Why is Itô Lemma?
The Ito’s lemma provides a framework to differentiate the functions of stochastic process and this is of particular significance to derivative pricing (before Ito’s work, people did not know how to do it). Ito’s lemma allows us to derive the stochastic differential equation (SDE) for the price of derivatives.
Why Ito integral is a martingale?
We give one and a half of the two parts of the proof of this theorem. If b = 0 for all t (and all, or almost all ω ∈ Ω), then F(T) is an Ito integral and hence a martingale. If b(t) is a continuous function of t, then we may find a t∗ and ǫ > 0 and δ > 0 so that, say, b(t) > δ > 0 when |t − t∗| < ǫ.
What is Ito calculus used for?
Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations.
Is stochastic integral martingale?
It is a local martingale, by definition, with quadratic variation given by Qt=∫t0H2udu. Now, QT is upper bounded by an almost surely finite random variable times T. So that the expectation of the quadratic variation is finite, because M is a martingale, and hence the stochastic integral is a martingale.
How is stochastic calculus different?
The fundamental difference between stochastic calculus and ordinary calculus is that stochastic calculus allows the derivative to have a random component determined by a Brownian motion. The derivative of a random variable has both a deterministic component and a random component, which is normally distributed.
Is stochastic calculus useful?
An important application of stochastic calculus is in mathematical finance, in which asset prices are often assumed to follow stochastic differential equations. In the Black–Scholes model, prices are assumed to follow geometric Brownian motion.
Is stochastic calculus used in trading?
The main use of stochastic calculus in finance is through modeling the random motion of an asset price in the Black-Scholes model. The physical process of Brownian motion (in particular, a geometric Brownian motion) is used as a model of asset prices, via the Weiner Process.