Itô's Lemma: Derivation and Applications

Hard·22 min read

Stochastic CalculusItô's LemmaStochastic Differential Equations

Quick Quiz

1. In the Itô multiplication table, what is $(dW_t)^2$ ?

0

dt

(dt)^2

dW_t

2. Apply Itô's lemma to $f(S_t) = \ln S_t$ where $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$ . What is $d(\ln S_t)$ ?

\mu \, dt + \sigma \, dW_t

\left(\mu - \frac{\sigma^2}{2}\right) dt + \sigma \, dW_t

\left(\mu + \frac{\sigma^2}{2}\right) dt + \sigma \, dW_t

\frac{1}{S_t} dS_t

3. The Itô isometry states that for a square-integrable adapted process $\sigma$ :

\mathbb{E}\left[\int_0^T \sigma_s \, dW_s\right] = \int_0^T \mathbb{E}[\sigma_s] \, ds

\mathbb{E}\left[\left(\int_0^T \sigma_s \, dW_s\right)^2\right] = \mathbb{E}\left[\int_0^T \sigma_s^2 \, ds\right]

\int_0^T \sigma_s^2 \, ds = T \cdot \mathbb{E}[\sigma_0^2]

\mathrm{Var}\left(\int_0^T \sigma_s \, dW_s\right) = \left(\int_0^T \sigma_s \, ds\right)^2

4. The Stratonovich integral $\int_0^T f(W_t) \circ dW_t$ satisfies the classical (Newton–Leibniz) chain rule, with no Itô correction.

truefalse

5. Under GBM $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$ , what is $S_T$ ?

S_0 e^{\mu T + \sigma W_T}

S_0 e^{(\mu - \sigma^2/2)T + \sigma W_T}

S_0 e^{(\mu + \sigma^2/2)T + \sigma W_T}

S_0 + \mu T + \sigma W_T

6. For which class of functions $f$ does the classical statement of Itô's lemma hold?

f

measurable

f

continuous

f \in C^{1,2}

(once in

t

, twice in

x

)

f

Lipschitz in

x