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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?

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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.

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  • $\begingroup$ Also, $f$ is defined on the whole line, so there is no concern of leaving the domain of the equation. (Seeing the log raises that concern at first.) $\endgroup$ – Ian Jul 12 '16 at 17:48
  • $\begingroup$ @Ian: I do not understand what do you mean "leaving the domain of the equation". By my understanding, as long as the Lipschitz condition is satisfied, there is a unique strong solution. $\endgroup$ – Gordon Jul 12 '16 at 17:51
  • $\begingroup$ @Ian: For the logrithemic function, as $1+x^2$ is applied, there is no domain problem. $\endgroup$ – Gordon Jul 12 '16 at 17:53
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    $\begingroup$ If the drift or diffusion is not defined on the whole space and is not sufficiently "contractive" into its domain, then it is possible for them to leave the domain of the equation in finite time. But yes, in this case the $1+x^2$ removes the concern, but the concern should arise when you see the log in the first place. $\endgroup$ – Ian Jul 12 '16 at 17:54

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