I'm trying to analyze the following Ito stochastic differential equation:(adsbygoogle = window.adsbygoogle || []).push({});

$$dX_t = \|X_t\|dW_t$$

where [itex]X_t, dX_t, W_t, dW_t \in \mathbb{R}^n[/itex]. Here, [itex]dW_t[/itex] is the standard Wiener process and [itex]\|\bullet\|[/itex] is the [itex]L^2[/itex] norm. I'm not sure if this has an analytical solution, but I am hoping to at least find an analytical expression for the expected value [itex]E[X_t][/itex].

In order to gain intuition for this problem, I'm considering the following ordinary differential equation:

$$\dot{z}(t) =\|z(t)\|b(t)$$

where [itex]z(t), b(t) \in \mathbb{R}^n [/itex] and everything is completely deterministic. Does anyone know the analytical solution to this second equation, and under what conditions it exists?

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# What is the solution to this ODE (and SDE)?

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