I encounter the following question
Let $X$ satisfy the SDE $$dX_s=k(\alpha-X_s)ds+\sigma\sqrt{X_s}dW_s$$ for $s\geq t$ with $X_t=x$, where $k,\alpha,\sigma$ are positive constants. Find the conditional expectation $$u(t,x)=\mathbb{E}[exp(\beta X_T)|\mathcal{F}_t]$$, where $\beta$ is a positive constant satisfying $k-\sigma^2\beta/2>0$.
$X$ obviously follows a CIR process and therefore $X_t$ has a chi-squared distribution and $u(t,x)$ is just the mgf of a chi-sq distribution (times some deterministic function). My question is: In an exam, if we don't know the density function of $X_t$, nor the mgf of $X_t$, is there any other way to solve for $u(t,x)$?