Differentiable dependence on the initial condition of the solution of a SDE

Let

• $$b,\sigma:\mathbb R\to\mathbb R$$ be differentiable and Lipschitz continuous
• $$(\Omega,\mathcal A,\operatorname P)$$ be a probability space
• $$(\mathcal F_t)_{t\ge0}$$ be a complete and right-continuous filtration on $$(\Omega,\mathcal A,\operatorname P)$$
• $$(W_t)_{t\ge0}$$ be an $$\mathcal F$$-Brownian motion on $$(\Omega,\mathcal A,\operatorname P)$$
• $$(X^x_t)_{(t,\:x)\in[0,\:\infty)\times\mathbb R}$$ be a real-valued continuous $$\mathcal F$$-adapted process on $$(\Omega,\mathcal A,\operatorname P)$$ with $$X^x_t=x+\int_0^tb(X^x_s)\:{\rm d}s+\int_0^t\sigma(X^x_s)\:{\rm d}W_s\tag1$$ for all $$t\ge0$$ almost surely for all $$x\in E$$

Are we able to show that $$\mathbb R\ni x\mapsto X^x_t(\omega)$$ is differentiable for all $$(\omega,t)\in(\Omega\setminus N)\times[0,\infty)$$ for some $$\operatorname P$$-null set $$N$$?

Note that the hypotheses are sufficient to guarantee the existence of the process $$X$$ above. However, I'm not sure if they are sufficient to guarantee the existence of a real-valued continuous $$\mathcal F$$-adapted process $$(Y^x_t)_{(t,\:x)\in[0,\:\infty)\times\mathbb R}$$ with $$Y^x_t=\int_0^tb'(X^x_s)Y^x_s\:{\rm d}s+\int_0^t\sigma'(X^x_s)Y^x_s\:{\rm d}W_s\tag2$$ for all $$t\ge0$$ almost surely for all $$x\in\mathbb R$$ as well. However, if this process $$Y$$ exists, then it's clearly a candidate for the desired process obtained by differentiating $$X$$ with respect to the spatial parameter.

Maybe it's useful to remember how the existence of $$X$$ is proved: For fixed $$T>0$$ and $$\xi\in L^2(\operatorname P)$$, we know that the map $$\Xi_T(Z):=\xi+\left(\int_0^tb(s,Z_s)\:{\rm d}s\right)_{t\in[0,\:T]}+\left(\int_0^t\sigma(s,Z_s)\:{\rm d}W_s\right)_{t\in[0,\:T]}\in\mathcal B_T$$ for $$Z\in\mathcal B_T$$, where $$\mathcal C_T:=\left\{Z:Z\text{ is a continuous }(\mathcal F_t)_{t\in[0,\:T]}\text{-adapted process on }(\Omega,\mathcal A,\operatorname P)\right\}$$ is equipped with $$\left\|Z\right\|_T:=\left\|\sup_{t\in[0,\:T]}Z_t\right\|_{L^2(\operatorname P)}\;\;\;\text{for }Z\in\mathcal C_T$$ and $$\mathcal B_T:=\left\{Z\in\mathcal C_T:\left\|Z\right\|_T<\infty\right\},$$ has a unique fixed point and there is a constant $$c\ge0$$, depending only on $$b$$ and $$\sigma$$, with $$\left\|\Xi_T(Z)\right\|_T^2\le3\left(\operatorname E\left[|\xi|^2\right]+c(T+4)\left(T+\int_0^T\left\|Z\right\|_t^2\:{\rm d}t\right)\right)\tag3$$ for all $$Z\in\mathcal B_T$$.

Maybe this can be used to show that $$\left\|\frac{X^{x+h}-X^x}h-Y^x\right\|_T\le c_1|h|\;\;\;\text{for all }T>0\text{ and }h\in\mathbb R\tag4$$ for some $$c_1\ge0$$, depending only on $$b$$ and $$\sigma$$, which should even allow for a stronger conclusion then the one asked for in the question.

• Have you looked at Kunita's st flour lecture notes on flows of diffeomorphism? Jun 9, 2020 at 15:10
• @oferzeitouni Yes, sure. But Kunita is hard to read and I think the presented theory is overkill for this rather simple scenario. Jun 9, 2020 at 15:11