# Dependence on the initial datum of the strong solution of a SDE

Let

• $$(\Omega,\mathcal A,\operatorname P)$$ be a probability space
• $$(\mathcal F_t)_{t\ge0}$$ be a complete and right-continuous filtration on $$(\Omega,\mathcal A)$$
• $$W$$ be a $$\mathcal F$$-Brownian motion on $$(\Omega,\mathcal A,\operatorname P)$$
• $$b,\sigma:[0,\infty)\times\mathbb R\to\mathbb R$$ be Borel measurable with $$\left|b(t,x)-b(t,y)\right|^2+\left|\sigma(t,x)-\sigma(t,y)\right|^2\le K|x-y|^2\;\;\;\text{for all }t\ge0\text{ and }x,y\in\mathbb R\tag1$$ and $$\left|b(t,x)\right|^2+\left|\sigma(t,x)\right|^2\le K\left(1+\left|x\right|^2\right)\;\;\;\text{for all }(t,x)\in[0,\infty)\times\mathbb R\tag2$$
• $$\mathcal V_s$$ denote the space of real-valued $$(\mathcal F_t)_{t\ge s}$$-adapted continuous processes $$(X_t)_{t\ge s}$$ on $$(\Omega,\mathcal A,\operatorname P)$$ with $$\left\|X\right\|_{\mathcal V_s}:=\operatorname E\left[\sup_{t\ge s}\left|X_t\right|^2\right]<\infty$$ for $$s\ge0$$

We know that $$\mathcal V_s$$ equipped with $$\left\|\;\cdot\;\right\|_{\mathcal V_s}$$ is a complete semi-normed space and that $$\Xi^x_s(X):=x+\left(\int_s^ub(t,X_t)\:{\rm d}t\right)_{u\ge s}+\left(\int_s^u\sigma(t,X_t)\:{\rm d}W_t\right)_{u\ge s}\in\mathcal V_s\;\;\;\text{for }X\in\mathcal V_s$$ has a unique fixed point $$X^{s,\:x}$$ for all $$x\in\mathbb R$$ and $$s\ge0$$. For simplicity, let $$X^x:=X^{0,\:x}$$ for $$x\in\mathbb R$$.

How do we need to understand the claim $$X_u^{s,\:X^x_s}=X_s^x+\int_s^ub\left(t,X^{s,\:X^x_s}_t\right)\:{\rm d}t+\int_s^u\sigma\left(t,X^{s,\:X^x_s}_t\right)\:{\rm d}W_t\tag3$$ for all $$u\ge s$$ almost surely for all $$(s,x)\in[0,\infty)\times\mathbb R$$ and how can we prove it?

I guess this can be proven in a similar way as we can prove $$X\int_{t_0}^t\Phi\:{\rm d}W=\int_{t_0}^tX\Phi\:{\rm d}W\;\;\;\text{for all }t\ge t_0\text{ almost surely}\tag4$$ for all $$(\mathcal F_t)_{t\ge t_0}$$-progressive processes $$(\Phi_t)_{t\ge t_0}$$ with $$\operatorname E\left[\int_{t_0}^t\left|\Phi_s\right|^2\:{\rm d}s\right]<\infty\;\;\;\text{for all }t\ge t_0\tag5$$ and bounded $$\mathcal F_{t_0}$$-measurable $$X:\Omega\to\mathbb R$$, for all $$t_0\ge0$$.

So, maybe $$(3)$$ follows by a more general result.