Sankhya: The Indian Journal of Statistics

1993, Volume 55, Series A, Pt. 2, 202--213

A TROTTER TYPE FORMULA FOR SEMIMARTINGALES

By

RAJEEVA L. KARANDIKAR, Indian Statistical Institute 

SUMMARY. Let \textbf{${S_1,S_2}$}be $d \times d$ matrix valued semimartingles and let \textbf{$Q_i$} $(s, t), s < t $ be given by \textbf{$Q_i$} $(s, t)= $ \textbf{$I$ + $f_{s}^{t}$} \textbf{$Q_i$} $(s, u-)d$ \textbf{$S_i(u)$}.This paper investigates the limits of $\prod_{i:{t_i}\leq 1}$ ( \textbf{$Q_i$} $(t_i,t_{i+1}\bigwedge t)Q_2 (t_i,t_{i+1}\bigwedge t))$ and $\prod_{i:{t_i}\leq 1}$ exp(\textbf{$S_1$} $(t_{i+1}\bigwedge t)-S_1(t_i))exp (S_2,(t_{i+1}\bigwedge t)-S_2(t_i))$ as the norm $sup_i(t_{i+1}-t_i)$ of the partition $t_0 = 0< l_1 < ...< l_i < .. t_i \uparrow \infty,$ goes to zero. Here $[t]$ are allowed to be stop times. The expressions are shown to converge in the Emery topology on the space of semimartingales. Conditions on $[t]$ are also given which ensure a.s. convergence.

.

AMS (1990) subject classification. 60G44, 60G44, 60H05, 60J57

Key words and phrases. Semimartingales, Trotter's formula, Emery topology

FULL PAPER.

This article in Mathematical Reviews.