s,x\right) \left( \mu \left( ds,dx\right) -ds\ \nu \left( dx\right) \right)
\},\end{array}$$and thus$$\begin{aligned}
\triangle D_{t} &=&D_{t-}\triangle \left( \int\limits_{]0,t]\times \mathbb{R}_{0}}\theta _{1}\left( s,x\right) \left( \mu \left( ds,dx\right) -ds\ \nu
\left( dx\right) \right) \right)  \\
&=&D_{t-}\triangle \left( \int\limits_{]0,t]\times \mathbb{R}_{0}}\theta
_{1}^{\prime }\left( s,x\right) \left( \mu \left( ds,dx\right) -ds\ \nu
\left( dx\right) \right) \right) .\end{aligned}$$Since $D_{t-}>0$ on $[\hspace{-0.05cm}[0,\tau _{0}[\hspace{-0.04cm}[,$ it follows that $$\triangle \left( \int\limits_{]0,t]\times \mathbb{R}_{0}}\theta _{1}\left(
s,x\right) \left( \mu \left( ds,dx\right) -ds\ \nu \left( dx\right) \right)
\right) =\triangle \left( \int\limits_{]0,t]\times \mathbb{R}_{0}}\theta
_{1}^{\prime }\left( s,x\right) \left( \mu \left( ds,dx\right) -ds\ \nu
\left( dx\right) \right) \right) .$$ Since two purely discontinuous local martingales with the same jumps are equal, it follows $$\int\limits_{]0,t]\times \mathbb{R}_{0}}\theta _{1}\left( s,x\right) \left(
\mu \left( ds,dx\right) -ds\ \nu \left( dx\right) \right)
=\int\limits_{]0,t]\times \mathbb{R}_{0}}\widehat{\theta }_{1}\left(
s,x\right) \left( \mu \left( ds,dx\right) -ds\ \nu \left( dx\right) \right)$$and thus $$\int D_{t-}d\{\int\limits_{]0,t]}\theta _{0}\left( s\right) dW_{s}\}=\int
D_{t-}d\{\int\limits_{]0,t]}\theta _{0}^{\prime }\left( s\right) dW_{s}\}.$$Then, $$0=\left[ \int D_{s-}d\left\{ \int\nolimits_{]0,s]}\left( \theta
_{0}^{\prime }\left( u\right) -\theta _{0}\left( u\right) \right)
dW_{u}\right\} \right] _{t}=\int\limits_{]0,t]}\left( D_{s-}\right)
^{2}\left\{ \theta _{0}^{\prime }\left( s\right) -\theta _{0}\left( s\right)
\right\} ^{2}ds$$and thus $\theta _{0}^{\prime }=\theta _{0}\ dt$-$a.s$ on $[\hspace{-0.05cm}[0,\tau _{0}]\hspace{-0.04cm}]$ for $\mathbb{P}$-almost all $\omega $.

On the other hand, $$\begin{aligned}
0 &=&\left\langle \int \left\{ \theta _{1}^{\prime }\left( s,x\right)
-\theta _{1}\left( s,x\right) \right\} \left( \mu 