## Abstract

This chapter provides a fundamental understanding of elementary stochastic calculus in relevance to modern finance, particularly to pricing financial derivatives. It introduces concepts such as conditional expectation with respect to a \(\sigma\)-algebra, filtrations, adapted processes, Brownian motion (BM), martingales, quadratic variation and covariation, the Itô integral with respect to BM, Itô’s lemma, Girsanov theorem for a single BM and geometric Brownian motion (GBM) model. GBM is used to model stock prices in the Black-Scholes-Merton model for option pricing. Options and option pricing will be discussed in later chapters. An important feature of this chapter is the balance between derivational approach and descriptive approach to abstract mathematical concepts.

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## Notes

- 1.
For our purpose, the index set is always a time index set although it is not necessarily so by definition.

- 2.
In mathematics, the set of objects that we are considering is often referred to as a space.

- 3.
There are different kinds of continuity for stochastic processes although we have only introduced one (because we will only use one). Other kinds such as continuity in mean, continuity in probability (or stochastic continuity), and cadlag continuity are also important concepts in the study of mathematical finance. We recommend that the reader who has a serious interest in mathematical finance begin with studying measure theory.

- 4.
A legitimate full description of the stochastic process is based on the notion of

*finite-dimensional distributions*. - 5.
A rigorous definition of conditional expectation can be found in standard graduate-level textbook of probability theory.

- 6.
We also say that \(\mathcal{F}_{t}\) consists of all events that are observable (i.e., \(\mathcal{F}_{t}\)-measurable) by time

*t*. - 7.
For this reason, a risk-neutral probability measure is referred to as an

*equivalent martingale measure*, which is a key concept in derivative pricing (see Remark 7.3, 2 on page 336). - 8.
A proof can be done either by applying Cauchy-Schwarz inequality or by using the result at the link http://mathworld.wolfram.com/RandomWalk1-Dimensional.html to show that

$$\displaystyle{\lim _{n\rightarrow \infty }\frac{\mathbb{E}(\vert S_{n}} {\sqrt{n}} = \sqrt{\frac{2} {\pi }} \,.}$$ - 9.
*White noise*thought of as the derivative of Brownian motion, \(\frac{d\mathfrak{B}_{t}} {dt}\), does not exist in the ordinary sense. It is related to the notion of*generalized stochastic process*, since \(\frac{d\mathfrak{B}} {dt}\) is well defined as a*generalized function*on an infinite dimensional space, which is a topic beyond the scope of this book. - 10.
Robert Brown (1773 - 1858).

- 11.
Nobert Wiener (1894 - 1964).

*Wiener process*is a more popular name among mathematicians than among physicists whereas*Brownian motion*is vice versa. - 12.
\(\mathfrak{B} =\{ \mathfrak{B}(t):\, t \geq 0\}\) on \((\varOmega, \mathfrak{F}, \mathbb{P})\) is often referred to as a

*ℙ-Brownian motion*where*ℙ*represents the probability measure in the real world (or physical world) in contrast to \(\mathbb{Q}\)*-Brownian motion*. \(\mathbb{Q}\)-Brownian motion means that \(\mathfrak{B} =\{ \mathfrak{B}(t):\, t \geq 0\}\) is a process on \((\varOmega, \mathfrak{F}, \mathbb{Q})\) where \(\mathbb{Q}\) represents the probability measure in the risk-neutral world. More detailed explanation is given in Section 6.8.3. - 13.
It is not obvious that all four properties in Definition 6.26 are compatible with each other. For instance, it is not obvious that stationary independent increments and sample continuity are compatible properties.

- 14.
The total variation is a way to measure the “variation” of a (deterministic) real-valued function (see Section (6.6)).

- 15.
In short, “a

*fractal*is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale.” For more explanation, we refer the reader to http://en.wikipedia.org/wiki/Fractal. - 16.
Note that 1 standard deviation from 0 is \(\sqrt{t}\).

- 17.
It is worth noting that both filtrations below are used frequently in the literature:

$$\displaystyle\begin{array}{rcl} (1)\ \mathcal{F}_{t}^{o} = \sigma (\mathfrak{B}_{s}: 0 \leq s \leq t),\quad \quad (2)\ \mathcal{F}_{t}^{+} = \cap _{ s\geq t}\mathcal{F}_{t}^{o},\quad \ \ \text{for}\ \ \forall \ t \geq 0.& & {}\\ \end{array}$$The first is the smallest filtration that makes \(\{\mathfrak{B}_{t}\}\) adapted. The second is an extension of the first by including some zero-probability subsets and has advantages of being

*complete*and*right-continuous*, which are convenient and important properties to have. The second filtration is referred to as the*Brownian filtration*. We use the first filtration in this example to suppress some conceptual and technical details involved in the second.In loose terms and for our purpose, both filtrations are denoted by \(\{\mathcal{F}_{t}\}\), where \(\mathcal{F}_{t}\) is interpreted as the set of information generated by the standard Brownian motion on the time interval [0,

*t*]. - 18.
Notice that, among different concepts of convergence of a sequence of random variables (i.e., convergence in probability, almost sure convergence, and convergence in mean square), convergence in probability is the weakest.

