Convergence of Numerical Schemes for Fractional Stochastic Differential Equations with Jumps and Applications in Finance

This article studies the convergence of numerical schemes for Fractional Stochastic Differential Equations (FSDEs) with jumps. Such equations provide a powerful framework for modeling complex phenomena with long memory, stochasticity, and jumps. We begin with the definition of fractional Brownian motion (fBm) and jump processes, focusing on the compound Poisson process. We then formulate a general FSDE with jumps. The analysis focuses on the convergence of Euler-Maruyama and Milstein schemes towards the equations. We identify the necessary conditions (Malliavin, coefficient regularity) and establish the convergence rates in Lp norms. We propose an application to option pricing in long memory jump markets (fractional Hestontype model with jumps) with numerical simulations demonstrating the convergence theorems and the efficiency of the method.