Long-Run Expectation and Absorption at Zero for a Stopped Diffusion

Summary

The document poses a stochastic-process question about a process that starts above zero, drifts downward at a constant rate, and has volatility proportional to its current level while positive. Once it reaches zero, it is held there. The central questions are what happens to its expected value over an infinite horizon and whether absorption at zero occurs almost surely.

No solution, derivation, or evidence is included, so the document does not establish the limiting expectation or the probability of eventual absorption. It is best understood as a prompt about long-run behavior of a diffusion with an absorbing boundary. Any resolution would need to account for the boundary condition and the relationship between the downward drift and state-dependent randomness.

Key ideas

• The process has a negative drift while positive and volatility proportional to its level.
• Zero is specified as an absorbing state.
• The question asks about the limiting expectation and almost-sure hitting of zero.
• The supplied text does not provide a derivation or answer.

Tags

# Expected value of a wiener process on an infinite time horizon with a barrier


# Expected value of a wiener process on an infinite time horizon with a barrier












Say I have a wiener process with $X(0) = X_0>0$ and the dynamics \begin{equation} dX(t) = \begin{cases} -\mu dt + \sigma X(t) dW(t)^{\mathbb{Q}} & \mathrm{for\ } X(t)>0\\ 0 & \mathrm{otherwise}\\ \end{cases} \end{equation} Where $\mu \geq 0$.

What can I say about the expected value of $E^{\mathbb{Q}}[X(T)]$ as $T \to \infty$? Naturally one would expect $X=0$ to happen eventually, but is it almost surely so?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.