Existence and Uniqueness for an SDE with an Absolute-Value Coefficient

Summary

The document poses a stochastic differential equation driven by standard Brownian motion, with a drift proportional to the state and a diffusion coefficient that depends on its absolute value. The author asks what sense of uniqueness applies and how to find the solution. No response or worked solution is included, so the document does not establish existence, uniqueness, or a closed-form process.

The questioner's proposed route is to check global Lipschitz and linear growth conditions for existence and pathwise uniqueness, then use an exponential substitution to remove the drift term. They report difficulty simplifying the transformed diffusion and applying Itô's formula because of the absolute-value term. The material is therefore useful as a statement of a stochastic-calculus problem and a tentative approach, but it provides no evidence that the conditions hold or that the substitution succeeds.

Key ideas

• The document asks for existence, uniqueness, and a solution to an SDE with an absolute-value-dependent diffusion coefficient.
• The questioner proposes checking global Lipschitz and linear growth conditions.
• An exponential substitution is suggested to remove the drift term.
• The document does not include an answer or establish a solution.

Tags

# Finding solution to a SDE with absolute value terms


# Finding solution to a SDE with absolute value terms












Let $B_t$ be a standard Brownian motion. Justify the following stochastic differential equation has only one solution (in which sense?) and find the solution.

$dX_t = X_t dt+(1-e^{-|X_t|})X_tdB_t$

My attempt is to use global Lipschitz and linear growth property to prove existence and pathwise uniqueness of the SDE, then use the substitution $Y_t =e^{-t}X_t$ to get rid of the $dt$ term, but this results in a complicated form in the $dB_t$ part. In particular, I don't see how I can use something like Ito's formula when there is absolute value involved...

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