3
$\begingroup$

Remark: I've asked this question on MSE as well.

Let

  • $T>0$
  • $I:=[0,T]$
  • $(\Omega,\mathcal A,\operatorname P)$ be a probability space
  • $(\mathcal F_t)_{t\in I}$ be a complete and right-continuous filtration of $\mathcal A$
  • $B$ be an $\mathcal F$-Brownian motion on $(\Omega,\mathcal A,\operatorname P)$
  • $b,\sigma:I\times\mathbb R\to\mathbb R$ be Borel measurable

Consider the Itō equation $${\rm d}X_t=\underbrace{b(t,X_t)}_{=:\:\varphi_t}{\rm d}t+\underbrace{\sigma(t,X_t)}_{=:\:\Phi_t}{\rm d}B_t\;\;\;\text{for all }t\in I\tag1$$ and the Stratonovich equation $${\rm d}X_t=b(t,X_t){\rm d}t+\sigma(t,X_t)\circ{\rm d}B_t\;\;\;\text{for all }t\in I\;.\tag2$$ Unfortunately, I wasn't able to find any book which rigorously introduces Stratonovich equations. In the best case, there is a tiny subsection which tells us what the Stratonovich integral is (most often without describing the class of possible integrands) and that any equation of the form $(1)$ can be translated into an equation of the form $(2)$ and vice versa.

My definition for a stochastic process $\Psi$ on $(\Omega,\mathcal A,\operatorname P)$ of being Stratonovich integrable is that $\Psi$ must be an Itō integrable (in the usual sense) $\mathcal F$-semimartingale. In that case, $$\int_0^t\Psi_s\circ{\rm d}B_s:=\int_0^t\Psi_s\:{\rm d}B_s+\frac12[\Psi,W]_t\;\;\;\text{for }t\in I\;,$$ where $[\Psi,B]$ denotes the quadratic covariation of $\Phi$ and $B$.

My problem is that I couldn't find any book which tells us which assumptions on the coefficients need to be imposed in order to ensure that (a) equation $(1)$ or $(2)$ is well-defined and (b) can be translated into an equation of the other form.

Clearly, in order for $X$ to be a well-defined solution of $(1)$ we need that

  1. $\varphi$ is $\mathcal F$-progressively measurable with $$\varphi(\omega)\in\mathcal L^1(\left.\lambda\right|_I)\;\;\;\text{for all }\omega\in\Omega\tag3\;,$$ where $\lambda$ denotes the Lebesgue measure on $\mathbb R$
  2. $\Phi$ is $\mathcal F$-progressively measurable and $$\Phi(\omega)\in\mathcal L^2(\left.\lambda\right|_I)\;\;\;\text{for }\operatorname P\text{-almost all }\omega\in\Omega\tag4$$

In that case and if $X$ satisfies $(1)$, then $X$ is an Itō process (and hence an $\mathcal F$-semimartingale) and the translation to an equation of the form $(2)$ can be achieved by an invocation of the Itō formula.

The crucial point is that in the above situation the quadratic covariation $[\Phi,B]$ is well-defined.

But which assumption on $\Phi$ do we need for the translation of $(2)$ into an equation of the form $(1)$?

I haven't find any book which treats this issue, but I don't see how $(2)$ is even well-defined unless $\Phi$ is an $\mathcal F$-semimartingale (for example, an Itō process).

Isn't that a quite limiting assumption? Unless $X$ is the solution of an Itō equation, I don't see any mild condition on $\sigma$ which would ensure that $\Phi$ is a $\mathcal F$-semimartingale.

So, is there anything I'm missing?

$\endgroup$

1 Answer 1

0
$\begingroup$

See the Proposition 2.21 of the book "Brownian Motion and Stochastic Calculus" by Ioannis Karatzas, Steven Shreve (Page 295).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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