Regime-Switching Diffusion

• Regime-switching diffusion is a continuous-time stochastic process combining SDE dynamics with Markovian regime changes to capture nonstationary behavior.
• It models state-dependent dynamics by interlacing SDE solutions with jumps in regimes, ensuring strong existence and uniqueness under Lipschitz conditions.
• Applications span finance, control theory, and physics, with robust numerical methods, Bayesian inference, and PDE links supporting practical analysis.

A regime-switching diffusion is a continuous-time stochastic process incorporating both diffusion dynamics, typically governed by stochastic differential equations (SDEs), and random regime switches dictated by a finite or countably infinite Markovian process. The regime process modulates the coefficients of the SDE (drift, diffusion, and possibly jump intensities), resulting in dynamic behavior that exhibit nontrivial temporal correlations and nonstationarities reflecting transitions between different regimes. These processes arise in a broad range of applications, including stochastic control, finance, filtering, and statistical physics, where systems may operate differently under varying external states or control policies.

1. Mathematical Formulation

Let $(X_t,\alpha_t)$ denote the joint process, where $X_t\in\R^d$ evolves according to an SDE and $\alpha_t$ is a pure-jump process on a (finite or countable) regime set $I$.

The canonical regime-switching diffusion in the absence of jumps is described by

$$dX(t) = b\bigl(t,X(t),u(t),\alpha(t_{-})\bigr)\,dt + \sigma\bigl(t,X(t),u(t),\alpha(t_{-})\bigr) \, dW(t), \qquad X(0) = x \in \R^d,$$

where $W$ is a $d$-dimensional Brownian motion, $u$ is a control, $b$ (drift) and $\sigma$ (volatility) are measurable coefficients, and $\alpha$ is a finite- or countable-state Markov chain with generator matrix $Q(x) = (q_{ij}(x))_{i,j}$, possibly depending on the current $X$. The switching process satisfies

$$\P\{\alpha(t+\Delta) = j \mid \alpha(t)=i, X(t)=x\} = q_{ij}(x)\Delta + o(\Delta), \quad i\neq j.$$

For regime-switching jump diffusions, jump terms of the form

$$\int_U c\bigl(X(t^-),\alpha(t^-),u\bigr)\, \tilde N(dt,du)$$

may be included, where $N$ is a Poisson random measure with compensator $\nu(du)\,dt$ (Xi et al., 2018).

The infinitesimal generator for such processes acts on test functions $f:\R^d\times I\to\R$ as:

$$\mathscr{L} f(x,i) = L_i f(\cdot,i)(x) + \sum_{j\neq i} q_{ij}(x)\bigl[f(x,j)-f(x,i)\bigr],$$

where $L_i$ is the diffusion generator associated to regime $i$.

2. Existence, Uniqueness, and Regularity

Strong solutions and pathwise uniqueness are established via an interlacing (or concatenation) construction. For each fixed regime, one solves the SDE for $X$ with frozen $\alpha = i$ up to the first switching time, sampled via exponential clocks determined by $q_{ij}(X(s))$. At each switching time, one updates the regime and continues, ensuring non-explosion via Lyapunov arguments (e.g., supermartingale control of growth functions) (Xi et al., 2018, Shao, 2015, Xi et al., 2017). Under global Lipschitz and growth assumptions on the coefficients, and mild control on $q_{ij}$ (e.g., $q_i(x) \leq H_i$), strong existence and uniqueness hold even in infinite regime spaces and with non-Lipschitz drift/diffusion (Xi et al., 2018, Shao, 2015).

Regularity of the Markov semigroup (Feller/strong Feller properties) is achieved via coupling constructions and resolvent expansions. If the SDE in each frozen regime is strong Feller and switching rates are sufficiently regular (local range, Lipschitz in $x$), the coupled process inherits the strong Feller property (Shao, 2015, Zhang, 2016).

3. Stability, Recurrence, and Ergodicity

Asymptotic stability, recurrence, and ergodicity are characterized by Lyapunov function techniques combined with algebraic criteria on the generator:

• Averaging Condition / Perron–Frobenius: For finite $I$, the process is (asymptotically) stable if there exists a Lyapunov function $V$ and constants $\beta_i$ such that for the invariant distribution $\mu$ of $Q$, $\sum_i \mu_i \beta_i < 0$ (Shao et al., 2015, Hu et al., 2014).
• $M$-Matrix and Finite Partition: For countable $I$, partitioning regimes into finitely many classes and aggregating the switching generator, stability (and exponential ergodicity) follows from the nonsingularity of certain $M$-matrices formed from $Q$ and the Lyapunov exponents in each class (Shao et al., 2015, Hu et al., 2014, Bao et al., 2014).
• Principal Eigenvalue: If $Q$ is reversible, principal eigenvalue methods on the Hilbert space $\ell^2(\pi)$, with quadratic form involving both switching and drift exponents, provide sharp spectral (gap) criteria (Shao et al., 2015, Hu et al., 2014).

