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