Csdrhrt

Let

• 
  $(Omega,mathcal A,operatorname P)$ be a probability space


• 
  $(mathcal F_t)_{tge0}$ be a complete and right-continuous filtration on $(Omega,mathcal A)$
  


• 
  $W$ be a $mathcal F$-Brownian motion on $(Omega,mathcal A,operatorname P)$
  


• 
  $b,sigma:[0,infty)timesmathbb Rtomathbb R$ be Borel measurable with $$left|b(t,x)-b(t,y)right|^2+left|sigma(t,x)-sigma(t,y)right|^2le K|x-y|^2;;;text{for all }tge0text{ and }x,yinmathbb Rtag1$$ and $$left|b(t,x)right|^2+left|sigma(t,x)right|^2le Kleft(1+left|xright|^2right);;;text{for all }(t,x)in[0,infty)timesmathbb Rtag2$$
  


• 
  $mathcal V_s$ denote the space of real-valued $(mathcal F_t)_{tge s}$-adapted continuous processes $(X_t)_{tge s}$ on $(Omega,mathcal A,operatorname P)$ with $$left|Xright|_{mathcal V_s}:=operatorname Eleft[sup_{tge s}left|X_tright|^2right]<infty$$ for $sge0$
  

We know that $mathcal V_s$ equipped with $left|;cdot;right|_{mathcal V_s}$ is a complete semi-normed space and that $$Xi^x_s(X):=x+left(int_s^ub(t,X_t):{rm d}tright)_{uge s}+left(int_s^usigma(t,X_t):{rm d}W_tright)_{uge s}inmathcal V_s;;;text{for }Xinmathcal V_s$$ has a unique fixed point $X^{s,:x}$ for all $xinmathbb R$ and $sge0$. For simplicity, let $X^x:=X^{0,:x}$ for $xinmathbb R$.

How do we need to understand the claim $$X_u^{s,:X^x_s}=X_s^x+int_s^ubleft(t,X^{s,:X^x_s}_tright):{rm d}t+int_s^usigmaleft(t,X^{s,:X^x_s}_tright):{rm d}W_ttag3$$ for all $uge s$ almost surely for all $(s,x)in[0,infty)timesmathbb R$ and how can we prove it?

I guess this can be proven in a similar way as we can prove $$Xint_{t_0}^tPhi:{rm d}W=int_{t_0}^tXPhi:{rm d}W;;;text{for all }tge t_0text{ almost surely}tag4$$ for all $(mathcal F_t)_{tge t_0}$-progressive processes $(Phi_t)_{tge t_0}$ with $$operatorname Eleft[int_{t_0}^tleft|Phi_sright|^2:{rm d}sright]<infty;;;text{for all }tge t_0tag5$$ and bounded $mathcal F_{t_0}$-measurable $X:Omegatomathbb R$, for all $t_0ge0$.

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