# Strict Local Martingales and the Khasminskii test for Explosions

## TL;DR

This paper provides conditions under which components of multidimensional SDEs with stochastic volatility are strict local martingales or martingales, extending understanding of their behavior and explosion criteria.

## Contribution

It introduces new sufficient conditions for local martingales and strict local martingales in multidimensional SDEs with stochastic volatility.

## Key findings

- Identifies conditions for local martingales to be strict or true martingales.
- Extends Khasminskii test to multidimensional SDEs with stochastic volatility.
- Provides criteria for explosion in stochastic differential equations.

## Abstract

We exhibit sufficient conditions such that components of a multidimensional SDE giving rise to a local martingale $M$ are strict local martingales or martingales. We assume that the equations have diffusion coefficients of the form $\sigma(M_t,v_t),$ with $v_t$ being a stochastic volatility term.