Research programme in relation to the state-of-the-art:
It is generally known that modeling of asset prices and their derivatives requires to consider more complicated noise consisting of both continuous part (Brownian motion) and discontinuous (jump part). Consequently, models with Levy has been considered as basic in mathematics of finance independently of the fact that such models include main insurance risk models. Such extension requires a number of technical problems to overcome. In particular, such necessity arises in term structure modeling and credit risk modeling (which are recently of great interest (see papers published in recent issues of Financial Mathematics and Finance and Stochastics).
Stability and closely related uniqueness of invariant measure are fundamental aspects considered in the study of long run asymptotics of stochastic models. In particular, asymptotics of filtering processes is the one of fundamental problems of recent systems theory (see papers by Van Handel 2008 and 2010).
There has been a lot of activities in mathematics of finance devoted to modeling and pricing of derivatives on idealistic liquid market without friction (see Delbaen Schachermayer "Mathematics of Arbitrage", Springer 2006). To make our modeling feasible, we have to consider more complicated models which involve transaction costs and an influence on the market of large transactions. Risk modeling, construction of proper measures of risk one of the most important problems of modern mathematics of finance.
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