If $X_t$ is a (continuous) diffusion process solving $$ dX_t = b(t,X_t)dt + \sigma(t,X_t)dW_t , $$ then its infinitesimal generator, denoted by $L$ is of the form by $$ L(f) = b(x) \cdot \nabla_{x} f(x) + \frac1{2} \big( \sigma(x) \sigma(x)^{\top} \big) : \nabla_{x} \nabla_{x} f(x). $$ Thus, $L$ is a second-order differential operator.

## Question

If $X_t$ is a *stochastic process* (not a diffusion), which admits an infinitesimal generator $L$, is it possible for $L$ to be of order greater than $2$?