Ziyuan Zhang
Ziyuan Zhang
Practice makes perfect

Stochastic Calculus Notes 01 | Probability Spaces, Filtrations, Conditional Expectation and Martingales

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Overview

These are my notes from QST MF 790 Stochastic Calculus at Boston University’s Questrom School of Business, cleaned up into 7 posts.

The whole course hangs on one sentence: “Let \(W\) be a Brownian motion adapted to a filtration \(\mathbb F\). Then \(W\) is a martingale.” This first post unpacks every word in it — probability space, random variable, filtration, adaptedness, conditional expectation and martingale — so that the rest of the series can just use them.

1. The probability space \((\Omega,\mathcal F,\mathbb P)\)

  • \(\Omega\): the sample space; an element \(\omega\in\Omega\) is one outcome (“one scenario of the world”).
  • \(\mathcal F\): the collection of events. An event is a set of outcomes, so each \(A\in\mathcal F\) is a subset of \(\Omega\).
  • \(\mathbb P\): the probability function, \(\mathbb P:\mathcal F\to[0,1]\).

A point that is easy to gloss over: \(\mathbb P\) is a function on \(\mathcal F\), not on \(\Omega\). So \(\mathbb P[\omega]\) is not defined, and \(\mathbb P[\{\omega\}]\) is defined only if \(\{\omega\}\in\mathcal F\).

Example: finite space, equal likelihood. \(\Omega=\{1,2,\dots,N\}\) and \(\mathcal F=2^\Omega\) (the power set, all subsets of \(\Omega\)). Here every singleton is an event, and for \(A\in\mathcal F\)

\[\mathbb P(A)=\frac{\#\{i\in A\}}{N},\qquad\text{in particular }\ \mathbb P[\{\omega\}]=\frac1N.\]

2. Random variables, expectation, independence

A random variable (r.v.) is a function \(X:\Omega\to\mathbb R\): feed it an outcome, it outputs a number.

  • Discrete: \(X\) only takes values in a countable set \(E_X=\{x_1,x_2,\dots\}\). Its pmf is \(p(x)=\mathbb P[\{\omega: X(\omega)=x\}]=\mathbb P[X=x]\), and \(p(x)=0\) for \(x\notin E_X\).
  • Continuous with pdf \(f\): \(\mathbb P(X\in E)=\int_E f(x)\,dx\) for every (Borel) set \(E\subseteq\mathbb R\).

The expectation is the \(\mathbb P\)-weighted average of \(X\):

\[E(X)=E^{\mathbb P}(X)=\int_\Omega \underbrace{X(\omega)}_{\text{size}}\ \underbrace{d\mathbb P(\omega)}_{\text{likelihood}}.\]

Mean \(=E(X)\), variance \(=E[(X-E X)^2]\), \(N\)-th central moment \(=E[\vert X-EX\vert ^N]\).

Independence. \(X\perp\!\!\!\perp Y\) iff for all (nice) functions \(f,g\)

\[E[f(X)g(Y)]=E[f(X)]\,E[g(Y)]\iff f_{X,Y}=f_Xf_Y.\]

Keep the superscript in \(E^{\mathbb P}\) in mind: in Part 06 we will change \(\mathbb P\), and every one of these averages will change with it.

3. Stochastic processes and sample paths

A stochastic process is a collection of random variables indexed by time:

  • discrete time: \(X=\{X_0,X_1,X_2,\dots\}\);
  • continuous time: \(X=\{X_t\}_{t\ge0}\).

It is cleanest to think of a process as a function of two arguments, \(X:[0,\infty)\times\Omega\to\mathbb R\):

  • fix \(t\): \(\omega\mapsto X_t(\omega)\) is a random variable;
  • fix \(\omega\): \(t\mapsto X_t(\omega)\) is a curve, called a sample path. Each scenario \(\omega\) draws its own curve, so a process is a “random curve”.

\(X\) has continuous sample paths if \(t\mapsto X_t(\omega)\) is continuous for every \(\omega\).

