Time-dependent weak rate of convergence for functions of generalized bounded variation

Let W den(te the Brownian m(tion. For any exponentially bounded Borel function g the function u defined by u(t, x) = E[g(x + sigma WT-t)] is the stochastic solution of the backward heat equation with terminal condition g. Let u(n)(t, x) den(te the corresponding approximation generated by a simple symmetric random walk with time steps 2T/n and space steps +/-sigma root T/n where sigma > 0: For a class of terminal functions g having bounded variation on compact intervals, the rate of convergence of u(n)(t, x) to u(t, x) is considered, and also the behavior of the error u(n)(t, x) - u(t, x) as t tends to T.

Automated summary

The text discusses the stochastic solution of the backward heat equation under Brownian motion and the convergence rate of the corresponding approximation, as well as the behavior of the error, which is applicable to terminal functions with bounded variation.