I'm reading the first chapter of Methods of mathematical finance from Karatzas as part of my B.Sc-thesis in mathematics and I try to show the uniqueness of the solution $$G(t) = S_0(t) \int_0^t \frac{1}{S_0(u)} \pi'(u) \mathrm d R(u) $$ of the following SDE $$ G(0) = 0, $$, $$\mathrm d G(t) = \frac{G(t)}{S_0(t)} \mathrm dS_0(t) + \pi'(t) \mathrm d R(t),$$ if $$ G(t) = \pi_0(t) + \pi'(t) \mathbb{1}$$ for all $t \in [0,T]$. Here is R a semimartingale , S has bounded variation on [0,T] and $\pi$ fulfills \begin{align*} \pi_0 (\cdot) \colon [0,T ] \times \Omega \longrightarrow R , \pi ( \cdot) := ( \pi_1(\cdot) , \ldots, \pi_N( \cdot) \big) \colon [0, T] \times \Omega \longrightarrow R^N \\ \text{with} \int_0^T | \pi_0(t) + \pi(t) \mathbb{1} | \big( r(t) \mathrm dt + \mathrm d \tilde{A}(t) \big) < \infty \text{ almost sure,} \\ \int_0^T \big| \pi(t) ( b(t) + \delta(t) - r(t) \mathbb{1} \big) \big| \mathrm dt < \infty \text{ a.s.,} \\ \text{ and } \int_0^T \| \sigma'(t) \pi (t) \|^2 \mathrm dt < \infty \text{ a.s. } \end{align*} where $b,\delta,\sigma$ have bounded variation on $[0,T]$.
I could show that G is a solution using Ito' partial integration for stochastic integrals from Philip E. Protter. Stochastic Integration and Differential Equations, II.6, theorem 22, corollary 2 and i used to show the uniqueness of the other stochastic differential equations of this chapter with the following theorem from Philip E. Protter. Stochastic Integration and Differential Equations, V.3, theorem 6
Let $Z_t$ be a semimartingale with $Z_0 = 0$ und let $f \colon R_{>0} \times \Omega \times R \longrightarrow R$ be such that, $$(i) \text{for fixed x } (t, \omega) \longrightarrow f(t, \omega, x) \text{ caglad is},$$ $$(ii) \text{for all } (t, \omega) \text{ and a finite random variable } K \text{ the estimation} $$$$ |f(t, \omega, x) - f(t, \omega, y)| \leq K(\omega) |x - y| \text{satisfied is.} $$ Further let $X_0$ be finite and $\mathcal{F}(0)$-measurable. Than the SDE \begin{align*} X_t = X_0 + \int_0^t f(s, \cdot, X_{s-}) \mathrm d Z_s \end{align*} has an unique solution which is a semimartingale.
My problem is how to choose $X_t$ and $Z_t$, the term $\pi'(t) \mathrm d R(t)$ is always a little troublemaker... I'd like to choose $$Z_t:= \int_0^t \frac{1}{S_0(u)} \pi'(u) \mathrm d R(u)$$ but it doesn't work, does it?
Please help, many, many thanks in advance!
Note that $S_0(t)$ is the money market value. Then we assume that $d\langle S_0, G\rangle_t = 0$ and that $d\langle S_0, S_0\rangle_t = 0$. Then, \begin{align*} d\left(\frac{G(t)}{S_0(t)}\right) &=\frac{1}{S_0(t)}dG(t) - \frac{G(t)}{S_0^2(t)}dS_0(t)\\ &=\frac{\pi'(t)}{S_0(t)}dR(t). \end{align*} Therefore, \begin{align*} G(t) = S_0(t)\int_0^t\frac{\pi'(u)}{S_0(u)}dR(u). \end{align*}