Feynman–Kac representations

Fix a finite terminal time and a nonexplosive diffusion from globally Lipschitz coefficients with linear growth. Let the terminal payoff and source be bounded and continuous, and let the killing rate be bounded, continuous, and nonnegative. Suppose a bounded function, once continuously differentiable in time and twice in space in the interior, solves the stated terminal PDE and is continuous up to its terminal data. These assumptions provide a clean class in which the representation and uniqueness can be verified.

Along a path, multiply the candidate solution by the exponential of minus the accumulated killing rate, then add the accumulated discounted source. Itô's formula shows that the PDE cancels the drift. The resulting process is initially a local martingale. Boundedness of the candidate, source, and finite horizon makes this process bounded, hence a true martingale after localization. Taking expectations and then the terminal limit proves the representation without silently assuming an arbitrary local martingale conserves its mean.

The representation has two terms: a discounted terminal payoff and an integral of discounted running source. Omitting the second term changes an inhomogeneous PDE into a homogeneous one. Since the killing rate is nonnegative, the discount factor is at most one; bounded payoff and source therefore give a finite expectation directly. Other growth conditions are possible, but an unbounded payoff or negative unbounded potential requires a separate integrability argument before the same expression can be asserted.

The verification theorem establishes uniqueness among bounded classical solutions satisfying its terminal condition. To conclude that the expectation itself is classical, one can impose, for example, smooth bounded coefficients and data with bounded derivatives and uniform ellipticity of covariance. Rougher data may yield a viscosity or distributional solution instead. The probabilistic representation does not erase these distinctions, and uniqueness can fail in overly large growth classes even for familiar heat equations.

On a bounded interval with absorbing boundary, stop the diffusion at its first exit time. A zero Dirichlet boundary condition replaces the terminal payoff by its contribution only on paths surviving to the horizon, and truncates the running-source integral at exit. Compatibility of continuous terminal data with the zero boundary value avoids a corner mismatch. Reflected diffusion would impose a different boundary rule and is not obtained by using the same killed-path indicator.

For nonzero prescribed boundary data, an additional discounted payoff is collected at exit before the terminal horizon. That term cannot be dropped merely because the process has stopped. The stopped Itô argument explains every contribution: what remains at the stopping time must match either terminal data or boundary data. On unbounded domains, controlling the limit of artificial spatial stopping times similarly requires the boundedness or growth assumptions that justify passing expectations to the limit.