Define $X_t$ as \begin{equation*} X_t=\int_0^t p\cdot\exp\left(a s + b W_s \right)ds+\int_0^t q \cdot\exp\left(a s + b W_s \right)dW_s, \end{equation*} where p,q,a,b are constants and $W_t$ is a standard Brownian motion.
Question 1: Is it always the case that the random variable $X_t$ has finite moments (that is, $\mathbb{E}[(X_t)^n]<\infty$ for $n=1,2,\ldots,$)?
Question 2: More generally, if we define $X_t$ as an Ito integral of the form \begin{equation*} X_t=\int_0^t f(s,W_s)ds+\int_0^t g(s,W_s)dW_s, \end{equation*} where $f$ and $g$ are integrable and continuous functions (as in question 1 above), is it always the case that the random variable $X_t$ has finite moments?
Note: Regarding question 2, one can see that this is true in many simple cases. For example, $X_t=\int_0^t g(s)dW_s$ is a normally distributed random variable with mean $0$ and variance $\int_0^t g(s)^2ds$. Likewise $X_t=\int_0^t W_sds$ is a normal random variable with mean $0$ and variance $(1/3)t^3$.