Yu. S. Mishura

Differentiability of Fractional Integrals Whose Kernels Contain Fractional Brownian Motions

We prove the stochastic Fubini theorem for Wiener integrals with respect to fractional Brownian motions. By using this theorem, we establish conditions for the mean-square and pathwise differentiability of fractional integrals whose kernels contain fractional Brownian motions.

Two-parameter Lévy processes: ItÔ formula, semigroups, and generators

We consider random Lévy fields, i.e., stationary fields continuous in probability and having independent increments. We prove that the trajectories of such fields have at most one jump on every line parallel to the axes. We derive an expression for the ItÔ change of variables for Lévy fields. We also consider semigroups generated by Lévy fields and their generators.

Optimal stopping times for solutions of nonlinear stochastic differential equations and their application to one problem of financial mathematics

We solve the problem of finding the optimal switching time for two alternative strategies at the financial market in the case where a random processXt,t ∈ [0, T], describing an investors assets satisfies a nonlinear stochastic differential equation. We determine this switching time τ∈[0,T] as the optimal stopping time for a certain processYt generated by the processXt so that the average investors assets are maximized at the final time, i.e.,EXT.

Method of successive approximations for abstract volterra equations in a banach space

We apply the method of successive approximations to abstract Volterra equations of the formx=f+a*Ax, whereA is a closed linear operator. The assumption is made that a kernela is continuous but is not necessarily of bounded variation.

The problem of extension for two-parameter kernels

We solve the problem of construction of a two-parame to either a multiplicative or a coordinatewise two-parameter semigroup. The construction is carried out on the basis of the “initial family of kernels.”

The Hille-Yosida theorem for resolvent operators of multiparameter semigroups

We consider multiparameter semigroups of two types (multiplicative and coordinatewise) and resolvent operators associated with such semigroups. We prove an alternative version of the Hille-Yosida theorem in terms of resolvent operators. For simplicity of presentation, we give statements and proofs for two-parameter semigroups.

Existence and properties of local times for Markov random fields

Random fields that have the “coordinatewise” Markov property are considered. The notions of an excessive function, a potential, and a continuous additive functional are introduced. Sufficient conditions for the existence of local time as a special form of continuous additive functional are formulated, and the uniqueness of this time to within a multiplicative constant is proved.