Does $E^{x,t}(f(X_T))$ solve a PDE if $f$ is not continuous?

Many books [see below for references] explore the connections between partial differential equations and expectation values.

Assume $X$ is a diffusion with generator $A$, then they conclude, that under certain conditions, the function $u(t,x):=E^{t,x}(f(X_T))$ is the solution to the following Cauchy problem:

$$\begin{gather} \frac{\partial u}{\partial t}+Au=0, \text{ on } [t,T)\times\mathbb R \\ u(T,x) = f(x), \text{ on } \mathbb R\end{gather} \tag{1}$$

More genereal versions of this are usually called the Feynman-Kac theorem.

Questions

Do you have a reference where $f$ is not required to be continuous?

I have never (expect for the case where $X$ is a Brownian Motion) found any theorem that did not require $f$ to be continuous. Why?

Apart from rigorous treatments (as in the mentioned books), most people seem not to care about $f$'s continuity. (e.g. the Wikipedia entry on the Kolmogorov backward equations, and many more). Are they wrong?

What about piecewise continous $f$? Is the problem to trivial and to specialized for the cited books?

My understanding

If for the process $X$ has a transition density $p(t,x,s,y)$ that is in $C^{1,2}$ for fixed $s,y$ (which seems to be the case under similar conditions as for the Feynman-Kac theorem), then

$$u(t,x) = \int f(y) p(t,x,T,y) dy$$

and $u$ is also in $C^{1,2}$ by Leibniz's integral rule for bounded $f$ and thus should also solve (1).

Is this correct?

References

Oksendal, Stochastic Differential Equations

Karatzas and Shreve, Brownian Motion and Stochastic Calculus

Friedman, Stochastic differential equations and applications

For example, Cerrai shows that if the coefficients of the generator are of class $C^k$, then for any $t>0$ and for any $f \in B_b(\mathbb{R}^n)$, the function $P_t f(x) = \mathbb{E}_x (f(X_t))$ is $k$-times differentiable and its derivatives up to order $k$ are bounded in the supremum norm; for precise assumptions see Hypotheses 1.1-1.3. Let me emphasize the proof of this regularizing property of the semigroup $P_t$ relies on the fact that the diffusion term is not degenerate, see Hypothesis 1.3. In addition, Theorem 1.6.2 states that the function $P_t f(x)$ solves (1), however, it requires that $f \in C_b(\mathbb{R}^n)$.
• Interesting! While I think I found the statements about the smoothing properties of the semigroup $P_t$ in Cerrai, I cannot find anything about $P_tf$ being the solution to a PDE if $f\in B_b$. Is it in Cerrai? – JSG Mar 14 '15 at 21:05