Ornstein–Uhlenbeck Probability of Hitting an Upper Barrier First

Summary

The document gives a closed-form expression for the probability that an arithmetic Ornstein–Uhlenbeck process reaches one of two barriers first. It considers a process starting between a lower and upper level, with positive mean-reversion speed, and expresses the result using the imaginary error function evaluated at scaled distances from the long-run mean.

The answer outlines the reasoning: transform the process with the imaginary error function, establish that the transformed process is a martingale using Itô’s lemma, and apply optimal stopping at the barriers. The displayed formula is for hitting the lower barrier first; the probability of hitting the upper barrier first follows by taking its complement, assuming one of the barriers is reached. The response offers no derivation details or numerical examples, so the martingale and stopping conditions are asserted rather than fully checked.

Key ideas

• A transformed OU process can be used to express its two-barrier hitting probabilities.

Tags

# What is the probability that a OU process hits an upper barrier U before a lower barrier L?


# What is the probability that a OU process hits an upper barrier U before a lower barrier L?












What is the probability that the arithmetic OU process $dx_t= \theta(\mu-x_t)dt+\sigma dW_t$ hits barrier $U$ before hitting barrier $L$ when $L<x_0<U$ ?

## Answer by M. Jeunesse (score 5, accepted)

https://quant.stackexchange.com/a/28086

Assuming $\theta>0$ (take $\tilde{X}=\mu-X$ if it is not the case)

Let us denote $\text{erfi}(x)$ the imaginary error function Let us denote $\tau_L$,resp.$\tau_U$ the hitting time of $L$resp.$U$ where $L<U$

1) Using Ito's lemma, prove that : $$Y_t = \text{erfi}\left(\sqrt{\frac{\theta}{\sigma^2}}\left(X_t-\mu\right)\right) \text{ is a martingale}$$

2) Using optimal stopping theorem, prove that : $$\mathbb{P}(\tau_L\leq \tau_U) = \frac{\text{erfi}\left(\sqrt{\frac{\theta}{\sigma^2}}\left(x_0-\mu\right)\right)-\text{erfi}\left(\sqrt{\frac{\theta}{\sigma^2}}\left(U-\mu\right)\right)}{\text{erfi}\left(\sqrt{\frac{\theta}{\sigma^2}}\left(L-\mu\right)\right)-\text{erfi}\left(\sqrt{\frac{\theta}{\sigma^2}}\left(U-\mu\right)\right)}$$

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