Abstract
The aim of this paper is to study the absolute continuity of a semimartingale (Xt) w.r.t. a continuous increasing and adapted process (Dt). Such problems arise in filtering theory. In that case dDs=ds.
Let (ω, F, P, (F t)0 ≤ t ≤ 1) be a filtered probability space satisfying the usual conditions. We suppose all processes are null at time 0 and they are indexed by [0, 1] rather then all of R. Assume (Bt) is a (F t) Brownian motion and (hs) is a predictable process such that ∫ 1o |hs| ds<+∞. It is well known that if for every sε[0, 1] E[|hs| |F Xs ]<+∞ where (F Xs ) denotes the natural filtration of (Xt), then the process vt=Xt − ∫ to E[hs| F Xs ]ds is a Brownian motion. One consequence of our main result will be that the condition ∫ 1o |hs| ds<+∞ implies the existence of a (F Xt ) predictable process (ĥs) such that ∫ 1o |ĥs| ds<+∞ and vt=Xt – ∫ to ĥs ds is a (F Xt ) Brownian motion.
In the last section we improve the Lindquist-Picci semimartingale representation theorem ([4]).
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© 1987 Springer-Verlag
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Stricker, C. (1987). Absolute continuity of a semimartin gale with respect to a continuous increasing and adapted process. In: Engelbert, H.J., Schmidt, W. (eds) Stochastic Differential Systems. Lecture Notes in Control and Information Sciences, vol 96. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0038953
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DOI: https://doi.org/10.1007/BFb0038953
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