I have the stochastic differential equation
$$dX_t = \ln(1+ X_t^2) \, dt + X_t \, dB_t$$
In this equation, $X_0 = x$, and $x \in\mathbb R$.
How can we show that this equation has a unique strong solution?
2026-03-26 22:17:34.1774563454
Proving an SDE has a unique strong solution
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Let $f(x) = \ln(1+x^2)$. Then \begin{align*} |f'(x)| = \frac{2|x|}{1+x^2} \le 1. \end{align*} Therefore the Lipschitz condition is satisfied, and there is a unique strong solution.