Projects per year
Abstract
Let z be a stochastic exponential, i.e., zt=1+∫t0zs−dMs, of a local martingale M with jumps △Mt>−1. Then z is a nonnegative local martingale with Ezt≤1. If EzT=1, then 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 EzT=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].
Original language | English |
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Pages (from-to) | 38-62 |
Number of pages | 25 |
Journal | Theory of Probability and its Applications |
Volume | 58 |
Issue number | 1 |
DOIs | |
Publication status | Published - 2014 |
Projects
- 2 Finished
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New Stochastic Processes with Applications in Finance
Klebaner, F. (Primary Chief Investigator (PCI)), Buchmann, B. (Chief Investigator (CI)) & Hamza, K. (Chief Investigator (CI))
Australian Research Council (ARC), Monash University
31/07/09 → 31/12/13
Project: Research
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Modelling with stochastic differential equations
Klebaner, F. (Primary Chief Investigator (PCI))
Australian Research Council (ARC), Monash University
1/06/08 → 30/06/11
Project: Research