Abstract
In a fully general setting, we study the relation between martingale spaces under two locally absolutely continuous probabilities and prove that the martingale representation property (MRP) is always stable under locally absolutely continuous changes of probability. Our approach relies on minimal requirements, is constructive and, as shown by a simple example, enables us to study situations which cannot be covered by the existing theory.
Highlights
Martingale representation results have fundamental applications in stochastic control, filtering, backward stochastic differential equations and mathematical finance, notably in connection with the property of market completeness
In a fully general setting, we study the relation between martingale spaces under two locally absolutely continuous probabilities and prove that the martingale representation property (MRP) is always stable under locally absolutely continuous changes of probability
If M = (Mt)t≥0 is an Rd-valued process such that M i ∈ Mloc(P ), for each i = 1, . . . , d, we denote by Lm(M, P ) the set of all Rd-valued predictable processes which are integrable with respect to M under the measure P in the sense of local martingales
Summary
Martingale representation results have fundamental applications in stochastic control, filtering, backward stochastic differential equations and mathematical finance, notably in connection with the property of market completeness. The crucial assumption in the above result is the requirement that [X, Z] has locally integrable variation under P (or, equivalently, that X is a special semimartingale under Q) This leaves open the question of whether, in the absence of such a condition, the MRP is preserved or not under an absolutely continuous change of probability. By relying on our main results, we address further issues, including the practically relevant case of locally equivalent probabilities and the dimension of martingale spaces under locally absolutely continuous probabilities (Section 2.3) These results enable us to provide a general solution to an open problem stated in [16].
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