Solving an SDE using Ito's Lemma

Question

Suppose that

$Z(t)=e^{-\int_0^t \theta'(s)dW(s)-\frac{1}{2}\int_0^t ||\theta(s)||^2ds}$

with $\theta()=\sigma^{-1}()[b()-r()]$, $\sigma()>0$ and invertable and $W()$ a Wiener process

There is also a process $V^{w,h}$ for describing the wealth of an investor such that

$\frac{V^{w,h}(t)}{B(t)}=w+\int_0^t\frac{h'(s)}{B(s)}\sigma(s)[dW(s)+\theta(s)ds]$

with $0\le t\le T$, $w$ being the initial wealth and $A(w)=\{h()/V^{w,h}()\ge 0\}$ almost surely.

Can you help me show that $\frac{V^{w,h}(t)}{B(t)}Z(t)=w+\int_0^t\frac{Z(t)}{B(s)}[V^{w,h}(s)(\theta(s))'-h'(s)\sigma(s)]dW(s)$ ?

I am new to stochastic calculus and i don't know how to correctly apply Ito's Lemma

Answer 1

Hints:

Use Ito theorem for $\ln Z_t$ to get:

$$dZ_t = -\theta_t Z_t dW_t $$

Then use the product rule to compute $d(U_tZ_t)$. I introduced $U_t:=V_t B_t^{-1}$ to keep things cleaner.

$$dU_t = h'_t B_t^{-1}\sigma_t dW_t + h'_t B_t^{-1}\sigma_t \theta_t dt $$

$$ dU_t\cdot dZ_t = -h'_t B_t^{-1}\sigma_t \theta_t Z_t dt $$

$$ d(U_tZ_t) = U_tdZ_t + Z_tdU_t +dU_t\cdot dZ_t$$