[PDF] stochastic-calculus-made-easy.pdf

So eq(ii) becomes

We can denote, ߂

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

As a->0, ஔర

Eq(iv) becomes,

Stochastic Differentiation

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:-

Stochastic Integrals Basics

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

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

1. Integrate ׬

Let f(W) = ܹ

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

Let's integrate this from 0 to T

4 = ׬

4൅6 {We know that W(0) = 0}

E[ଵ

6(ܹ

6 (E[ܹ

6(T - T) = 0

So the Expectation of the above Integration = 0

Var(ଵ

2. Integrate ׬

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 = ׬

ݐெ is nothing but T

Now we need to simplify this equation

ݐଵെ6;

Hence, ׬

1. E[׬

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