4. Brownian motion (first look)

A process \(W=\{W_t\}_{t\ge0}\) on \((\Omega,\mathcal F,\mathbb P)\) is a Brownian motion if

  1. \(W_0=0\) with probability one;
  2. the sample paths \(t\mapsto W_t\) are continuous with probability one;
  3. for \(0\le s<t\): \(W_t-W_s\sim N(0,\,t-s)\);
  4. for \(0\le s<t\le u<v\): \(W_v-W_u\) and \(W_t-W_s\) are independent.

(Throughout the series \(N(\mu,\sigma^2)\) means mean \(\mu\) and variance \(\sigma^2\). The increment over a window of length \(t-s\) has variance \(t-s\), i.e. standard deviation \(\sqrt{t-s}\).)

5. Filtrations: information that only grows

Discrete time. \(\mathbb F=\{\mathcal F_n\}_{n\ge0}\) is a filtration if each \(\mathcal F_n\subseteq\mathcal F\) is a \(\sigma\)-algebra and \(\mathcal F_m\subseteq\mathcal F_n\) for \(m\le n\).

Continuous time. \(\mathbb F=\{\mathcal F_t\}_{t\ge0}\) with \(\mathcal F_s\subseteq\mathcal F_t\subseteq\mathcal F\) for \(0\le s\le t\).

Read \(\mathcal F_t\) as “everything we know at time \(t\)”. The only requirement is that we never forget.

The filtration generated by a process. \(\mathcal F^X_n:=\sigma(X_0,\dots,X_n)\), or in continuous time \(\mathcal F^X_t:=\sigma(X_s:0\le s\le t)\) — the smallest \(\sigma\)-algebra containing what you learn by watching \(X\) up to now.

6. Adaptedness and measurability

\(X\) is adapted to \(\mathbb F\) if the information in \(\mathbb F\) is enough to know \(X\) at every time:

  • \(X_t\) is \(\mathcal F_t\)-measurable (“mbl”), i.e.
  • \(\{\omega: X_t(\omega)\in E\}\in\mathcal F_t\) for every nice \(E\subseteq\mathbb R\).

Intuition: at time \(t\), if you know \(\mathcal F_t\) you can read off the number \(X_t\). No peeking into the future is needed.

A filtration for a Brownian motion. \(\mathbb F\) (not necessarily \(\mathbb F^W\)) is a filtration for \(W\) if

  • \(W\) is adapted to \(\mathbb F\), and
  • \(W_t-W_s\) is independent of \(\mathcal F_s\) for all \(0\le s<t\).

The second condition says: everything we know today tells us nothing about tomorrow’s increment.

7. Conditional expectation

Fix a r.v. \(X\) and a sub-\(\sigma\)-algebra \(\mathcal G\subseteq\mathcal F\). The conditional expectation \(E[X\mid\mathcal G]\) is a random variable with two properties:

  1. Measurability: \(E[X\mid\mathcal G]\) is \(\mathcal G\)-measurable (knowing \(\mathcal G\) is enough to know it);
  2. Partial averaging: for every \(A\in\mathcal G\),
\[E[\mathbf 1_A X]=E\big[\mathbf 1_A\,E(X\mid\mathcal G)\big].\]

Interpretation: \(E[X\mid\mathcal G]\) is our best guess of \(X\) given the information in \(\mathcal G\).

Intuition for partial averaging. Think of \(\mathcal G\) as a way of chopping \(\Omega\) into pieces you can tell apart. On each piece \(A\), the guess \(E[X\mid\mathcal G]\) must have the same average as \(X\) itself. So the conditional expectation is “\(X\), averaged within each piece you can distinguish” — it keeps exactly the detail that \(\mathcal G\) can see and smooths away the rest. Example: two coin tosses, \(X=\) number of heads, \(\mathcal G=\) “the first toss”. On \(\{H\ast\}\) the guess is \(1.5\), on \(\{T\ast\}\) it is \(0.5\); check \(E[\mathbf 1_{\{H\ast\}}X]=\tfrac14(2+1)=\tfrac34=\tfrac12\cdot1.5\).

