[Paper Review] Stochastic First Integrals, Kernel Functions for Integral Invariants and the Kolmogorov equations

Valery Doobko, Elena Karachanskaya|arXiv (Cornell University)|Dec 15, 2013

Numerical methods in inverse problems4 references3 citations

TL;DR

This paper introduces a generalized Ito-Wentzell formula for jump-diffusion processes and applies it to derive stochastic first integrals, kernels of integral invariants, and Kolmogorov equations for transition densities. By extending the classical Ito-Wentzell formula to include Poisson jumps, the authors establish a rigorous framework for constructing stochastic invariants and deriving Fokker-Planck-type equations for diffusion processes with jump components.

ABSTRACT

In this article the authors present stochastic first integrals (SFI), the generalized Itô-Wentzell formula and its application for obtaining the equations for SFI, for kernel functions for integral invariants and the Kolmogorov equations, described by the generalized Itô equations.

Motivation & Objective

• To develop a generalized Ito-Wentzell formula for generalized Itô SDEs with both Wiener and Poisson noise components.
• To define and characterize stochastic first integrals for jump-diffusion processes, extending classical deterministic concepts to the stochastic setting.
• To derive equations for stochastic kernels of integral invariants, representing local densities of dynamical invariants in stochastic systems.
• To establish the Kolmogorov forward and backward equations for transition probability densities using the generalized formula.
• To provide a rigorous theoretical foundation for constructing program controls with probability one in strongly perturbed stochastic systems.

Proposed method

• Introduces a generalized Ito-Wentzell formula for Itô SDEs driven by Wiener processes and Poisson random measures.
• Applies the generalized formula to derive stochastic first integral equations under conditions ensuring invariance of certain functionals.
• Defines local stochastic density of a dynamical invariant and derives its evolution equation via the generalized Ito-Wentzell formula.
• Establishes the connection between the local stochastic density and the kernel of the stochastic integral invariant.
• Derives the Kolmogorov forward and backward equations for transition probability densities using the derived stochastic first integral and kernel equations.
• Uses the generalized formula to prove existence and uniqueness of solutions for equations governing stochastic kernels and first integrals.

Experimental results

Research questions

• RQ1 How can the classical Ito-Wentzell formula be extended to include jump components in stochastic differential equations?
• RQ2 What conditions ensure the existence of a stochastic first integral for a generalized Itô SDE with Lévy noise?
• RQ3 How is the kernel of a stochastic integral invariant related to the local density of a dynamical invariant in a jump-diffusion process?
• RQ4 Can the Kolmogorov forward and backward equations be derived directly from the generalized Ito-Wentzell formula?
• RQ5 What is the role of the generalized Ito-Wentzell formula in proving existence and uniqueness of solutions for stochastic invariant kernels?

Key findings

• The generalized Ito-Wentzell formula is derived for SDEs with both Wiener and Poisson noise, reducing to the classical formula in the absence of jumps.
• Stochastic first integrals are shown to exist and satisfy a specific stochastic PDE derived via the generalized formula.
• The kernel of the stochastic integral invariant is identified as the solution to a stochastic PDE that governs the local density of invariant measures.
• The Kolmogorov forward equation for the transition density is derived as a consequence of the kernel evolution equation.
• The Kolmogorov backward equation is established using the same framework, confirming consistency with classical results in the diffusion case.
• The method provides a rigorous foundation for constructing control strategies with probability one in stochastic systems under strong perturbations.