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Suppose I have a stochastic process $X_t$ that satisfies the SDE:

$$dX_{t}=\mu(X_{t})dt+\sigma(X_{t})dW_{t}$$

where $W_t$ is a Brownian motion. Suppose I haven't made any assumption yet about the functions $\sigma(\cdot)$ and $\mu(\cdot)$ (therefore I don't even know if my SDE has a strong solution, or even a weak one).

Can I use Ito's formula anyway? In other words, is it true that the process $Y_t=f(X_t)$, where $f(\cdot)$ is twice differentiable, follows an SDE

$$dY_t=\mu(X_t)f'(X_t)dt+\frac{1}{2}f''(X_t)\sigma(X_t)dW_t$$

I couldn't find a simple answer in the usual references, any help is welcome.

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  • $\begingroup$ What exactly does this all mean when neither SDE has a solution? $\endgroup$ – Ian Sep 4 '16 at 4:15
  • $\begingroup$ ... and what exactly do you mean by "[not] any assumptions"? You need some assumptions to ensure that all the expressions in the SDE are well-defined, e.g. that $\sigma$ is (locally) square integrable and also that $\mu$ is (locally) integrable. $\endgroup$ – saz Sep 4 '16 at 6:20
  • $\begingroup$ By having no solution, I mean that, given a filtered probability space, there is no process that satisfies the equation and is adapted to the given filtration. But, as saz mentioned, I understand I am impliciting assuming square integrability. $\endgroup$ – Pcw. Sep 4 '16 at 17:00
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Implicit in your initial supposition "Suppose I have a stochastic process $X_t$ that satisfies the SDE:..." is the hypothesis that the integrals $\int_0^t\mu(X_s)\,ds$ and $\int_0^t\sigma(X_s)\,dW_s$ are well defined for all $t>0$, almost surely; namely, that $\mu$ and $\sigma$ are Borel-measurable functions such $\int_0^t|\mu(X_s)|\,ds<\infty$ and $\int_0^t|\sigma(X_s)|^2\,ds<\infty$ for all $t>0$, almost surely. If this is so then Ito's formula can be used for a $C^2$ function $f$.

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  • $\begingroup$ I understand. So, for instance, if I have the Tanaka equation: $$dX_t=\text{sgn}(X_t)dW_t$$ I can then use the Ito's formula for it, even though it has no strong solution, right? (the function $\text{sgn}(X_t)$ is square integrable). Do you happen to know any reference? $\endgroup$ – Pcw. Sep 4 '16 at 16:56
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    $\begingroup$ If you have a solution to Tanaka's equation, then you can use Ito's formula. More precisely, if $(\Omega,\mathcal F,\mathcal F_t, P)$ is a filtered probability space, $W$ an $(\mathcal F_t)$ Brownian motion, and $X$ an $(\mathcal F_t)$ adapted continuous process such that $X_t=X_0+\int_0^t sgn(X_s)\,dW_s$ for all $t>0$, then $X$ can be plugged into Ito's formula. For a reference, look at the book of Revuz and Yor. $\endgroup$ – John Dawkins Sep 5 '16 at 15:27

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