Absolute Continuity of Semimartingales

David Criens; Kathrin Glau

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arXiv · 1706.04944

Absolute Continuity of Semimartingales

David Criens·Kathrin Glau

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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 It\^o-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 multi-dimensional It\^o-diffusions and derive linear growth conditions for the martingale property of stochastic exponentials.

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David Criens, Kathrin Glau. 2017-06-15. Absolute Continuity of Semimartingales. https://arxiv.org/abs/1706.04944

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