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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.

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