**Foundations of Stochastic Calculus**

Stochastic Calculus is a branch of mathematics that deals with random processes. Beyond probabilities it also has links with differential equations, and is widely used in finance particularly for option modelling.

Kiyosi Itô (1915-2008) pioneered the field inventing the concept of stochastic integral and stochastic differential equations.

**Itô Stochastic Integral**

In order to define a stochastic integral, we first define it for elementary stochastic processes, which can be expressed as a sum of stochastic variables (A_{i})_{i=1..n} times the indicator functions on time intervals [t_{i-1},t_{i}): 1_{[ti-1,ti)}(t)=1 if t∈[t_{i-1},t_{i}).

A stochastic process is elementary if it can be approximated by a step function:

For such process the stochastic integral between 0 and T is defines as following, with the Brownian motion W_{t}:

For a general process S_{t} the stochastic integral is defined as the limit of the integral for an elementary process converging to S_{t}:

If we calculate the stochastic integrale between 0 and t of the square of dW_{t} we find the quadratic variation of the brownian motion. It is equal to t:

So by taking the differential we can write the following expression, very useful in stochastic differential equations and stochastic integrals:

**Itô Isometry**

The Ito Isometry allows to calculate the variance and the covariance of random variables which are defined by an Ito integral.

The expectation of the product of two integrals over a Brownian motion W_{t} is equal to the expectation of the integral of the product by replacing the square of dW_{t} by dt.

We have as well:

**Itô Process**

An Itô process X_{t} has the following form:

Or by taking the differential:

µ_{t} and σ_{t} are random processes, W_{t} is a Brownian motion.

**Itô Lemma**

Itô Lemma is one of the main result in stochastic calculus.

The derivative of a function f twice differentiable in x and differentiable in t is equal to:

X_{t} being an Itô process.

In order to demonstrate it, we do a Taylor development of f at the second order, dX^{2} being equal to σ^{2}.dt:

**Itô’s Multiplication Table**

The following multiplication table is a good summary of Ito’s lemma and useful when resolving stochastic differential equations or stochastic integrals.

**Stochastic Differential Equations**

A stochastic differential equation (SDE) has the following form:

This is a generalisation of standard differential equations, where one or more terms is a stochastic process.

We only consider continuous processes, we do not consider processes with jumps here.

**Example 1: a, σ constant**

A first simple example of SDE is with a and σ constant:

By integrating between 0 and T we get the following expression for S_{T}.

S_{T} follows a gaussian distribution with a mean equal to S_{0} + a.T and a variance equal to σ^{2}.T

**Example 2: Geometric Brownian Motion**

Another important example in finance is the geometric brownian motion, often used to model the dynamic of stock prices.

This is typically the case in the famous Black-Scholes model.

The solution of this SDE has the following expression, S_{T} follows a log-normal distribution:

To demonstrate it, we apply the Ito’s lemma to the fonction logarithm ln.

The first derivatives is dln(x) / dx = 1 / x while the second derivatives is d^{2}ln(x) / dx^{2} = – 1 / x^{2}:

After regrouping the terms in dt and the terms in dW_{t} we get the following expression for the differential of ln(S_{t}), and we integrate it between 0 and T to finish the demonstration.

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