I've asked this question here at Physics SE, but I figured that some parts would be more appropriate to ask here. So I'm rephrasing the question again.
We know that for option value calculation, path integral is one way to solve it. But the solution I get from the Black-Scholes formula (derived from the above question):
$$\begin{array}{rcl}\mathbb{E}\left[ F(e^{x_T})|x(t)=x \right] & = & \int_{-\infty}^{+\infty} F(e^{x_T}) p(x_T|x(t)=x) dx_T \\ & = & \int_{-\infty}^{+\infty} F(e^{x_T}) \int_{\tilde{x}(t)=x}^{\tilde{x}(T)=x_T} p(x_T|\tilde{x}(\tilde{t})) p(\tilde{x}(\tilde{t})|x(t)=x) d\tilde{x}(\tilde{t}) dx_T \end{array}$$
is very cryptic and simply unusable on a computer.
My question is, how can we program this solution? Or more generally, how can we devise computer algorithms to solve path integral problem in quantitative finance?