Optimal control by random sequences with constraints

Translated from Matematicheskie Zametki, Vol. 49, No. 6, pp. 143–145, June, 1991.     

A New Martingale Representation Theorem (Discrete Time)

In this paper, we prove a new theorem concerning representation of discrete time martingales, which is based on solving a specially chosen problem of optimal control with random sequences. We establish a relationship between the solution of the Bellman equation and the martingale representation. We illustrate the results with an example of calculating a European type option.     

On the Uniqueness of the Optional Decomposition of Semimartingales

Mathematical Notes -     

Existence Conditions for Extremal Probability Measures on Polish Spaces and Some of Their Properties

Mathematical Notes - In this paper, the properties of the primitives of Legendre polynomials on the interval $$[0;1]$$ are studied. It is proved that the Legendre polynomials form an...     

Asymptotics of the Solution to the Cauchy Problem for Linear Parabolic Equations of Second Order with Small Diffusion

This paper is devoted to constructing an asymptotics of the solution to the Cauchy problem for a linear parabolic equation of second order with variable coefficients containing a small parameter at the highest derivative. Sufficient conditions for the existence and uniqueness of the “multiplicative” asymptotic expansion of the global solution of the problem are given.