Abstract
We derive equivalent conditions for the (local) absolute continuity of two laws of semimartingales on random sets. Our result generalizes previous results for classical semimartingales by replacing a strong uniqueness assumption by a weaker uniqueness assumption. The main tool is a generalized Girsanov’s theorem, which relates laws of two possibly explosive semimartingales to a candidate density process. Its proof is based on an extension theorem for consistent families of probability measures. Moreover, we show that in a one-dimensional Ito-diffusion setting our result reproduces the known deterministic characterizations for (local) absolute continuity. Finally, we give a Khasminskii-type test for the absolute continuity of multidimensional Ito-diffusions and derive linear growth conditions for the martingale property of stochastic exponentials.
Highlights
In the 1970s, probabilists studied conditions under which laws of semimartingales are absolutely continuous
The most general results were obtained by Jacod and Memin [11] and Kabanov, Lipster and Shiryaev [13, 14] under a strong uniqueness assumption, called local uniqueness in the monograph of Jacod and Shiryaev [12]
In this article we provide equivalent statements for the absolute continuity of semimartingales on random sets under the assumption that the dominated law is unique
Summary
In the 1970s, probabilists studied conditions under which laws of semimartingales are (locally) absolutely continuous. Mijativic and Urusov [20] proved equivalent conditions for the martingale property of stochastic exponentials In both cases, the proofs are based on an extension of stopping times and different from ours. In a multidimensional Ito-diffusion setting, Ruf [26] proved equivalent conditions for the martingale property of stochastic exponentials using an extension argument similar to ours. As a second application of our main result, we generalize Benes’s [1] linear growth condition for the martingale property of stochastic exponentials to continuous Ito-process drivers. We emphasis that this application differs from the others, because no uniqueness argument is necessary. All standing assumptions are imposed only for the section they are stated in
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