  Let $W$ denote the Brownian motion. For any exponentially bounded Borel
function $g$ the function $u$ defined by $u(t,x)= \mathbb{E}[g(x{+}\sigma
W_{T-t})]$ is the stochastic solution of the backward heat equation with
terminal condition $g$. Let $u^n(t,x)$ denote the corresponding approximation
generated by a simple symmetric random walk with time steps $2T/n$ and space
steps $\pm \sigma \sqrt{T/n}$ where $\sigma > 0$. For quite irregular terminal
conditions $g$ (bounded variation on compact intervals, locally H\"older
continuous) 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 $T$

Probability