Can somone gives me an introduction about the so-called Dynkin Operator ? (finance student means that I know stochastic calculus only at an introductory level without a solid background in funtional analysis, measure theory, ...) \begin{equation} D(X(t)):=\frac{d}{dt}{E}(X(t)) \end{equation} I have met it studying stochastic calculus for finance to compute the moments of Ito processes, using this equality: \begin{equation} \frac{E(d(x(t))}{dt}=\frac{dE(X(t))}{dt} \end{equation} (I am taking it from Baz,J.,Chacko,G., Financial derivatives p. 227)
For instance, here's an example with an Ornstein-Uhlenbeck process. \begin{equation} dX(t)=k[\theta-X(t)]dt+\sigma dW(t) \end{equation} From the previous equality follows that \begin{equation} \frac{dE(X(t))}{dt}=k \theta-k E(X(t)) \end{equation} This is a simple ODE that can be solved subject to $E(X(0))=x_0$ and so we obtain \begin{equation} E(X(t))=\theta+(x_0-\theta)e^{-kt} \end{equation}
However, I do not know anything about its interpretation, about the reason beacuse the above equality it's true and moreover I usually see it written in a much more complicated form. For instance, it seems to me that it is related to the infinitesimal generator, defined as \begin{equation} \mathcal{A}f(x)= \lim_{t \to 0} \frac{E^x[f(X_T)]-f(x)}{t} \end{equation}
UPDATE
I've discovered reading Bjork,T., Arbitrage theory in continuous time, that Dynkin operator is also called Infinitesimal operator, or Ito operator or Kolmogorov backward operator. He defines it in the following way: given \begin{equation} dX_t=\mu (t,X_t) dt + \sigma (t,X_t),dW_t \end{equation} the infinitesimal operator of $X$ is defined for any function $h(x) \in C^2(R^n)$ by \begin{equation} \mathcal{A}h(t,x)= \sum_{i=1}^n \mu_i(t,x) \frac{\partial h}{\partial x_i} (x)+\frac{1}{2} \sum_{i,j=1}^n C_{ij}(t,x) \frac{\partial h}{\partial x_i \partial x_j}(x) \end{equation} where \begin{equation} C(t,x)=\sigma(t,x)\sigma^*(t,x) \end{equation}
Thank you for your help.