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So I'm looking at a diffusion process with killing with a state- and time-dependent killing rate. This is described in Oksendal's Stochastic differential equations pages 143-145 "The Feynman-Kac Formula. Killing". Basically, you have a generator $$ L f = -\sum_i \frac{\partial }{\partial x_i} A_i(x,t) f(x) + \frac{1}{2} \sum_{i,j} \frac{\partial^2}{\partial x_i \partial x_j} B_{i,j}(x,t) f(x) - c(x,t)f(x), $$ and this corresponds to a process with drift $A$, diffusion $B$ and a killing rate given by $c(x,t)$.

However, no one mentions if the killing rate can be time-dependent. The demonstration uses the stochastic process $$ Z_t = \exp(-\int_0^t c(X_s)) ds, $$ with $dZ_t$ given by $$ dZ_t = -Z_t c(X_t) dt. $$ However, if $c$ is a function of time too, there would be an additional term in the differential, right? Wikipedia's Feynman-Kac formula page states the problem with a time-dependent potential $V(x,t)$ but then goes on to drop this dependence throughout the page. Can I go on to use the Feynman-Kac formula if the potential is time-dependent? Do I have to include some additional terms somewhere? My hunch is yes, but I'm not sure how to derive the correct formulation!

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    $\begingroup$ No, I would say there are no additional terms. For a precise statement (and proof) of the Feynman-Kac formula with killing and time dependent coefficients you may look at "Brownian Motion and Stochastic calculus" by Karatzas-Shreve (2nd edition), Theorem 7.6 on page 366. $\endgroup$
    – Hans
    Commented Feb 22, 2012 at 12:47
  • $\begingroup$ if $c$ is a function of time too, just consider $$Z_t=\exp(-\int_0^tc(s,X_s))ds$$ then you have $dZ_t=-Z_tc(s,X_s)dt$ and the same reasoning applies. $\endgroup$
    – Hicham
    Commented Jun 13, 2013 at 14:52

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I was probably a bit confused because of the operators used. The operator shown in the question is the Kolmogorov forward operator. The backwards operator is the adjoint of $L$, given by $$ L^* f = \sum_i A_i(x,t) \frac{\partial f}{\partial x_i} + \sum_{i,j}\frac{B_{i,j}(x,t)}{2}\frac{\partial^2 f}{\partial x_i \partial x_j} - c(x,t) f. $$ Furthermore, the operator only helps to define the process $X_t$, which will lead to the Ito formula and then to the Feynman-Kac formula.

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