Original scientific paper

Stable convergence in law in approximation of stochastic integrals with respect to diffusions

Snježana Lubura Strunjak ; Faculty of Science, University of Zagreb, Zagreb, Croatia

Abstract

We assume that the one-dimensional diffusion X satisfies a stochastic differential equation of the form:
\(dX_t=\mu(X_t)dt+\nu(X_t)dW_t$, $X_0=x_0\), \(t\geq 0\). Let \((X_{i\Delta_n},0\leq i\leq n)\) be discrete observations along fixed time interval [0,T]. We prove that the random vectors which j-th component is \(\frac{1}{\sqrt{\Delta_n}}\sum_{i=1}^n\int_{t_{i-1}}^{t_i}g_j(X_s)(f_j(X_s)-f_j(X_{t_{i-1}}))dW_s\), for \(j=1,\dots,d,\) converge stably in law to mixed normal random vector with covariance matrix which depends on path \((X_t,0\leq t\leq T)$, when $n\to\infty\). We use this result to prove stable convergence in law for \(\frac{1}{\sqrt{\Delta_n}}(\int_0^Tf(X_s)dX_s-\sum_{i=1}^nf(X_{t_{i-1}})(X_{t_i}-X_{t_{i-1}}))\).

Keywords

asymptotic mixed normality, diffusion processes, approximations of stochastic integrals

Hrčak ID:

321270

URI

https://hrcak.srce.hr/321270

Publication date:

7.10.2024.

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