Stochastic Calculus
DISCLAIMER THERE IS AT TIMES LOT OF HAND WAVING ,THIS IS NOT AS MATHEMATICALLY RIGOROUS AS YOU WOULD EXPECT LIKE IN A PURE MATH TEXTBOOK.
Resources
These are list of books , videos and courses that I am currently referencing for content on this page , for more detailed list please check out the resource section above.
Mathematical modelling and computation in finance by Cornerlis Ooseterlee and Lech Grzelak (Main Text)
Stochastic CalcuIus for Finance II Continuous-Time Models by Steven Shreve (Main Reference text)
Stochastic Differential Equations By Bernt Oksendal
Preliminaries
Definition 1 (\sigma algebra) : Let \Omega be a nonempty set , and let \mathcal{F} be a set of collections of subsets of \Omega. We say that \mathcal{F} is a \sigma algebra provided that
- The empty set \emptyset belongs to \mathcal{F}
- A and A^{c} belong to \mathcal{F} whenever A \in \mathcal{F}
- whenever A_1 ,A_2 ,A_3 ..... \in \mathcal{F} then \bigcup_{i=1}^\infty A_{i} \in \mathcal{F}
Definition 2 (Probability measure) : Let \Omega be a nonempty set , and let \mathbb{F} be \sigma algebra of subsets of \Omega. A probability measure \mathbb{P} is a function that, to every set A \in \mathcal{F} , assigns a number in [0,1] called the probability of A written as \mathbb{P}(A) .We require
\mathbb{P}(\Omega) = 1 and
(Countable Additivity) Whenever A_1,A_2,A_3.... is a sequence of disjoint sets in \mathcal{F} , then \mathbb{P}(\bigcup_{n=1}^\infty A_n) = \sum_{n=1}^{\infty} \mathbb{P}(A_n)
The Triple (\Omega, \mathcal{F},\mathbb{}P) is called probability measure
From the above it follows that for finitely many disjoint sets A_1, A_2,A_3....A_n \mathbb{P}(\bigcup_{n=1}^N) = \sum_{n=1}^N \mathbb{P}(A_n)
and \mathbb{P}(A^c) = 1 - \mathbb{P}(A)
Example 1 (Uniform(Lebesgue) measure on [0,1])
\Omega = [0,1]
\mathbb{P}[a,b] = b-a ,0 \leq a \leq b \leq1
single points have zero probability and
(a,b) can be written as \bigcup_{n=1}^\infty[a+\frac{1}{n} , b -\frac{1}{n}]
The \sigma-algebra beginning with closed intervals and adding everything else necessary in order to have a \sigma- algebra is called a Borel \sigma-algebra and denoted by \mathcal{B}[0,1]. Borel \sigma-algebras are \sigma-algebra generated by open sets and equivalently closed sets over any topological space.
Definition 3 let (\Omega,\mathcal{F},\mathbb{P}) be a probability space. If a set A \in \mathcal{F} satisfies \mathbb{P}(A) = 1, we say the event A occurs almost surely
Example 2 Let \mathbb{P} be the uniform measure as defined in example 1. Define X(\omega) = \omega and Y(\omega) = 1 - \omega for all \omega \in [0,1] , then the distribution measure of X is uniform \mu_{X}[a,b] = \mathbb{P}\{\omega;a\leq X(w)\leq b\} = \mathbb{P}[a,b] = b-a, 0\leq a \leq b \leq 1 by definition of \mathbb{P}.
Its easy to see to see that X and Y have the same distribution. But under the probability measure\mathbb{\widetilde{P}} on [0,1] defined by \mathbb{\widetilde{P}[a,b]} = \int_a^b 2\omega \, d\omega = b^2-a^2 , 0 \leq a\leq b \leq1 X and Y have different distributions.
Definition 4 (cumulative distribution function and density function) F(x) = \mathbb{P}\{X \leq x\} , x \in \mathbb{R} \mu_X[a,b] = \mathbb{P}\{a \leq X \leq b \} = \int_a^b f(x) \, dx , -\infty < a \leq b < \infty f(x) is called the density funtion and F(x) is called the cummulative distribution function.
Example 3 (Standard normal random variable) Let \phi(x) = \frac {a}{\sqrt{2\pi}}e^{\frac{-x^2}{2}} be the standard normal density and definiing the cummulative normal distribution function as N(x) = \int_{-\infty}^x \phi(\xi)\,d\xi The function N(x) is strictly increasing function which is surjective onto (0,1) from \mathbb{R}, so it has a strictly increasing inverse function N^{-1}(y),y \in (0,1). Let Y be a uniformly distributed random variable defined on some probability space (\Omega,\mathcal{F},\mathbb{P}) and set X = N^{-1}(Y) then
\begin{align*} \mu_X[a,b] &= \mathbb{P}\{\omega \in \Omega ; a \leq X(\omega) \leq b\} \\ &= \mathbb{P}\{\omega \in \Omega ; a \leq N^{-1}(Y(\omega)) \leq b\}\\ &= \mathbb{P}\{\omega \in \Omega ; N(a) \leq Y(\omega) \leq N(b) \} \\ &= N(b) - N(a) \\ &= \int_a^b \phi(x) \, dx \end{align*}Any random variable that has this distribution regardless of the probability space is called standard normal distribution.
Thing to note the use of uniformly distributed random variable for generating a standard random variable is called probability integral transform and this is commonly used in Monte Carlo simulation