Feynman-Kac And Infinite Time Horizon

Question

(1) is there literature about a Feynman-Kac formula on an infinite time horizon?

(2) This means:

... Let $X$ be the process satisfying the SDE \begin{equation} dX(t) = \mu(t,X(t)) dt + \sigma(t,X(t)) dW(t),\quad t\ge 0, \end{equation} $\mu,\sigma$ satisfying the usual Lipschitz conditions, $W$ being a Wiener process. Let $\mathcal A$ denote the infinitesimal generator of this SDE, and let be $f,k: \mathbb R_+ \times \mathbb R_+^\ast\longrightarrow \mathbb R$, sufficiently integrable (...), and such that $k\ge0$.

... Now, let $v \in C^{1,2}(\mathbb R_+ \times \mathbb R_+^\ast)$ be a solution to the PDE \begin{equation} \frac{\partial v}{\partial t} + \mathcal A v-k v+f =0, \end{equation} with terminal condition (?), then \begin{equation} v(t,x) = \mathbb E\Big[\int_t^\infty \beta^{t,X}(s)f(s,X(s)) ds + (??) \mid X(t)=x\Big], \quad (t,x) \in \mathbb R_+ \times \mathbb R_+^\ast, \end{equation} where $\beta^{t,X}(s)=e^{-\int_t^s k(u,X(u))~du}$, $s\ge t$.

(3) Now the question is: What can we take for (?) and (??) in order to make the statement true?

I suggest: (?) = $\sup_{x\ge 0}|v(t,x)|$ has at most polynomial growth as $t\rightarrow \infty$, e.g. $\lim_{t\rightarrow \infty} \sup_{x\ge 0} v(t,x) = 0$, (??) = $0$.

(4) Better offers?

Yours,

K.

Answer 1

I have came across the following Feynmann-Kac formulation.

Let $Z$ be a stochastically continuous Feller process and $c: \mathbb{R}\rightarrow\mathbb{C}$ be uniformly bounded and continuous function. Then for each $f\in Dom(\mathcal{A})$ $$ v(t, x) := \mathbb{E}_xf(Z_t)\exp\left(\int_0^tc(Z_s)ds\right) $$ is the unique solution of $$ \frac{\partial}{\partial t}v=Av+cv, \quad v(0, x)=f(x) $$ in a class of functions $v$ such as:

• $\sup_x |v(t, x)|\leq Ce^{\gamma t}$, for some constants $C, \gamma$,
• both $v$ and $\frac\partial{\partial t}v$ are continuous as functions from $\mathbb{R}_+$ to $C_0$. ($\mathbb{R}_+ \rightarrow C_0$ mapping is between functions on state space).

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Answering your question, I think at most exponential growth and convergence of $v$ w.r.t. to $t$ is enough and (??) becomes something connected with the limit. The formulation above is taken from lecture notes, non-english unfortunately. Hope this helps.