Ramponi, 2012, Wolf et al., 2024). The Feynman–Kac formulas precisely link SDE solutions to IPDEs for pricing and control (Zhu et al., 2017). For mean field and controlled settings, viscosity solutions to master equations on the space of probability measures characterize optimal controls in the many-agent limit (Bayraktar et al., 2021).

Stability and control properties are often reduced to solving systems of Riccati ODEs or BSDEs with jump and switching couplings (Shi et al., 2024). Risk-sensitive optimal investment and portfolio strategies in these models lead to high-dimensional ODE or PDE systems reflecting both continuous and regime risks (Andruszkiewicz et al., 2014, Das et al., 2016).

6. Applications and Model Extensions

Regime-switching jump diffusions are foundational in finance (option pricing under stochastic volatility/regimes, risk management, portfolio optimization), insurance (ruin with environmental regime shifts), engineering (hybrid control, network systems), and biology (epidemic transitions, ecological shifts). Semi-Markov extensions model memory and duration effects in regime residence times, improving fit to empirical data (e.g., business cycle, volatility clustering) (Goswami et al., 2018, Das et al., 2016). Bayesian inference for discretely observed data is tractable via (retrospective) data augmentation and bridge sampling, yielding consistent estimators for switching rates, volatility functions, and jump laws (Stumpf-Fétizon et al., 13 Feb 2025, Das et al., 2019).

7. Asymptotic and Scaling Results

In singular limits, e.g., fast switching regimes, the process behavior averages over the stationary regime distribution. For example, a diffusion with rapidly switching regimes converges to a process whose coefficients are averaged accordingly. Occupation times of domains converge weakly to nontrivial limiting distributions such as the Lévy arcsine law under null-recurrence and symmetric averaging (Zhu et al., 2017). Such scaling results are essential in approximating long-time behavior and justifying reduced modeling approaches.

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References:

• "Feynman-Kac Formulas for Regime-Switching Jump Diffusions and their Applications" (Zhu et al., 2017)
• "Almost Sure and Moment Exponential Stability of Regime-Switching Jump Diffusions" (Chao et al., 2017)
• "Stability of Nonlinear Regime-switching Jump Diffusions" (Yang et al., 2014)
• "On Feller and Strong Feller Properties and Exponential Ergodicity of Regime-Switching Jump Diffusion Processes with Countable Regimes" (Xi et al., 2017)
• "Regime-Switching Jump Diffusions with Non-Lipschitz Coefficients and Countably Many Switching States" (Xi et al., 2018)
• "On Feller and Strong Feller Properties and Irreducibility of Regime-Switching Jump Diffusion Processes with Countable Regimes" (2002.01422)
• "Risk Sensitive Portfolio Optimization in a Jump Diffusion Model with Regimes" (Das et al., 2016)
• "Consistent asset modelling with random coefficients and switches between regimes" (Wolf et al., 2024)
• "Fourier Transform Methods for Regime-Switching Jump-Diffusions and the Pricing of Forward Starting Options" (Ramponi, 2011)
• "Option Pricing in a Regime Switching Jump Diffusion Model" (Goswami et al., 2018)
• "Inference of Binary Regime Models with Jump Discontinuities" (Das et al., 2019)
• "Computing Quantiles in Regime-Switching Jump-Diffusions with Application to Optimal Risk Management: a Fourier Transform Approach" (Ramponi, 2012)
• "Risk-sensitive investment in a finite-factor model" (Andruszkiewicz et al., 2014)
• "Constrained stochastic linear quadratic control under regime switching with controlled jump size" (Shi et al., 2024)
• "Periodic solutions of hybrid jump diffusion processes" (Guo et al., 2019)
• "Exact Bayesian inference for Markov switching diffusions" (Stumpf-Fétizon et al., 13 Feb 2025)
• "Mean field control and finite dimensional approximation for regime-switching jump diffusions" (Bayraktar et al., 2021)
• "A Games-in-Games Paradigm for Strategic Hybrid Jump-Diffusions: Hamilton-Jacobi-Isaacs Hierarchy and Spectral Structure" (Pan et al., 19 Dec 2025)