NDLI: When a Stochastic Exponential Is a True Martingale. Extension of the Beneš Method

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Liptser, R. Klebaner, F.
Copyright Year	2014
Abstract	Let ${\frak z}$ be a stochastic exponential, i.e., ${\frak z}_t=1+\int_0^t{\frak z}_{s-}\,dM_s,$ of a local martingale $M$ with jumps $\triangle M_t>-1$. Then ${\frak z}$ is a nonnegative local martingale with ${\bf E}\,{\frak z}_t\le 1$. If ${\bf E}\,{\frak z}_{_T}= 1$, then ${\frak z}$ is a martingale on the time interval $[0,T]$. The martingale property plays an important role in many applications. It is therefore of interest to give natural and easily verifiable conditions for the martingale property. In this paper, the property ${\bf E}\,{\frak z}_{_T}=1$ is verified with the so-called linear growth conditions involved in the definition of parameters of $M$, proposed by Girsanov [Theory Probab. Appl., 5 (1960), pp. 285--301]. These conditions generalize the Beneš idea [SIAM J. Control, 9 (1971), pp. 446--475] and avoid the technology of piecewise approximation. These conditions are applicable even if the Novikov [Theory Probab. Appl., 24 (1979), pp. 820--824] and Kazamaki [Tôhoku Math. J., 29 (1977), pp. 597--600] conditions fail. They are effective for Markov processes that explode, Markov processes with jumps, and also non-Markov processes. Our approach is different from the recently published paper [P. Cheredito D. Filipovićbͅ and M. Yor, Ann. Appl. Probab., 15 (2005), pp. 1713--1732] and preprint [A. Mijitović and M. Urusov, arXiv:0905.3701v1[math.PR], 2009].
Starting Page	38
Ending Page	62
Page Count	25
File Format	PDF
ISSN	0040585X
DOI	10.1137/S0040585X97986382
e-ISSN	10957419
Journal	Theory of Probability and its Applications (TPRBAU)
Issue Number	1
Volume Number	58
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2014-01-01
Access Restriction	Subscribed
Subject Keyword	Beneš method exponential martingale diffusion process with jumps Girsanov theorem
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Statistics, Probability and Uncertainty

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