Ruin Probability for a Geometric Ornstein–Uhlenbeck Process

Summary

The document asks how to calculate the probability that a Geometric Ornstein–Uhlenbeck process reaches a default boundary before a specified horizon. It gives the process with mean-reversion parameters and multiplicative state-dependent diffusion, focusing on the case where the diffusion exponent is one. The quantity of interest is the chance that the default time occurs before the horizon.

The author contrasts this case with the standard Ornstein–Uhlenbeck and Cox–Ingersoll–Ross processes, for which they have found closed-form results. The document provides no derivation, proposed solution, numerical example, or evidence that a closed form exists for the geometric case. It is therefore a research question that identifies the model and target probability, rather than a complete method. Any calculation would also depend on details not supplied here, including the starting value and the definition of the default boundary.

Key ideas

• The target is the probability that default occurs before a finite time horizon.
• The process combines mean reversion with diffusion proportional to the state when the exponent is one.
• The author contrasts this case with standard Ornstein–Uhlenbeck and Cox–Ingersoll–Ross models.
• The document poses the problem but does not provide a solution or establish whether a closed form exists.

Tags

# What is the probability of ruin of a Geometric Ornstein-Uhlenbeck process?


# What is the probability of ruin of a Geometric Ornstein-Uhlenbeck process?












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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