Applying Itô’s Product Rule to a Discounted Wealth Process

Summary

The document asks how to derive an identity involving an investor’s discounted wealth process and an exponential process defined from a Wiener process. The response suggests first applying Itô’s theorem to the logarithm of the exponential process to obtain its stochastic differential. It then defines discounted wealth as a separate process and writes its differential using the portfolio, bond, volatility, and market-price-of-risk terms.

The key calculation is the Itô product rule: combine each process’s differential with their quadratic covariation. The response explicitly computes the cross term, which contributes a drift term, and lays out the product differential. However, the excerpt stops before integrating the result or displaying the final identity. It is a useful outline of the method, but a learner must complete the simplification and check the notation and assumptions against the original model.

Key ideas

• Applying Itô’s theorem to the logarithm yields the differential of the exponential process.
• The discounted wealth process can be represented by its own stochastic differential.
• The product differential includes the quadratic covariation term between the two processes.
• The supplied answer gives calculation hints but does not complete the requested integrated identity.

Tags

# Solving an SDE using Ito's Lemma


# 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

## Answer by ir7 (score 3, accepted)

https://quant.stackexchange.com/a/65524

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.

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

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

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

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.