The Euler-Maruyama approximations for the CEV model

TL;DR

This paper proves the weak convergence of Euler-Maruyama approximations for the CEV model with non-Lipschitz diffusion, introduces a new continuous approximation method, and discusses implications for ruin probability estimation.

Contribution

It provides a new continuous approximation approach for the CEV model, relaxing previous technical conditions, and analyzes convergence issues in ruin probability estimates.

Findings

Euler-Maruyama approximations converge weakly in the Skorokhod metric.

A new continuous process approximation facilitates convergence proofs.

Simulation-based ruin probabilities may not converge without considering the Levy metric.

Abstract

The CEV model is given by the stochastic differential equation $X_t=X_0+\int_0^t\mu X_sds+\int_0^t\sigma (X^+_s)^pdW_s$, $\frac{1}{2}\le p<1$. It features a non-Lipschitz diffusion coefficient and gets absorbed at zero with a positive probability. We show the weak convergence of Euler-Maruyama approximations $X_t^n$ to the process $X_t$, $0\le t\le T$, in the Skorokhod metric. We give a new approximation by continuous processes which allows to relax some technical conditions in the proof of weak convergence in \cite{HZa} done in terms of discrete time martingale problem. We calculate ruin probabilities as an example of such approximation. We establish that the ruin probability evaluated by simulations is not guaranteed to converge to the theoretical one, because the point zero is a discontinuity point of the limiting distribution. To establish such convergence we use the Levy metric, and also confirm the convergence numerically. Although the result is given for the specific model, our method works in a more general case of non-Lipschitz diffusion with absorbtion.