We consider the stochastic system $$\frac{dS_t}{S_t}=-R_t\,dW_t,$$ with $$dR_t=-R_t\,dt-R_t\,dW_t, \quad R_0>0.$$ We conjecture, and would like to show that $$\mathbb{E}[S_t^2] = S_0^2\,\mathbb{E}\left[\exp\left(-2\int_0^t R_s\,dW_s-\int_0^t R_s^2\,ds\right)\right]=+\infty \quad \text{for all } t>0.$$ A natural idea is to localize the stochastic exponential by setting $$\tau_n:=\inf\left\{u\ge0:\int_0^u R_s^2\,ds\ge n\right\}\wedge t,$$ and to perform a Girsanov change of measure on $[0,\tau_n]$. Writing $$Z_t^{(n)} := \exp\left(-2\int_0^{t\wedge\tau_n}R_s\,dW_s-2\int_0^{t\wedge\tau_n}R_s^2\,ds\right),$$ we have $$S_{t\wedge\tau_n}^2 = S_0^2\,Z_t^{(n)}\exp\left(\int_0^{t\wedge\tau_n}R_s^2\,ds\right),$$ and $Z^{(n)}$ is a true martingale. Defining $\mathbb Q_n$ by $$\frac{d\mathbb Q_n}{d\mathbb P}=Z_t^{(n)},$$ the Brownian motion is shifted into $$\widetilde W_u^{(n)}:=W_u+2\int_0^{u\wedge\tau_n}R_s\,ds,$$ and the process $R$ solves on $[0,\tau_n]$ $$dR_u=(-R_u+2R_u^2)\,du-R_u\,d\widetilde W_u^{(n)}.$$ The extra quadratic drift $2R_u^2$ implies that the tilted dynamics explodes: immediate application of the Feller criteria, which in turn indicates that the stopped moments $\mathbb E[S_{t\wedge\tau_n}^2]$ should become arbitrarily large. However, the main difficulty is to remove the localization, since from $S_{t\wedge\tau_n}^2\to S_t^2$ almost surely one only gets $$\mathbb E[S_t^2]\le \liminf_{n\to\infty}\mathbb E[S_{t\wedge\tau_n}^2].$$
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