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Dynamic risk measuring has been developed in recent years in the setting of a filtered probability space (Ω,(Ft)0≤t, P). In this setting the risk at time t is given by a Ft-measurable function defined as an ”ess-sup” of conditional expectations. The property of time consistency has been characterized in this setting. Model uncertainty means that instead of a reference probability easure one considers a whole set of probability measures which is furthermore non dominated. For example one needs to deal with this framework to make a robust evaluation of risks for derivative products when one assumes that the underlying model is a diffusion process with uncertain volatility. In this case every possible law for the underlying model is a probability measure solution to the associated martingale problem and the set of possible laws is non dominated.
In the framework of model uncertainty we face two kinds of problems. First the Q-conditional expectation is defined up to a Q-null set and second the sup of a non-countable family of measurable maps is not measurable. To encompass these problems we develop a new approach [1, 2] based on the “Martingale Problem”.
The martingale problem associated with a diffusion process with continuous coefficients has been introduced and studied by Stroock and Varadhan [4]. It has been extended by Stroock to the case of diffusion processes with Levy generators [3]. We study [1] the martingale problem associated with jump diffusions whose coefficients are path dependent. Under certain conditions on the path dependent coefficients, we prove existence and uniqueness of a probability measure solution to the path dependent martingale problem. Making use of the uniqueness of the solution we prove some ”Feller property”. This allows us to construct a time consistent robust evaluation of risks in the framework of model uncertainty [2].
References
[1] Bion-Nadal J., Martingale problem approach to path dependent diffusion processes with jumps, in preparation.
[2] Bion-Nadal J., Robust evaluation of risks from Martingale problem, in preparation.
[3] Strook D., Diffusion processes asociated with Levy generators, Z. Wahrscheinlichkeitstheorie verw. Gebiete 32, pp. 209-244 (1975).
[4] Stroock D. and Varadhan S., Diffusion processes with continuous coefficients, I and II, Communications on Pure and Applied Mathematics, 22, pp 345-400 (1969).
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