# Solving an SDE using Ito's Lemma

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

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.

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

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

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