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

Question

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

Answer 1

For a probabilistic and analytic treatment to your questions check out:

1. Chapter 1 of Second Order PDE's in Finite and Infinite Dimension. S. Cerrai. Springer, 2001

2. Chapter 2 of Analytical Methods for Markov Semigroups. L. Lorenzi, M. Bertoldi, CRC Press, 2006.

respectively. I believe both references treat the Kolmogorov equation where time flows forward, not backwards as in (1), but that is a minor difference.

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 $f \in C_b(\mathbb{R}^n)$.