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 socalled linear growth conditions involved in the definition of parameters of M, proposed by Girsanov [Theory Probab. Appl., 5 (1960), pp. 285301]. These conditions generalize the Beneš idea [SIAM J. Control, 9 (1971), pp. 446475] and avoid the technology of piecewise approximation. These conditions are applicable even if the Novikov [Theory Probab. Appl., 24 (1979), pp. 820824] and Kazamaki [Tôhoku Math. J., 29 (1977), pp. 597600] conditions fail. They are effective for Markov processes that explode, Markov processes with jumps, and also nonMarkov processes. Our approach is different from the recently published paper [P. Cheredito D. Filipovićbͅ and M. Yor, Ann. Appl. Probab., 15 (2005), pp. 17131732] and preprint [A. Mijitović and M. Urusov, arXiv:0905.3701v1[math.PR], 2009].
Original language  English 

Pages (fromto)  3862 
Number of pages  25 
Journal  Theory of Probability and its Applications 
Volume  58 
Issue number  1 
DOIs  
Publication status  Published  2014 
Projects
 2 Finished

New Stochastic Processes with Applications in Finance
Klebaner, F., Buchmann, B. & Hamza, K.
Australian Research Council (ARC), Monash University
31/07/09 → 31/12/13
Project: Research

Modelling with stochastic differential equations
Australian Research Council (ARC), Monash University
1/06/08 → 30/06/11
Project: Research