Convergence of discretized stochastic (interest rate) processes with stochastic drift term

Abstract

For applications in finance, we study the stochastic differential equation dXs = (2βXs + δs) ds + g(Xs) dBs with β a negative real number, g a continuous function vanishing at zero which satisfies a Hölder condition and δ a measurable and adapted stochastic process such that ∫t0 δu du < ∞ a.e. for all t ∈ ℝ+ and which may have a random correlation with the process X itself. In this paper, we concentrate on the Euler discretization scheme for such processes and we study the convergence in L1-supnorm and in ℋ︁1-norm towards the solution of the stochastic differential equation with stochastic drift term. We also check the order of strong convergence. © 1998 John Wiley & Sons, Ltd.