Exit Probability for a Diffusion via Its Generator ODE
Summary
The document derives a boundary-value equation for the probability that a one-dimensional diffusion exits an interval through its upper endpoint. For a process with drift g(x) and diffusion coefficient equal to the square root of the state, the desired function is the probability of reaching the upper boundary before the lower one. The answer uses the diffusion’s scale function: its normalized change across the interval gives the exit probability.
Applying Itô’s formula to the scale function makes its drift vanish, yielding a second-order ordinary differential equation. The exit probability satisfies the same equation, with boundary values zero at the lower endpoint and one at the upper endpoint. The response notes regularity caveats: standard Lipschitz conditions may fail for the square-root diffusion coefficient, while a Hölder condition can still apply; it also questions whether boundedness alone is enough for the drift. The derivation relies on diffusion theory and does not settle those assumptions for every bounded drift.
Key ideas
- The probability of exiting through the upper endpoint can be represented using a normalized scale function.
- Itô’s formula shows that the scale function has zero drift inside the interval.
- The exit probability solves a second-order generator ODE with absorbing boundary values.
- For the stated diffusion, the equation combines the drift term with one-half the state times the second derivative.
- Existence and regularity depend on assumptions that may require more than a bounded drift.
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Full text
# SDE into ODE problem
# SDE into ODE problem
Let S be the solution of the SDE: $dS_t = g(S_t)dt + \sqrt S_tdW_t, \; S_0 ∈ (1, 2)$, where $g(·)$ is a bounded function. Let $τ$ be the exit time $τ = min(t ≥ 0 : S_t ≥ 2 \; or \; S_t ≤ 1)$. Obtain ODE for the function $f(x), x ∈ (1, 2)$, such that $f(S_0) = P(S_τ = 2)$.
Any help is greatly appreciated.
## Answer by ir7 (score 3)
https://quant.stackexchange.com/a/66097
(My attempt is based on $f(S_t)$, $0\leq t \leq \tau$, being a local martingale.)
We first note that: $$\tau = \min \{\tau_1, \tau_2 \} $$ where $$\tau_1 = \inf \{t\geq 0 | S_t =1 \}, \; \; \tau_2 = \inf \{t\geq 0 | S_t =2 \}$$ and $$ \{ S_\tau = 2\} = \{ \tau = \tau_2 \} = \{\tau_2 < \tau_1 \}. $$ Aslo, we note that the standard Lipschitz conditions for a time-homogenous Markov diffusion are not met, but more refined ones (Yamada-Watanabe) hold. In particular, the diffusion coefficient is Holder continuous with exponent $1/2$ as $|\sqrt{x} -\sqrt{y}|\leq \sqrt{|x-y|}$ for all $x\geq 0, y\geq 0$. (Caveat: I'm not sure if the boundness of the drift coefficient function $g$ is sufficient; it might need to be assumed Lipschitz.)
From Revuz & Yor, Continous Martingales and Brownian Motion, Chapter VII, Propositions (3.2) and (3.5), we have:
- There exists a continuous, strictly increasing function $\phi$, called scale function, such that, for the solution $(S^x_t)_{t\geq 0}$ for SDE started at time $0$ with $S_0=x\in (1,2)$, $$ f(x) := P(S^x_\tau=2)=P_x(\tau_2 < \tau_1) = \frac{\phi(x) - \phi(1)}{\phi(2) -\phi(1)} $$
- The process $\phi(S_t)$, $0\leq t < \tau$, is a local martingale.
Applying Ito Lemma to $\phi(S_t)$ we get: $$ d\phi(S_t) = \phi^{'}(S_t)dS_t + 0.5\phi^{''}(S_t)(dS_t)^2$$ $$ = \left( g(S_t)\phi^{'}(S_t) + 0.5S_t\phi^{''}(S_t) \right) dt + \sqrt{S_t}\phi^{'}(S_t)dW_t,$$ whose drift needs to be $0$, leading to the ODE: $$ g(x)\phi^{'}(x) + 0.5x\phi^{''}(x) =0 $$ and $$ g(x)f^{'}(x) + 0.5xf^{''}(x) =0, $$ $$ \lim_{x\rightarrow 1^+} f(x) = 0, \; \; \lim_{x\rightarrow 2^-} f(x) = 1. $$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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