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


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


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

  • $\begingroup$ 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

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 $f$.

| cite | improve this answer | |
  • $\begingroup$ 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
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.