Fix $\alpha >0$, and for $h > 0$, consider the Markov kernel $K_h$ derived by composing the following two `moves':
- From $x^t$, move to $x^{t+1/2} = x^t + y^t$, where $y^t \sim \text{Gamma}(\alpha h, 1)$ .
- To fix notation, by this I mean $p(y) \propto y^{\alpha h-1} \exp(-y)$.
- From $x^{t+1/2}$, move to $x^{t+1} = x^t \cdot z^t$, where $z^t \sim \text{Beta}(\alpha, \alpha h)$.
- Again, for clarity, by this I mean $p(z) \propto z^{\alpha-1} (1-z)^{\alpha h-1}$.
I use $K_h$ to denote the composite Markov kernel which takes me from $x^t$ to $x^{t+1}$.
I know that this chain has $\text{Gamma}(\alpha, 1)$ as an invariant measure, and moreover, that the chain is reversible with respect to this measure.
I want to compute the behaviour as $h \to 0^+$ of this chain. Initially, I thought that the chain would admit a diffusion limit (something like the CIR process), but after carrying out some simulations, it appears more likely that it's something like a jump-diffusion, or even a pure jump process.
Anyhow, I'm not sure how I should go about i) identifying a limiting process, and ii) proving rigorously that it is the true limit of these Markov kernels. I can identify that if I define
$$B(x;h) \triangleq \mathbf{E}_{K_h} [x^{t+1} - x^t | x^t = x]$$
$$V(x;h) \triangleq \textbf{Var}_{K_h} [x^{t+1} - x^t | x^t = x], $$
then
\begin{align} B(x;h) &= h (\alpha - x ) + o(h) \\ V(x;h) &= h \left\{ \alpha^2 + \frac{\alpha}{\alpha + 1} x^2 \right\} + o(h). \end{align}
I initially though that this would mean the limiting process would be
$$dX_t = ( \alpha - X_t) dt + \sqrt{\alpha^2 + \frac{\alpha}{\alpha + 1} x^2 } dW_t$$
but this both i) doesn't have $\text{Gamma}(\alpha, 1)$ as a stationary measure, and ii) doesn't account for the jump behaviour I observe in the simulations.
Any advice would be well-received.