Simulation Algorithms for the Second-Order Parabolic Cauchy Problem

Monte Carlo algorithms, which solve boundary value problems for the heat equation whose elliptic part is the Laplace operator, have been known for a long time [1], [2]. They essentially use the explicit form of a fundamental solution and cannot be transferred to equations containing higher derivatives with nonconstant coefficients. A simulation method for solving the Cauchy problem for a second-order parabolic equation with smooth coefficients is proposed and thoroughly studied. Unbiased estimators for both the solution of the Cauchy problem and functionals of this solution are constructed.