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I have an Ito's stochastic differential equation

[itex] dX_t = a(X_t,t) dt + b(X_t,t) dB_t[/itex]

where [itex]a(X_t,t)[/itex] and [itex]b(X_t,t)[/itex] satisfy the Lipschitz condition for existence and uniqueness of solutions.

Given a function [itex]f(X_t,t) \in C^2[/itex] using Ito's formula I can derive the SDE

[itex] df = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial x} dX_t + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} dX_t^2[/itex]

where [itex]dX_t^2[/itex] is computed using Ito's lemma.

The question is: are there any requirements that [itex] f(X_t,t)[/itex] must satisfy to guaranty

the existence and uniqueness of solutions (I would say yes).

Any reference is welcome. Thanks in advance.

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# Existence/uniqueness of solution and Ito's formula

Can you offer guidance or do you also need help?

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