Bene\check\bf S condition for discontinuous exponential martingale

2009/11/03 by Liptser, R.
#60G45 #60G46 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.0911.0641

Abstract

It is known the Girsanov exponent \mathfrakz_t, being solution of Doleans-Dade equation \mathfrakz_t=1+∫₀^tα(ω,s)dB_s generated by Brownian motion B_t and a random process α(ω,t) with ∫₀^tα²(ω,s)ds0, holds true. In this paper, we show B_t can be replaced by by a homogeneous purely discontinuous square integrable martingale M_t with independent increments and paths from the Skorokhod space \mathbbD_[0,∞) having positive jumps \triangle M_t with \E∑_s∈[0,t](\triangle M_s)³0. The method of proof differs from the original Bene\rm \checks one and is compatible for both setting with B_t and M_t.

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