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I would like to calculate the probability of ruin (or, default), i.e. $$\text{Pr}(\tau<T),$$ where $\tau$ is the default time and $X_t$ follows the Geometric Ornstein-Uhlenbeck (O-U) process

$$dX_t=\kappa (\theta - X_t) dt + \sigma X_t^{\delta}dW_t,$$ with $\delta=1$.

I can find closed-form solutions for the cases $\delta=0$ and $\delta=1/2$, which correspond to the standard O-U and CIR processes, for example here. Does a similar solution exist for the $\delta=1$ case?

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    $\begingroup$ Where is the default boundary - a) exactly at zero or b) strictly positive? $\endgroup$ – LocalVolatility Jun 18 '17 at 21:51
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    $\begingroup$ $B = 0$, thanks for asking, I should have clarified it. $\endgroup$ – Vasilis D Jun 18 '17 at 21:52
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    $\begingroup$ Here is my intuition: Consider $\delta = 1$ and $\theta = 0$. Then we are back in the standard GBM case. For $X_0 > 0$, zero cannot be attained in finite time. Adding back $\kappa \theta > 0$, this shouldn't change anything. $\endgroup$ – LocalVolatility Jun 18 '17 at 22:04
  • $\begingroup$ True, so let's consider the case where $\kappa<0$ and $\theta >0$. In this case, my intuition is that default is possible. Think of $\kappa=-1$ for simplicity, and of $\theta$ as the rate of consumption per unit of time from an initial endowment $X_0>0$ whose growth follows a GBM. For a high enough consumption rate, financial ruin is probable! $\endgroup$ – Vasilis D Jun 18 '17 at 22:12
  • $\begingroup$ I tried to answer the question, but I am having difficulty interpreting the $\delta$ parameter. Intuitively, wouldn't $\delta = 1$ correspond with GBM? $\endgroup$ – David Addison Jan 31 '18 at 21:47

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