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The Kolmogorov forward equation for a distributed model of regime-switching diffusions

Alexander S. Bratus, Olga S. Rozanova

TL;DR

The paper formulates a continuum of hidden states for regime-switching diffusions via an integro-differential forward equation for densities $p(t,x,s)$ on $s\in[0,1]$, and develops a constructive algorithm to solve the Cauchy problem using a Fourier transform in $x$ and an $L_2(0,1)$ basis in $s$. By expanding the kernel $K(s,\xi)$ and the initial data in an orthonormal basis, the authors reduce the problem to an infinite (or finite) system of modal equations, from which $p(t,x,s)$ can be reconstructed as a series $p(t,x,s)=\sum_l B_l(t,x,s) X_l(s)$. They provide explicit solutions in several cases, including uniform Gaussian and delta initial data, and demonstrate how discrete finite-state models arise as approximations of the continuous-state model, with concrete results for the generalized Ornstein-Uhlenbeck setting where the density converges to a nontrivial steady state with multi-peak structure. The approach is demonstrated for both $M=1$ and $M=3$ hidden-state configurations, yielding closed-form expressions or tractable estimates for the density components and establishing stability via negative eigenvalues of the governing modal matrices. The method offers a constructive pathway to analyze regime-switching systems beyond finite-state approximations and can be extended to higher dimensions and multivariate contexts.

Abstract

For the regime-switching diffusion process with and without advection term we propose an integro-differential equation describing the densities of states continuously distributed over a segment. We demonstrate that there exists a constructive algorithm for solving the Cauchy problem. We then show that for some initial distributions of states, the solution can be found explicitly. We also discuss how a model with a discrete number of hidden states can be approximated by a model with continuously distributed states.

The Kolmogorov forward equation for a distributed model of regime-switching diffusions

TL;DR

The paper formulates a continuum of hidden states for regime-switching diffusions via an integro-differential forward equation for densities on , and develops a constructive algorithm to solve the Cauchy problem using a Fourier transform in and an basis in . By expanding the kernel and the initial data in an orthonormal basis, the authors reduce the problem to an infinite (or finite) system of modal equations, from which can be reconstructed as a series . They provide explicit solutions in several cases, including uniform Gaussian and delta initial data, and demonstrate how discrete finite-state models arise as approximations of the continuous-state model, with concrete results for the generalized Ornstein-Uhlenbeck setting where the density converges to a nontrivial steady state with multi-peak structure. The approach is demonstrated for both and hidden-state configurations, yielding closed-form expressions or tractable estimates for the density components and establishing stability via negative eigenvalues of the governing modal matrices. The method offers a constructive pathway to analyze regime-switching systems beyond finite-state approximations and can be extended to higher dimensions and multivariate contexts.

Abstract

For the regime-switching diffusion process with and without advection term we propose an integro-differential equation describing the densities of states continuously distributed over a segment. We demonstrate that there exists a constructive algorithm for solving the Cauchy problem. We then show that for some initial distributions of states, the solution can be found explicitly. We also discuss how a model with a discrete number of hidden states can be approximated by a model with continuously distributed states.
Paper Structure (12 sections, 1 theorem, 79 equations, 3 figures)

This paper contains 12 sections, 1 theorem, 79 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Construction of a solution
  3. Formal scheme.
  4. Existence and uniqueness.
  5. One hidden state: $M=1$
  6. Approximation by two first modes.
  7. Examples of orthonormal systems
  8. Examples of solutions
  9. Uniform Gaussian initial distribution
  10. Stepwise Gaussian initial distribution
  11. A four-state switching diffusion, $M=3$
  12. Discussion

Key Result

Proposition 1

Let $b(s)$, $c(s)$, $R(s)>0$, $K(s,\xi)$ and $\Phi(x,s)\ge 0$ are bounded and Lipschitz continuous functions on $[0,1]$, $[0,1]\times [0,1]$ and ${\mathbb R}\times [0,1]$, respectively. Then there exist a unique classical nonnegative solution to the problem 3, 4. It is bounded, continuous on $[0,T]

Figures (3)

  • Figure 1: The dynamics of densities of main (left) and hidden (right) states for $b=0$, other parameters are given in \ref{['param']}; $t=0$ (thin solid line), $t=1$ (dots), $t=10$ (thick solid line). The initial data are \ref{['ID1']}.
  • Figure 2: The dynamics of densities of main (left) and hidden (right) states for $b<0$, the parameters are given in \ref{['param']}; $t=0.1$ (thin solid line), $t=0.5$ (dots), $t=100$ (thick solid line). The initial data are \ref{['IDel']}.
  • Figure 3: The dynamics of densities of main (top left) and hidden states 1, 2, 3 (top right, bottom left, bottom right, respectively) for parameters given in \ref{['param4']}; $t=0$ (thin solid line), $t=3$ (dots), $t=15$ (thick solid line).

Theorems & Definitions (2)

  • Proposition 1
  • Remark 1