The probability of ruin for a Geometric Ornstein-Uhlenbeck (O-U) process with (\delta = 1) (i.e., (dX_t = \kappa (\theta - X_t) dt + \sigma X_t dW_t)) does not generally admit a closed-form solution, unlike the cases for (\delta = 0) (standard O-U) or (\delta = 1/2) (CIR process).

Key Analysis:

1. Process Dynamics: The given process is a mean-reverting diffusion with volatility p...

Assume default/ruin means hitting zero: τ = inf{t ≥ 0 : Xt ≤ 0}, starting from X0 > 0.

For δ = 1 the SDE is dXt = κ(θ − Xt) dt + σ Xt dWt.

This linear SDE has an explicit pathwise solution via variation of constants: Xt = Φt [ X0 + κ θ ∫0^t Φs^{-1} ds ], with Φt = exp(σ Wt − (κ + 0.5 σ^2) t). Hence Φt > 0 a.s. and Xt = exp(σ Wt − (κ + 0.5 σ^2) t) [ X0 + κ θ ∫0^t exp((κ + 0.5 σ^2) s − σ Ws) ds ]....

The Geometric Ornstein-Uhlenbeck process with δ = 1 that you've specified:

is particularly interesting because it combines mean-reversion with multiplicative noise. Unfortunately, there is no known closed-form solution for the probability of ruin for this process, unlike the standard O-U (δ = 0) and CIR (δ = 1/2) cases.

Why is this case...

For the special “geometric‐OU’’ drift‐diffusion

dXₜ = κ(θ–Xₜ) dt + σ Xₜ dWₜ,

and if you insist on calling “ruin’’ the first time Xₜ hits 0, the answer is in fact trivial:

• Xₜ never actually hits zero in finite time. By the usual Feller‐test of boundaries (or by writing down the explicit solution via an integrating factor) one shows that zero is a natural (inaccessible) boundary. Equival...

Let

 X₀ = x > 0 ,  τ = inf{ t ≥ 0 : X_t = 0 } ,

and let

 dX_t = κ ( θ – X_t ) dt + σ X_t dW_t ,  σ > 0 , κ > 0 (1)

(the geometric Ornstein–Uhlenbeck diffusion, i.e. δ = 1).

The question is to evaluate

 P_x ( τ ≤ T ) for a fixed horizon T > 0.  (2)

The answer is extremely simple: for every x>0 and every finite T,

 P_x ( τ ≤ T ) = 0 ...

<think> We are given a Geometric Ornstein-Uhlenbeck process (with δ=1) defined by: dX_t = κ(θ - X_t) dt + σ X_t dW_t

We wish to compute the probability of ruin (default) before time T, i.e., Pr(τ < T), where τ is the first time that X_t hits 0 (or a lower barrier, but typically 0 for ruin).

Note: In the standard Ornstein-Uhlenbeck process (δ=0), the volatility is constant and the process can b...

Let

  dX_t = κ ( θ – X_t ) dt + σ X_t dW_t ,  κ , θ , σ > 0, X_0 = x_0 > 0 (1)

(the case δ = 1 of the family dX = κ(θ–X)dt + σ X^δ dW).

Denote by

  τ := inf { t ≥ 0 : X_t = 0 } (the “ruin’’ time).

We want P( τ < T ).
For (1) the answer is simply 0 : the process can never hit 0 in finite time.
Hence a “closed-form’’ for a non–existent event is not needed.

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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...

Probability of Ruin for Geometric Ornstein-Uhlenbeck Process

The question asks about finding a closed-form solution for the probability of ruin (default) for a Geometric Ornstein-Uhlenbeck process with :

This is an interesting question, especially since the questioner notes that closed-form solutions exist f...