We study the Darcy problem with log-normal permeability, modeling the fluid flow in a heterogeneous porous medium. A perturbation approach is adopted, expanding the solution in Taylor series around the nominal value of the permeability. The resulting recursive deterministic problem satisfied by the expected value of the stochastic solution, analytically derived and studied in [4], is discretize...

In this paper we study backward stochastic differential equations with general terminal value and general random generator. In particular, we do not require the terminal value be given by a forward diffusion equation. The randomness of the generator does not need to be from a forward equation, either. Motivated from applications to numerical simulations, first we obtain the Lp-Hölder continuity...

This paper studies two non-zero-sum stochastic differential games of control and stopping. One game has interaction in the players’ stopping rules, whereas the other does not. Solutions to backward stochastic differential equations (BSDEs) will be shown to provide the value processes of the first game. A multi-dimensional BSDE with reflecting barrier is studied in two cases for its solution: ex...

In this paper we consider robust and risk sensitive control of discrete time nite state systems on an in nite horizon. The solution of the state feedback robust control problem is characterized in terms of the value of an average cost dynamic game. The risk sensitive stochastic optimal control problem is solved using the policy iteration algorithm, and the optimal rate is expressed in terms of ...

This paper presents some new concepts and procedures for financial risk management. To complement the use of value at risk a new concept, upside potential or opportunity value as means to weigh opportunity loss versus risk reduction as well as an area ratio are introduced and discussed. Upper and lower bounds for risk curves corresponding to the optimal stochastic solution are developed, the ap...

This work develops a new framework for a class of stochastic control problems with optimal stopping. One of our main motivations stems from dealing with the option pricing of American type. The value function is characterized as the unique solution of a partial differential equation in a Sobolev space. Together with certain regularities and estimates of the value function, the existence of the ...

Optimality functions define stationarity in nonlinear programming, semi-infinite optimization, and optimal control in some sense. In this paper, we consider optimality functions for stochastic programs with nonlinear, possibly nonconvex, expected value objective and constraint functions. We show that an optimality function directly relates to the difference in function values at a candidate poi...

In this paper, a numerical efficient method based on two-dimensional block-pulse functions (BPFs) is proposed to approximate a solution of the two-dimensional linear stochastic Volterra-Fredholm integral equation. Finally the accuracy of this method will be shown by an example.

In this paper, stochastic generalizations of some fixed point for operators satisfying random contractively generalized hybrid and some other contractive condition have been proved. We discuss also the existence of a solution to a nonlinear random integral equation in Banah spaces.