Approximation schemes associated to a differential equation governed by a Hölderian function; the case of fractional Brownian motion.
The author considers stochastic differential equations of the form \[ dx_t=\sigma(x_t)dg_t + b(x_t) dt,\;t\in [0,1],\quad x(0)=x_0,\tag{1} \] where \(g:[0,1] \rightarrow R\) is a Hölder function with index \(\alpha \in(0,1]\). He provides a definition of the stochastic integral represented by the term \(\sigma(x_t)dg_t\) in (1) and also of the solution of (1). Two approximation schemes, the Euler and the Milstein method, are proposed and a proof of their convergence is sketched. The case of \(g\) being a trajectory of fractional Brownian motion is discussed as a special case of the previous presentations.
stochastic integral with Hölder function
scientific article; zbMATH DE number 2175628