These methods extend classical results to nonlinear systems, countably infinite regime spaces, and state-dependent $Q$. For state-dependent switching, pathwise comparison arguments enable control under minimal “local range” and boundedness assumptions (Shao, 2022).

4. Numerical Methods and Simulation

Discretization schemes for regime-switching diffusions (notably Euler–Maruyama) have been rigorously analyzed in both finite and countable regime settings:

• Under the averaging condition, the EM scheme admits a unique numerical invariant measure converging to the true invariant measure in Wasserstein distance, with explicit $O(\delta^{p/2})$ rates (Bao et al., 2014). For reversible regime chains and under principal eigenvalue criteria, sharper spectral gap bounds are available.
• For state-dependent $Q$, strong $L^1$ convergence at order $O(\Delta)$ is achieved via careful Skorokhod coupling of the discrete chain and the original process, controlling the regime-mismatch probability (Shao, 2017).
• For sampling from mixture distributions, regime-switching diffusions engineered via balance conditions on $q_{ij}(x)$ yield exact invariant laws for arbitrary mixture targets, with Euler schemes retaining weak order 1 convergence (Tretyakov, 18 Jul 2024).

5. Control Theory and Stochastic Optimization

Regime-switching diffusions are directly relevant for stochastic control with Markovian regime uncertainty. The optimal control problem is formulated over controlled regime-switching diffusions, leading to value functions governed by coupled systems of Hamilton-Jacobi-Bellman (HJB) equations, or equivalently, to stochastic maximum principles with regime-switching backward SDEs (BSDEs) (Donnelly, 2010).

For quadratic (mean-variance-type) cost functionals, explicit solutions for the optimal control can be constructed by solving backward systems of ODEs in each regime, coupled via the generator $Q$, yielding closed-form, regime-dependent feedback laws (Donnelly, 2010).

6. Feynman–Kac Representation and PDE Connections

Regime-switching (jump) diffusions provide probabilistic representations for systems of coupled PDEs or PIDEs, such as Feynman-Kac formulae for backward Cauchy problems, Dirichlet boundary value problems, and terminal-value problems:

• Value functions are characterized by expectations over regime-switching diffusions with suitable exponential weights, regardless of whether the switching is Markov, semi-Markov (age-dependent), or governed by jump processes (Zhu et al., 2017, Goswami et al., 2018, Ocejo, 2019).
• For jump diffusions, the generator yields coupled integro-differential operators, and the associated boundary and terminal value problems admit explicit probabilistic solutions under global Lipschitz/coercivity conditions (Zhu et al., 2017).

Integral equation characterizations (e.g., for option pricing) allow direct approximation of solution via Picard iteration exploiting knowledge of fundamental solutions in each regime (Ocejo, 2019).

7. Statistical Inference and Bayesian Methodology

Recently, exact Bayesian inference has been developed for regime-switching diffusions observed at discrete time points, avoiding bias from SDE discretization:

• The exact posterior (on both latent regime path and SDE parameters) is sampled by retrospective Poisson simulation techniques, employing two-coin Bernoulli factory and Lamperti transformation for the diffusion component, ensuring convergence to the continuous-time law (Stumpf-Fétizon et al., 13 Feb 2025).
• Such methods rely on conjugacy in the Markov jump generator, Barker-within-Gibbs steps for parameters, and Poisson coin rejection for path-integral weightings, yielding scalable and unbiased inference for large time series and complex regime-switching models.

8. Applications and Impact

Regime-switching diffusions have a central role in quantitative finance (e.g., regime-switching option pricing, mean-variance portfolio selection), engineering (feedback control under intermittent or uncertain environments), and stochastic modeling of systems with dynamic switching—enabling explicit analysis and simulation of stabilization, recurrence, and occupation-time phenomena (e.g., arcsine laws in null-recurrent diffusion limits) (Goswami et al., 2018, Zhu et al., 2017).

Key advances include explicit characterizations of (algebraic) decay rates, robust numerical discretization, sharp control and comparison theorems for state-dependent and countably infinite regimes, and the development of MCMC and MCEM inference algorithms targeting exact posteriors for observed trajectory data.

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Principal references: (Donnelly, 2010, Xi et al., 2018, Shao et al., 2015, Ocejo, 2019, Li et al., 2015, Xi et al., 2017, Shao, 2015, Shao, 2022, Bao et al., 2014, Nguyen et al., 2017, Zhang, 2016, Hu et al., 2014, Shao, 2014, Tretyakov, 18 Jul 2024, Shao, 2017, Goswami et al., 2018, Zhu et al., 2017, Stumpf-Fétizon et al., 13 Feb 2025).