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

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.

### MSC:

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