Properties (use these constantly)

  1. Notation. If \(\mathcal G=\sigma(Y)\), write \(E[X\mid Y]\).
  2. Independence: if \(X\perp\!\!\!\perp\mathcal G\) then \(E[X\mid\mathcal G]=E[X]\).
  3. Known: if \(X\) is \(\mathcal G\)-measurable then \(E[X\mid\mathcal G]=X\).
  4. Linearity: \(E[\alpha X+\beta Y\mid\mathcal G]=\alpha E[X\mid\mathcal G]+\beta E[Y\mid\mathcal G]\).
  5. Taking out what is known (TOWK): if \(X\) is \(\mathcal G\)-measurable, \(E[XY\mid\mathcal G]=X\,E[Y\mid\mathcal G]\).
  6. Tower: if \(\mathcal H\subseteq\mathcal G\subseteq\mathcal F\), then \(E\big[E(X\mid\mathcal G)\mid\mathcal H\big]=E[X\mid\mathcal H]\).

Memory trick for Tower: the smaller information set wins, and in time-indexed form the later time sits inside.

On processes. With \(X\) adapted and \(0\le s<t<u\):

\[\text{TOWK: } E[f(X_s)g(X_t)\mid\mathcal F_s]=f(X_s)\,E[g(X_t)\mid\mathcal F_s],\qquad\text{Tower: } E[X_u\mid\mathcal F_s]=E\big[E(X_u\mid\mathcal F_t)\mid\mathcal F_s\big].\]

At time \(s\), \(X_s\) behaves like a constant; Tower is time-consistency of averaging.

8. Martingales

Definition (continuous time). An adapted \(X\) is a martingale if for all \(0\le s<t\)

\[E[X_t\mid\mathcal F_s]=X_s\quad\text{(with probability one).}\]

Discrete time is the same with \(n,m\); there it is enough to check one step: \(E[X_{n+1}\mid\mathcal F_n]=X_n\) for all \(n\).

  • Super-martingale (\(\searrow\)): \(E[X_t\mid\mathcal F_s]\le X_s\) — the future is smaller on average.
  • Sub-martingale (\(\nearrow\)): \(E[X_t\mid\mathcal F_s]\ge X_s\).

Taking expectations (Tower), \(t\mapsto E[X_t]\) is non-increasing for a super-martingale, constant for a martingale, non-decreasing for a sub-martingale.

Intuition: a martingale is a fair game. Given everything you know, your best forecast of tomorrow’s value is today’s value.

Brownian motion is a martingale

Let \(\mathbb F\) be a filtration for \(W\) and \(s<t\):

\[\begin{aligned} E[W_t\mid\mathcal F_s]&=E[W_t-W_s\mid\mathcal F_s]+E[W_s\mid\mathcal F_s]&&\text{(linearity)}\\ &=E[W_t-W_s]+W_s&&(W_t-W_s\perp\!\!\!\perp\mathcal F_s;\ W_s\text{ known})\\ &=W_s&&(W_t-W_s\sim N(0,t-s)). \end{aligned}\]

The trick — write \(W_t=(W_t-W_s)+W_s\), “future increment plus what we know” — will be used again and again.

Series contents

  1. 01 · Probability Spaces, Filtrations, Conditional Expectation and Martingales (this post)
  2. 02 · Random Walks, Quadratic Variation and Brownian Motion
  3. 03 · Stochastic Integration
  4. 04 · Itô’s Formula, Itô Processes and Geometric Brownian Motion
  5. 05 · Black–Scholes: Replication, the PDE and the Greeks
  6. 06 · Risk-Neutral Pricing, Change of Measure and Girsanov
  7. 07 · SDEs, Feynman–Kac and Stopping Times

Next: Random Walks, Quadratic Variation and Brownian Motion →

This series collects my notes from QST MF 790 Stochastic Calculus at Boston University’s Questrom School of Business, in 7 parts. The content follows the order of the course; wording, intuition and a few corrections are my own. Throughout, \(N(\mu,\sigma^2)\) uses the variance as its second argument. Corrections are welcome. Chinese version of this post: 中文版.

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