arXiv subjects

J. A. León

Publications and source records attributed to J. A. León.

3 recordsLinked to original sources

arXiv2019

Numerical scheme for stochastic differential equations driven by fractional Brownian motion with 1/4 < H < 1/2

In this article, we study a numerical scheme for stochastic differential equations driven by fractional Brownian motion with Hurst parameter H in (1/4; 1/2). Towards this end, we apply Doss-Sussmann representation of the solution and an approximation of this representation using a first order Taylor expansion. The obtained rate of convergence is n^(2H+rho), for rho small enough.

math.PR↗

arXiv2011

Approximations of Fractional Stochastic Differential Equations by Means of Transport Processes

We present strong approximations with rate of convergence for the solution of a stochastic differential equation of the form $$ dX_t=b(X_t)dt+σ(X_t)dB^H_t, $$ where $b\in C^1_b$, $σ\in C^2_b$, $B^H$ is fractional Brownian motion with Hurst index $H$, and we assume existence of a unique solution with Doss-Sussmann representation. The results are based on a strong approximation of $B^H$ by means of transport processes of Garzón et al (2009). If $σ$ is bounded away from 0, an approximation is obtained by a general Lipschitz dependence result of Römisch and Wakolbinger (1985). Without that assumption on $σ$, that method does not work, and we proceed by means of Euler schemes on the Doss-Sussmann representation to obtain another approximation, whose proof is the bulk of the paper.

math.PR↗

arXiv2008

The Local Time of the Classical Risk Process

In this paper we give an explicit expression for the local time of the classical risk process and associate it with the density of an occupational measure. To do so, we approximate the local time by a suitable sequence of absolutely continuous random fields. Also, as an application, we analyze the mean of the times $s \in [0,T]$ such that $0\leq X_{s} \leq X_{s+ε} $ for some given $ε>0$.

math.PR↗