The following aspects of stochastic calculus, statistics and limit theorems for stochastic processes and fields will be investigated under this Project.

1. Non-Ito stochastic integration, stochastic differential equations and their intersection with financial mathematics. Sample paths properties, type of energy, properties of quadratic variation for stochastic integrals with respect to fractal processes, arbitrage possibilities for fractal integrals and mixed models will be considered; moment and maximal inequalities, Fubini theorem, Ito formula, general Girsanov theorem for fractal integrals will be proved; Malliavin calculus for fractal integrals will be constructed. Existence, uniqueness and smoothness of the solutions of stochastic differential equations (SDE) with mixed fractal differentials will be established; changing of measure for equations without arbitrage possibilities will be realized, the conditions of economic equilibrium and financial applications will be presented.

2. Properties and asymptotic behaviour of SDE and stochastic partial differential equations (SPDE) with different kinds of perturbing terms. Existence, uniqueness, stability of solutions, stability properties of the equilibrium, structural stability properties via random attractions of SPDE of the reaction-diffusion type will be proved, random invariant measures will be constructed, stochastic bifurcations will be considered, applications to climate dynamics will be presented. Quasi-geostrophic approximation of the Navier-Stokes equation for atmosphere and ocean will be constructed, where a stationary stochastic field perturbs the driving wind field. The localization of the solution around the source, localization and propagation phenomena of the temporal evolution will be investigated. Asymptotics of SDE (SPDE) with perturbing term represented by rapidly oscillating process (field) will be established and applied to generalize Black-Scholes model. Asymptotical stochastic stability of random evolution processes that are the solutions of stochastic integral equations will be established and applied to financial market and to semi-Markov risk processes. General comparison theorem and convergence of solutions of backward SDE will be established.

3. Aspects of Malliavin calculus. Malliavin calculus associated with the processes having time-dependent coefficients (diffusion models, solutions of Volterra equations, Heath-Jarrow-Morton model) will be constructed.

4. Asymptotical statistics of stochastic processes (including financial applications). Laws of large numbers, large deviation theorems, weak convergence theorems for the logarithm of the local density processes for semimartingales with discontinuous characteristics will be proved, rate of decrease of error probabilities for Bayes, minimax and Neyman-Pearson tests, weak convergence for Bayes and maximum-likelihood estimates of unknown parameters for observed semimartingales will be established. General limit theorems and asymptotic expansions for likelihood ratio processes will be obtained. Necessary and sufficient conditions for a sequence of filtered statistical experiments to be approximated by a sequence of exponential filtered experiments will be proved and applied for estimation "Greek letters" of risk-management models.

5. Statistics of stochastic fields with applications. The conditions of absolute continuity and singularity of the measures corresponding to random fields with independent increments will be presented in terms of Hellinger field. Nonparametric estimation of intensity function of nonhomogeneous Poisson field and mixed diffusion-Poisson model will be constructed, its asymptotic properties will be established and applied to ecology. Consistent estimations of regression of linear models of fields arising in geostatistics will be investigated.

6. Mathematical models in finance. The moments of optimal switching for financial investor will be found, optimal stopping domains for the buyer of American call options will be constructed, representation theory will be applied to classical American put options, Malliavin calculus and enlargement of filtrations will be applied to the problem of additional expected utility of an insider which he can get from his extra information at financial market with two agents. The estimation of intensity and of value of jumps will be realized in the following problem: to construct the histogram of the income of the buyer of call option and to find an equilibrium option price. The methods of calculations of premium in finance and insurance will be studied and compared. The structure of hedges for quintile hedging in complete and incomplete markets, markets with constrains and transaction costs will be investigated. Pricing of insurance derivatives, namely, equity-linked insurance contracts, CAT-options and CAT-futures will be studied.