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    A Hoeffding-Azuma Type Inequality for Random Processes
    (Etamaths Publ, 2023) Hasanov, Mahir
    The subject of this paper is a Hoeffding-Azuma type estimation for the difference between an adapted random process and its conditional expectation given a related filtration.
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    Initial value problems for fractional p-Laplacian equations with singularity
    (Amer Inst Mathematical Sciences-Aims, 2024) Hasanov, Mahir
    We have studied initial value problems for Caputo fractional differential equations with singular nonlinearities involving the p -Laplacian operator. We have given a precise mathematical analysis of the equivalence of the fractional differential equations and Volterra integral equations studied in this paper. A theorem for the global existence of the solution was proven. In addition, an example was given at the end of the article as an application of the results found in this paper.
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    A PDE Approach to the Problems of Optimality of Expectations
    (Etamaths Publ, 2023) Hasanov, Mahir
    Let (X, Z) be a bivariate random vector. A predictor of X based on Z is just a Borel function g(Z). The problem of least squares prediction of X given the observation Z is to find the global minimum point of the functional E[(X - g(Z))2] with respect to all random variables g(Z), where g is a Borel function. It is well known that the solution of this problem is the conditional expectation E(X|Z). We also know that, if for a nonnegative smooth function F : RxR & RARR; R, arg ming(Z)E[F (X, g(Z))] = E[X|Z], for all X and Z, then F (x, y) is a Bregmann loss function. It is also of interest, for a fixed & phi; to find F (x, y ), satisfying, arg ming(Z)E[F (X, g(Z))] = & phi;(E[X|Z]), for all X and Z. In more general setting, a stronger problem is to find F (x, y) satisfying arg miny & ISIN;RE[F (X, y)] = & phi;(E[X]), & FORALL;X. We study this problem and develop a partial differential equation (PDE) approach to solution of these problems.