Let's consider Dothan's model of the short-time interest rate:
$$d r_t = \mu r_t dt + \sigma r_t dW_t,$$ where
$r_0 = r$, $\sigma>0$, $\mu \in \mathbb{R}$.
Prove that: $\mathbb{E}(B_t)= \infty$, where $B_t$ is the banking account process.
My first idea was to find the solution of SDE $d r_t = \mu r_t dt + \sigma r_t dW_t$, which is: $r_t = r_0 \exp((\mu-\frac{\sigma^2}{2})t + \sigma W_t)$. Knowing that $W_t \sim N(0,t)$ and $B_t = \exp(\int\limits_0^t r_u du$), I tried to calculate $E(B_t)$, but I failed.
One of the big drawback of lognormal models is the explosion of the bank account. Let $\Delta t$ be small, then $$\mathbb{E}[B_{\Delta t}]=\mathbb{E}\left[\exp\left(\int_0^{\Delta t}r_u du\right)\right]\approx\mathbb{E}\left[\exp\left(\frac{r_0+r_{\Delta t}}{2}\Delta t\right)\right]$$ We have $$\mathbb{E}[B_{\Delta t}]\approx\mathbb{E}[\exp(\exp(Y))]$$ where Y is Gaussian distributed. But such an expectation is infinite. This means that in arbitrarily small time the bank account growths to infinity in average.