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
- $\varphi$ 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 $\mathbb R$
- $\Phi$ 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?
