Probabilistic representation of solutions to parabolic PDEs. For V(t,x) solving partial_t V + μ(x)partial_x V + tfrac12σ^2(x)partial_xx V - rV = 0 with terminal condition V(T,x) = h(x), the solution is V(t,x) = mathbbE!left[e^-r(T-t),h(X_T),big|,X_t = xright], where X follows dX = μ(X)dt + σ(X)dW. The bridge between the BSM PDE and the risk-neutral expectation pricing formula — and the theoretical basis of Monte Carlo pricing.