Convergence of discretized stochastic (interest rate) processes with stochastic drift term. (English) Zbl 0915.60064

Deelstra, G.; Delbaen, F.

Appl. Stochastic Models Data Anal. 14, No. 1, 77-84 (1998).

The authors study the stochastic differential equation \[ dX_{s}= (2\beta X_{s}+\delta_{s})ds+ g(X_{s})dB_{s}, \quad s\in \mathbb{R}^{+}, \] with \(X_0 \geq 0\), \(\beta \leq 0\), where \(g\) is a function vanishing at zero which satisfies the Hölder condition: \(| g(x)-g(y)|\leq b| x-y|^{1/2}\) and \(\delta_{s}\) is a measurable and adapted stochastic process such that \(\int_0^{t}\delta_{u} du<\infty\) a.e. The authors discuss the Euler discretization scheme for this stochastic differential equation with a drift term which may depend on a stochastic process with random correlation. It is shown that the approximating solution converges in \(L^1\)-supnorm and \(H^1\)-norm to the solution of this differential equation. The authors find conditions under which the Euler scheme strongly converges with order \(v=0.5\) at time \(T\). The authors note that this stochastic differential equation may be applied in finance.

Reviewer: Yu.V.Kozachenko (Kyïv)

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
62P20	Applications of statistics to economics

Keywords:

stochastic differential equation; stochastic drift term; Hölder condition; Euler discretization scheme; strong convergence