Basic Understanding: Explain Ito's Lemma and discuss its significance

in stochastic calculus.

Simple Application: Given a standard Brownian motion $W_t,$ use Ito's

Lemma to find the differential $d(e^{W_t}).$

Function Application: Let $f(t,x)=t{x}^2.$ Apply Ito's Lemma to find $df$

when $x$ is a geometric Brownian motion given by $d{x}_t=x_{t}dt+x_{t}d{W}_t,$

where $W_t$ is a standard Brownian motion.

Comparison with Classical Calculus: Compare and contrast Ito's

Lemma with the chain rule in classical calculus. Provide examples where

each is used.

Proof and Derivation: Prove Ito's Lemma for a twice continuously

differentiable function $f(t,x_t),$ where $x_t$ is an Ito process.

Multidimensional Ito's Lemma: State and prove Ito's Lemma for a

multidimensional Ito process. Provide an example with two stochastic

processes.

Application in Option Pricing: Explain how Ito's Lemma is used in

the derivation of the Black$-$Scholes equation for option pricing.

Practical Scenario: Consider an asset whose price follows a stochastic

process described by $d{S}_t=S_t(dt+d{W}_t).$ Use Ito's Lemma to find the

differential of $ln(S_t).$

Advanced Application: Given a stochastic process $x_t=sin(W_t),$

where $W_t$ is a standard Brownian motion, apply Ito's Lemma to find

$d{x}_t.$

Integration with Other Concepts: Discuss how Ito's Lemma inte$-$

grates with other concepts in stochastic calculus, such as martingales and

stochastic integrals.