Stochastic Calculus Made Easy

Most of us know how standard Calculus works. We know how to differentiate, how to integrate etc.

But stochastic calculus is a totally different beast to tackle; we are trying to play with the calculus of

Random Variables. It's a field where Probability Theory and Calculus meet. Let's start the journey͗-

We will denote W(t) as the Standard Brownian Motion. Some properties are as follows:-

1. W(t) ± W(s) is normally distributed with mean 0 and variance t-s, for s random variables
3. W(0) = 0

4. The sample paths are continuous function of t
draw a tangent at any point in time to time s7. Martingale Property:

Some Stochastic Calculus

What is d(ܹ

Calculus, we are dealing with a Random Variable, so this is clearly not the correct answer, but it is

more or less in line if not exactly similar. When we are dealing with Stochastic Calculus we always need to go to 2nd order terms which were not necessary in Standard Calculus. This because of Quadratic Variation of Brownian Motion, which states:-

For smooth differentiable functions this relationship will be = 0. Now I will try to prove that this relationship

actually holds

If I prove that eq(i) has Expected Value = 0 and Variance = 0, then I can say Almost Surely (a.s) that the Quadratic

9MULMPLRQ UHOMPLRQVOLS OROGVB 6R OHP·V VHH ORR RH ŃMQ GR POMPB

We know from the Properties of Brownian Motion that, W(t) ± W(s) is normally distributed with mean 0

and variance t-s, for sE[W(t) ± W(s)] = 0

So eq(ii) becomes

We can denote, ߂

௔೙ E[݁௔௓], as a ->0. Basically we need to do partial differentiate n times

With the help of Moment Generating Functions, we know that - E[݁௔௓], where Z~N(0,1) = ݁

.௔మ = a݁ .௔మ = (1 + ܽ .௔మ = (2a) ݁ .௔మ + (a + ܽ .௔మ = ( 3a + ܽ .௔మ = (3 + 3ܽ

As a->0, ஔర

.௔మ = 3. ܧ

Eq(iv) becomes,

So we have proved the relationship of Quadratic Variation of Brownian Motion.

Stochastic Differentiation

Taylor Series is the key. A simple stochastic differential equation will be of the form:-

on a ͞drift" term which eǀolǀes with time and a random ǀariable whose eǀolution is unknown to us.

If there is another function ͞V", such that V(S,t), how do we find dV? The answer is Taylor Series:-

6 6 dt.dt -> 0, dt.dW -> 0, dW.dW -> dt After applying the above rule eq(v) reduces to a simplified form and which is also called as 6 6 So with the help of above rules we can find the SDE of any process as long as we know the base asset price dynamics SDE.

Stochastic Integrals Basics

A general stochastic Integral is of the form I(t) = ׬ 4

As this is not standard calculus we can't use the standard rules of calculus. But there is another way

to look at this which was shown to us by Ito. We have to take a partition and break the above integral as a Riemann Sum,

Here ܻ

0, G(t) -> I(t)

How to solve Stochastic Integrations

This is a very huge topic and it's often difficult to comprehend. I will try to present a few examples

which may make the learning a little easier

1. Integrate ׬

4 Solution: - This is 1 of the most famous Stochastic Integrals and its essential for our learning. I will show you the easiest way to look at this.

Let f(W) = ܹ

Let's do a Taylor Eǆpansion of f(WнdW) around dW df = ఋ௙ 6 df = 2W dW + ଵ

6*2 dt, df = 2W dW + dt

dܹ

Let's integrate this from 0 to T

4 = ׬

4 + T

4൅6 {We know that W(0) = 0}

Now the 2nd question is, as this is Stochastic, this will have an Expectation and a Variance, what are those?

E[ଵ

6(ܹ

6 (E[ܹ

6(T - T) = 0

So the Expectation of the above Integration = 0

Var(ଵ

2. Integrate ׬

4 Solution :- We will use the same steps that we had used before, using Taylor series:- dܹଷ = 3ܹ

Integrating both sides from 0 to T

4 + 3׬

4 ଵ

And now the question is, what is׬

4? How do we solve this?

Let ܣ் = ׬

4 time before t=0 it will always be 0}

ݐெ is nothing but T

Now we need to simplify this equation

ݐଵെ6;

integration. distributed. 4

Hence, ׬

So now we have seen 2 interesting Stochastic Integrals, and there are many more complex ones out there. This is just giving you a flavour of the unknown. When you work with Stochastic Integrals, you need to be careful regarding the Expectation and Variance of the Integral because the integral itself is Stochastic in nature. The below 2 properties are extremely useful:-

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