Absolute continuity of a semimartin gale with respect to a continuous increasing and adapted process

Abstract

The aim of this paper is to study the absolute continuity of a semimartingale (X_t) w.r.t. a continuous increasing and adapted process (D_t). Such problems arise in filtering theory. In that case dD_s=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 (B_t) is a (F _t) Brownian motion and (h_s) is a predictable process such that ∫ ¹_o |h_s| ds<+∞. It is well known that if for every sε[0, 1] E[|h_s| |F ^X_s ]<+∞ where (F ^X_s ) denotes the natural filtration of (X_t), then the process v_t=X_t − ∫ ^t_o E[h_s| F ^X_s ]ds is a Brownian motion. One consequence of our main result will be that the condition ∫ ¹_o |h_s| ds<+∞ implies the existence of a (F ^X_t ) predictable process (ĥ_s) such that ∫ ¹_o |ĥ_s| ds<+∞ and v_t=X_t – ∫ ^t_o ĥ_s ds is a (F ^X_t ) Brownian motion.

In the last section we improve the Lindquist-Picci semimartingale representation theorem ([4]).