Stochastic viability and comparison theorems for mixed stochastic differential equations

TL;DR

This paper establishes viability and comparison theorems for mixed stochastic differential equations involving Wiener and H"older processes, with applications to option pricing.

Contribution

It introduces new stochastic viability and comparison theorems for mixed SDEs with H"older continuous processes, extending existing results.

Findings

Proved a stochastic viability theorem for mixed SDEs.

Derived a pathwise comparison theorem.

Applied results to estimate option prices.

Abstract

For a mixed stochastic differential equation containing both Wiener process and a H\"older continuous process with exponent $\gamma>1/2$, we prove a stochastic viability theorem. As a consequence, we get a result about positivity of solution and a pathwise comparison theorem. An application to option price estimation is given.