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Random walk path methods including walk on spheres and walk on rectangles have been used to solve elliptic and parabolic partial differential equations (PDEs). These methods are able to provide not only the pointwise solutions to the linear PDEs but also contributions of boundaries and all source/sink terms as an analytical solution does. However, due to difficulty in dealing with heterogeneity, these methods cannot be applied to groundwater flow problems in highly heterogeneous aquifers. A novel method called walk on grid (WOG) is proposed based on lattice random walk to overcome the difficulty. WOG algorithm is verified in a 1D homogeneous transient problem, a 1D heterogeneous steady-state problem, a 2D heterogeneous transient problem and a 3D optimization problem. It is demonstrated that WOG is effective in solving groundwater flow problems in highly heterogeneous confined aquifers. Probabilities of walkers arriving at prescribed boundaries (terminal weights) and source counts may be useful for characterization of medium heterogeneity. WOG method sheds a new light on solving the PDEs of complicated groundwater problems in a changing environment and on analyzing medium heterogeneity. [ABSTRACT FROM AUTHOR]

In this article, we account for the liquidity risk in the underlying assets when pricing European exchange options, which has not been considered in the literature. An Ornstein–Uhlenbeck process with the mean-reversion property is selected to model the market liquidity risk, whose impacts on the underlying assets are assumed to be imposed with a liquidity discount factor. Under this framework, we develop a simplified approach to obtain the pricing formula, a distinguishing feature of which is that it does not require any numeraire change. We first transform the exchange option's pricing PDE into a pricing PDE of the European vanilla option, which is then solved in closed-form using the characteristic function approach. Finally, we present accuracy tests and sensitivity analysis to demonstrate the correctness of the formula and the influence of introducing the liquidity risk on exchange options, respectively. • We account for the liquidity risk for pricing European exchange options. • We develop a novel approach when deriving the closed-form solution. • We carry out sensitivity analysis to study the effect of liquidity risk. [ABSTRACT FROM AUTHOR]

We present the idea of intertwining of two diffusions by Feynman–Kac operators. We present implications of the method and give its applications. The examples give new results on stochastic processes including a generalized squared Bessel processes. We present a version of the method and its applications to PDE of the second order. A new dependence between diffusions and solutions of hyperbolic PDE is presented. The version of Feynman–Kac representation for hyperbolic PDE is given. It is presented the simple form of Laplace transform of wave equation with axial symmetry. [ABSTRACT FROM AUTHOR]

The Feynman–Kac theorem and Bogolyubov inequality are applied to obtain a lower bound and an upper bound to the free energy of the s–d Hamiltonian with locally smeared interactions between electrons and impurities. The two bounds, which express in terms of the free energy of impurities in a mean field and electrons in a field of barriers and wells localized at the impurity sites, are almost equal if the impurity concentration is sufficiently small or s–d coupling sufficiently weak. [Copyright &y& Elsevier]