Introduction to Stochastic Calculus

Back in undergraduate days, when I took my first module on financial mathematics, my professor introduced us by that the most important things are the following

$(\Omega, \mathcal{F}, \mathcal{P})$

This is a probability triple where
1. $\mathcal{P}$ is the ‘true’ of physical probability measure
2. $\Omega$ is the universe of possible outcomes.
3. $\mathcal{F}$ is the set of possible events where an event is a subset of $\Omega$ .

There is also a filtration $\{\mathcal{F}_t\}_{t \ge 0}$ , that models the evolution of information through time. For example, if by time $t$ , we know that event $\mathcal{E}$ has occurs, then $\mathcal{E} \in \mathcal{F}_t$ . In the case of a finite horizon from $[0,T]$ , then $\mathcal{F} = \mathcal{F}_T$

A stochastic process $X_t$ is $\mathcal{F}_t$ -adapted if the value of $X_t$ is know at time $t$ when the information represented by $\mathcal{F}_t$ is known. Most of the times, we have sufficient information at present.

In the continuous-time model, $\{\mathcal{F}_t\}_{t \ge 0}$ will be the filtration generated by the stochastic processes (usually a brownian motion, $W_t$ ), based on the model’s specification.

Next, we review some martingales and brownian motion, alongside with quadratic variation here.

About

KS Teng

KS has been teaching H2/H1 Mathematics and IB mathematics for the more than 10 years. Having taught students from all various Junior Colleges, KS adapts to students' abilities and help them better understand the topics. As someone who loves to teach mathematics and sees it as a truly useful tool in life, KS seeks to enable students to appreciate math. Therefore, his tuition mission is to motivate and cultivate students to be independent and confident thinkers.

Introduction to Stochastic Calculus

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