Maximal inequalities for a series of continuous local martingales

Abstract

Let $\{X^j=(X_t^j,{ℱ}_t),j\ge 1\}$ be a sequence of continuous local martingales and $\{\langle X^j\rangle \}$ the corresponding sequence of their quadratic variation processes and let $H_n(x,y),n=1,2,\dots$ be the Hermite polynomials with parametric variable $y$.

In this paper, we consider the series ${\displaystyle \sum _{j=1}^{\infty }{H}_n^2}(X^j,\langle X^j\rangle )$ of the continuous local martingales

$$H_n(X^j,\langle X^j\rangle )={(H_n(X_t^j,{\langle X^j\rangle }_t),{ℱ}_t)}_{t\ge 0},j=1,2,\dots ,$$

and its discrete analogue, and obtain some maximal inequalities.