Using Itô's Lemma for Log Prices and Quadratic Covariation

# Clarification of Ito's lemma


# Clarification of Ito's lemma












I was looking at the various examples provided in the discussion Worked examples of applying Ito's lemma

One such example is 9.1 (c). This states that -

if $S_t =\! S_0 + \int\limits_{0}^{t} \mu_u S_u du + \int\limits_{0}^{t} \sigma_u S_u dW_u$ with $\mu=\left(\mu_t \right)_{t\geq0}, \sigma=\left(\sigma_t \right)_{t\geq0}, \int\limits_{0}^{T} |\mu_s| + |\sigma_s|^2 ds < \infty$. Then $\int\limits_{0}^{T} \sigma^2_s ds = -2\log \frac{S_T}{S_0} + \int\limits_{0}^{T} \frac{2}{S_u} dS_u$

Then it says $\frac{S_T}{S_0} = e^{\int\limits_{0}^{T} \sigma_s dW_s - \int\limits_{0}^{T} \left(0.5\sigma_s^2 - \mu_s \right) ds}$, which I understand the derivation.

I then failed to grasp the remaining part which shows that : $\log S_T - \log S_0 = \int\limits_{0}^{T} \frac{1}{S_u} dS_u -0.5 \int\limits_{0}^{T} \sigma_u^2 du$

2nd example goes for 4. This states that -

if $X_t =\! e^{W_t+0.5t} + e^{W_t-0.5t}$, then $dX_t =\! X_t dW_t + e^{W_t+0.5t}dt$.

To prove this, it is taken that $X_t=Z_tY_t, Z_t = e^{W_t-0.5t}, Y_t = e^t + 1$. It sates that the process $Z_t$ is continuous semi-martingale and $Y_t$ is continuous semi-martingale of bounded variation. Therefore it holds that $\left[ ZY \right]=0$. My questions are

- Why $Z$ is continuous semi-martingale and $Y$ is continuous semi-martingale with bounded variation? What is required to prove them so?

- How to show exactly that $\left[ZY\right] = 0$

Your pointer will be highly helpful

## Answer by ir7 (score 4)

https://quant.stackexchange.com/a/55918

For the first one, we have:

$$ dS_t = \mu_t S_t dt + \sigma_tS_t dW_t $$ and note that

$$ (dS_t)^2 = \sigma_t ^2 S_t^{2} dt. $$

We apply Ito formula to

$$ f(S_t) = \ln S_t. $$

As $f'(x) = x^{-1}$ and $f^{''}(x)= -x^{-2}$, we get:

$$ d \ln S_t = S_t^{-1} dS_t - 0.5 S_t^{-2} (dS_t)^2 $$

which is equivalent to

$$ d \ln S_t = S_t^{-1} dS_t - 0.5 \sigma_t^2 dt. $$

Integration from $0$ to $T$ gives:

$$ \ln S_T - \ln S_0 = \int_0^T S_t^{-1} dS_t - 0.5 \int_0^T \sigma_t^2 dt. $$

For the second one:

A process has bounded variation if almost all its paths are functions with bounded variation.

This needs to be proven for $Y_t$.

For the rest of the questions, you can take as facts (proven in many books on stochastic calculus) that:

- Ito processes are continuous semimartingales (with the two pieces, continuous local martingale and continuous finite variation process, visible).

- Finite variation processes (bounded variation on every finite time interval, with probability $1$) have $0$ quadratic variation.

- The quadratic covariation between a finite variation process and a continuous semimartingale is $0$.