Finite Time Explosion of Stochastic Differential Equations: A survey into Khasminskii's Lyapunov Method and its Consistency with the Osgood Criterion

TL;DR

This survey reviews Khasminskii's Lyapunov method for classifying explosive behaviors in solutions of stochastic differential equations, highlighting its advantages over Feller's test and extending it to jump processes.

Contribution

It provides a comprehensive overview of Khasminskii's Lyapunov method, compares it with Feller's test, and extends its application to jump processes with Poisson jumps.

Findings

Khasminskii's Lyapunov method effectively classifies explosion types.

Feller's test has notable limitations in explosion classification.

Extensions to jump processes broaden the method's applicability.

Abstract

Solutions of Stochastic Differential Equations can have three types of explosive behaviors: almost-sure non-explosive, explosion with positive probability, and almost sure explosion. In this paper, we will provide a survey of Khasminskii's Lyapunov method for classifying explosive behaviors of solutions of stochastic differential equations. We will embark our expedition by examining the renowned Feller's test for explosion and observing its shortfalls. Afterwards, we will present Khasminskii's Lyapunov method for almost-sure non-explosion, explosion with positive probability, and almost-sure explosion. Ample examples will be provided to illuminate the power of Khasminskii's Lyapunov methods. Furthermore, quick layovers will be made to extend Khasminskii's Lyapunov method for almost-sure non-explosion and explosion with positive probability for jump processes with constant Poisson intensities.