On Some Inequalities of Local Times of Iterated Stochastic Integrals

Volume 3, Issue 4, 2002

Article 62

ON SOME INEQUALITIES OF LOCAL TIMES OF ITERATED STOCHASTIC INTEGRALS

DEPARTMENT OF MATHEMATICS
FACULTY OF SCIENCE
TOYAMA UNIVERSITY
3190 GOFUKU, TOYAMA 930-8555
JAPAN
E-Mail: yan@math.toyama-u.ac.jp

Received 06 July, 2001; Accepted 25 June, 2002.
Communicated by: N.S. Barnett

ABSTRACT. Let $X=(X_t,{\mathcal F}_t)_{t\geq 0}$ be a continuous local martingale with quadratic variation process $\langle X \rangle$ and

. Define iterated stochastic integrals $I_n(X)=\left(I_n(t,X),{\mathcal F}_t\right)$ $(n\geq 0)$ , inductively by

$\displaystyle I_n(t,X)=\int_0^tI_{n-1}(s,X)dX_s$

with

and

. In this paper, we obtain some martingale inequalities for

, $n=1,2,\ldots$ and their local times at any random time.

Key words:
Continuous local martingale, Continuous semimartingale, Iterated stochastic integrals, Local time, Random time, Burkholder-Davis-Gundy inequalities, Barlow-Yor inequalities.

2000 Mathematics Subject Classification:
60H05, 60G44, 60J55.

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