Well-posedness of a stochastic differential equation in the Stratonovich sense

Question

Let

• $T>0$
• $(\Omega,\mathcal A,\operatorname P)$ be a probability space
• $(\mathcal F_t)_{t\in[0,\:T]}$ be a complete filtration on $(\Omega,\mathcal A)$
• $(B_t)_{t\in[0,\:T]}$ be an $(\mathcal F_t)_{t\in[0,\:T]}$-Brownian motion on $(\Omega,\mathcal A,\operatorname P)$
• $X_0$ be a real-valued $\mathcal F_0$-measurable random variable on $(\Omega,\mathcal A,\operatorname P)$
• $\varphi:\Omega\times[0,T]\times\mathbb R\to\mathbb R$
• $\Phi:\Omega\times[0,T]\times\mathbb R\to\mathbb R$

I want to consider the equation $$X_t=X_0+\int_0^t\varphi(s,X_s)\:{\rm d}s+\int_0^t\Phi(s,X_s)\:\circ{\rm d}B_s\;\;\;\text{for all }t\in[0,T]\tag 1$$ where the last integral has to be understood in the Stratonovich sense.

Question: Under which condition on $\Phi$ is $(1)$ well-defined?

If $(Y_t)_{t\in[0,\:T]}$ is a stochastic process on $(\Omega,\mathcal A,\operatorname P)$, then $$\int_0^tY_s\:\circ{\rm d}B_s:=\int_0^tY_s\:{\rm d}B_s+\frac12\langle Y,B\rangle_t\;\;\;\text{for }t\in[0,T]\tag 2$$ where the integral on the right-hand side is understood in the Itō sense and $(\langle Y,B\rangle_t)_{t\in[0,\:T]}$ denotes the covariation of $Y$ and $B$. Of course, in order for $(2)$ to be well-defined we need that

1. $Y$ is integrable in the Itō sense, i.e. $Y$ is $(\mathcal F_t)_{t\in[0,\:T]}$-progressively measurable and $$\operatorname P\left[\int_0^TY_s^2\:{\rm d}s<\infty\right]=1\tag 3$$
2. the covariation of $Y$ and $B$ exists, i.e. $Y$ is an $(\mathcal F_t)_{t\in[0,\:T]}$-semimartingale

So, $(1)$ is well-defined if and only if 1. and 2. are satisfied by $$Y_t:=\Phi(t,X_t)\;\;\;\text{for }t\in[0,T]\;.$$ Since $X$ is the unknown solution (whose existence is not even known and cannot be proved until $(1)$ is at least well-posed), we somehow need to ensure this by a suitable assumption for $\Phi$. The question is which assumption this is.

Answer 1

As mentioned in this answer, the conversion to Stratonovich SDE is the following:

You are considering the Stratonovich SDE $$ dX = h(X) \, dt + \gamma (X) \circ dW, $$ where I suppose the following hold:

1. $h \colon \mathbb{R}^n \to \mathbb{R}^n$,
2. $\gamma = (\gamma_{ij})_{i,j = 1}^{n,m} \colon \mathbb{R}^n \to \mathbb{R}^{n\times m} $,
3. $W$ is an $m$-dimensional Brownian motion.

Such a SDE is equivalent to the Itô SDE $$ dX = \left( h(X) + \frac{1}{2} c(X) \right) dt + \gamma (X) \, dW, \tag{1} \label{1}$$ where $c = (c_i)_{i=1}^n \colon \mathbb{R}^n \to \mathbb{R}^n$ is defined by $$ c_i (x) = \sum_{k=1}^{m} \sum_{j=1}^{n} \frac{\partial \gamma_{ik}}{\partial x_j} (x) \, \gamma_{jk} (x).$$ In order to obtain \eqref{1}, apply exactly the conversion formula that you can find, e.g., at page 123 of the book An introduction to stochastic differential equations by Lawrence Evans.I hope that this can help to clarify what the notes you are looking at do, but perhaps you have to give some more details about the functions appearing into the SDE.

So here we can use the existence conditions for the Ito sde: Weak solutions of linear parabolic PDEs and corresponding SDEs