Excellent question. The process you're describing is a specific case of a mean-reverting Constant Elasticity of Variance (CEV) process, often called the Geometric Ornstein-Uhlenbeck (G-OU) or Geometric Mean Reversion (GMR) process.

The SDE is:

You are correct that for the Ornstein-Uhlenbeck process () and the Cox-Ingersoll-Ross (CIR) process (), closed-form or semi-closed-form solutions for first-passage time probabilities exist, often involving the error function or the non-central chi-squared distribution, respectively.

The case for is fundamentally different and, in general, does not have a simple closed-form solution for the finite-horizon ruin probability. Let's break down why and what solutions are available.

Part 1: The Definition of "Ruin" at the Boundary

First, let's consider the literal interpretation of ruin as hitting the boundary .

For the G-OU process, with standard assumptions for financial applications (), the process can never reach the zero boundary.

Why? Let's examine the behavior of the drift and diffusion coefficients at :

• Drift at 0: . The drift is strictly positive, strongly pushing the process away from zero.
• Diffusion at 0: . The diffusion term vanishes at zero, meaning the random fluctuations disappear as the process approaches the boundary.

Because the diffusion term dies off and the drift becomes strongly positive, the process is repelled from the zero boundary. In the language of stochastic calculus, the point is a natural boundary, which is inaccessible to the process starting from in finite time.

Conclusion: If "ruin" means hitting exactly 0, then for the G-OU process:

Part 2: A More Practical Definition: Ruin at a Barrier

In practice, default or ruin occurs when an asset's value falls below a certain positive threshold (e.g., a debt level). The problem then becomes calculating: for some barrier and a starting point .

This is a first-passage time problem for the G-OU process. Unlike the O-U and CIR cases, there is no known simple closed-form solution for this probability in a finite time horizon .

The standard approach for this kind of problem is to solve the associated Kolmogorov Backward Equation. Let . This function solves the partial differential equation (PDE):

with the following boundary conditions:

1. Terminal Condition: for (If you reach time T without hitting the barrier, you haven't been ruined).
2. Boundary Condition: for (If you are at the barrier, ruin is certain).
3. Upper Boundary: (If the value is very large, ruin is very unlikely).

While this PDE does not have a simple solution in terms of elementary functions, we can solve it using several methods.

Part 3: Solutions and Approaches

1. The Infinite Horizon Case ()

If you are interested in the probability of ever hitting the barrier , i.e., , a closed-form solution exists and can be derived using the scale function of the diffusion.

For a general diffusion , the probability of hitting a level before a level (with ) is given by: where is the scale function density, defined as .

For our G-OU process:

The integral in the exponent is:

So, the scale function density is:

The probability of ever hitting the barrier starting from is the probability of hitting before hitting . This is: These integrals do not have a simple elementary form but can be expressed in terms of the incomplete gamma function. This is a semi-closed-form solution.

2. Solutions for the Finite Horizon ()

Since there is no simple closed-form solution, the standard approaches are:

• Numerical PDE Solvers: Use a finite difference method (like Crank-Nicolson) to solve the Kolmogorov backward equation numerically on a grid for . This is a very common and robust industry practice.

• Monte Carlo Simulation: This is the most straightforward approach.

  1. Discretize the SDE using a scheme like Euler-Maruyama: where .
  2. Simulate a large number of paths for from to .
  3. For each path, check if drops to or below at any point in time.
  4. The probability is estimated as the fraction of paths that hit the barrier.

• Analytical Approximations / Spectral Expansions: For more advanced applications, researchers like V. Linetsky have developed methods to solve the PDE using eigenfunction expansions (also known as spectral methods). The solution for is expressed as an infinite series. The terms in the series involve special functions (Whittaker functions). While not "closed-form" in a simple sense, this provides a very accurate and efficient computational method once the framework is implemented.

Summary