Expected value formula holds for family of functions.

Question

Does Dynkin's formula hold for any function $f: \mathbb{R}^d \to \mathbb{R}$ such that $f$ and all of its partial derivatives of order $\le 2$ have at most polynomial growth at $\infty$?

Answer 1

Yes, it does. Let $f \in C_{\infty}^2$ (i.e. $f$ and its derivatives of order $\leq 2$ vanish at $\infty$). Then Dynkin's formula is a consequence of the fact that $f$ is in the domain of the generator $A$,

$$A := \left\{f \in C_{\infty}(\mathbb{R}^d); \exists h \in C_{\infty}(\mathbb{R}^d): \lim_{t \to 0} \left\| \frac{\mathbb{E}f( \bullet + W_t)-f(\bullet)}{t} -h(\bullet) \right\|_{\infty}=0 \right\}.$$

Here, $$C_{\infty}(\mathbb{R}^d) := \{f: \mathbb{R}^d \to \mathbb{R}; f \, \text{continuous}, \lim_{|x| \to \infty} f(x)=0\}.$$

Now the idea is to consider weighted spaces $C_{\infty,g}$ instead of $C_{\infty}$:

$$C_{\infty,g}(\mathbb{R}^d) := \left\{f: \mathbb{R}^d \to \mathbb{R}; f \, \text{continuous}, \lim_{|x| \to \infty} \left| \frac{f(x)}{g(x)} \right| = 0 \right\}$$ $$\|f\|_{\infty,g} := \sup_{x \in \mathbb{R}^d} \left| \frac{f(x)}{g(x)} \right|$$

for "nice" functions $g$. Since Brownian motion has exponential moments, one can show that $g(x) := \exp(|x|)$ is admissible and that any function $f \in C_{\infty,g}$ such that the derivatives of order $\leq 2$ are also contained in $C_{\infty,g}$ is in the domain of $A_g$,

$$A_g := \left\{f \in C_{\infty,g}(\mathbb{R}^d); \exists h \in C_{\infty,g}(\mathbb{R}^d): \lim_{t \to 0} \left\| \frac{\mathbb{E}f( \bullet + W_t)-f(\bullet)}{t} -h(\bullet) \right\|_{\infty,g}=0 \right\},$$

and $A_g f = \frac{1}{2} \Delta f$ (as expected). Using standard techniques, this yields Dynkin's formula for such functions $f$.