- 19.
The random

*n*-vector \(\mathbf{X}^{\intercal } = [X_{1}\ X_{2}\ldots X_{n}]\) is (or the random variables \(X_{1},X_{2},\ldots,X_{n}\) are) said to have a*multivariate normal distribution*if and only if all linear combinations of \(X_{1},X_{2},\ldots,X_{n}\) are normally distributed. When*n*= 2, \(\mathbf{X}^{\intercal }\) is said to have a*bivariate normal distribution*. - 20.
*Brownian noise*(also known as*brown noise*or*red noise*) is the kind of signal noise produced by Brownian motion. Naturally, it is also called*random walk noise*as a Brownian motion can be viewed as a limit of random walks. Note that a random walk noise is not a white noise (see Example 6.27). - 21.
That is, following an approximation procedure:

*Step 1.*Divide the interval into finitely many subintervals (the partition).*Step 2.*Construct a simple function (use step functions for intuition) that has a constant value on each of the subintervals of the partition (the upper and lower sums).*Step 3.*Define integrals of simple functions (simple processes are random step functions).*Step 4.*Take the limit of these simple functions as more and more dividing points are added to the partition. If the limit exists, it is called the Riemann integral and the function is called the Riemann integrable (Itô integrability requires convergence in mean square).Note that a

*simple function*is a finite linear combination of indicator functions of measurable sets. Thus all step functions are simple functions. - 22.
With probability 1 the Brownian path is non-differentiable in the ordinary sense, but in the context of

*generalized stochastic process*, \(\frac{d\mathfrak{B}} {dt}\) is well defined as a*generalized function*on an infinite dimensional space, which is a topic beyond the scope of this book. - 23.
Itô’s lemma ensures that

*Y*in (6.58) is an Itô process. Thus, both drift process \(\mu _{_{Y }}\) and volatility process \(\sigma _{_{Y }}\) are adapted to the Brownian filtration by definition. - 24.
*(a) Let*\(X_{t} = -\int _{0}^{t}\theta (s)\,d\mathfrak{B}(s)\)*. A straightforward computation leads to*\(D(t) =\mathrm{ e}^{X_{t}-\frac{1} {2} [X]_{t}}\)*and*\(W(t) = \mathfrak{B}(t) - [\mathfrak{B},\,X]_{t}\)*. (b) If*\(\theta (t) =\theta\)*is constant, then*\(D(t) =\mathrm{ e}^{-\theta \,\mathfrak{B}(t)-\frac{1} {2} \theta ^{2}t}\)*and*\(W(t) =\theta t + \mathfrak{B}(t)\). - 25.
Given a filtered probability space \((\varOmega, \mathfrak{F},\{\mathcal{F}_{t}\}, \mathbb{P})\) and a probability measure \(\mathbb{Q}\) defined on measurable space \((\varOmega, \mathfrak{F})\), \(\mathbb{Q}\) is said to be

*absolutely continuous*with respect to probability measure*ℙ*if any \(A \in \mathfrak{F}\) with*ℙ*(*A*) = 0 implies \(\mathbb{Q}(A) = 0\) (i.e., every*ℙ*-null event is a \(\mathbb{Q}\)-null event). This is one of the reasons we prefer another filtration (infinitesimally larger than the natural filtration) for the Brownian motion (see the footnote on page 289) and work with complete probability space. - 26.
Two measures

*ℙ and*\(\mathbb{Q}\)*are equivalent*if for any \(A \in \mathfrak{F}\), \(\mathbb{P}(A) = 0\ \Leftrightarrow \ \mathbb{Q}(A) = 0\). - 27.
This is significant because martingales model a fair game, and security valuation is the determination of the fair price of a security.

- 28.
The subscript

*n*is dropped from p_{ n }, u_{ n }, and d_{ n }to ease the notation. - 29.
More precisely speaking, we assume that \(\mathfrak{B}_{1}\) and \(\mathfrak{B}_{2}\) are defined on the same filtered probability space \(\{\varOmega,\, \mathfrak{F},\,\{\mathcal{F}_{t}\},\, \mathbb{P}\}\) and adapted to the filtration \(\{\mathcal{F}_{t}\}\).

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Petters, A.O., Dong, X. (2016). Stochastic Calculus and Geometric Brownian Motion Model. In: An Introduction to Mathematical Finance with Applications. Springer Undergraduate Texts in Mathematics and Technology. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3783-7_6

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