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Existence/uniqueness of solution and Ito's formula

  1. Jun 6, 2013 #1


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    Hi everybody,

    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.
  2. jcsd
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