Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
http://www.math.unl.edu
Voice: 402-472-3731
Fax: 402-472-8466
Stochastic Processes and
Advanced Mathematical Finance
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The Deﬁnition of Brownian Motion and the Wiener Process
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Mathematically Mature: may contain mathematics beyond calculus with proofs.
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Some mathematical objects are deﬁned by a formula or an expression. Some other mathematical objects are deﬁned by their properties, not explicitly by an expression. That is, the objects are deﬁned by how they act, not by what they are. Can you name a mathematical object deﬁned by its properties?
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Brownian motion is the physical phenomenon named after the English botanist Robert Brown who discovered it in 1827. Brownian motion is the zig-zagging motion exhibited by a small particle, such as a grain of pollen, immersed in a liquid or a gas. Albert Einstein gave the ﬁrst explanation of this phenomenon in 1905. He explained Brownian motion by assuming the immersed particle was constantly buﬀeted by the molecules of the surrounding medium. Since then the abstracted process has been used for modeling the stock market and in quantum mechanics. The Wiener process is the mathematical deﬁnition and abstraction of the physical process as a stochastic process. The American mathematician Norbert Wiener gave the deﬁnition and properties in a series of papers starting in 1918. Generally, the terms Brownian motion and Wiener process are the same, although Brownian motion emphasizes the physical aspects and Wiener process emphasizes the mathematical aspects. Bachelier process means the same thing as Brownian motion and Wiener process. In 1900, Louis Bachelier introduced the limit of random walk as a model for prices on the Paris stock exchange, and so is the originator of the mathematical idea now called Brownian motion. This term is occasionally found in ﬁnancial literature.
Previously we considered a discrete time random process. That is, at discrete times $n=1,2,3,\dots $ corresponding to coin ﬂips, we considered a sequence of random variables ${T}_{n}$. We are now going to consider a continuous time random process, a function $W\left(t\right)$ that is a random variable at each time $t\ge 0$. To say $W\left(t\right)$ is a random variable at each time is too general so we must put more restrictions on our process to have something interesting to study.
Deﬁnition (Wiener Process). The Standard Wiener Process is a stochastic process $W\left(t\right)$, for $t\ge 0$, with the following properties:
Note that property 2 says that if we know $W\left(s\right)={x}_{0}$, then the independence (and $W\left(0\right)=0$) tells us that further knowledge of the values of $W\left(\tau \right)$ for $\tau <s$ give no added knowledge of the probability law governing $W\left(t\right)-W\left(s\right)$ with $t>s$. More formally, this says that if $0\le {t}_{0}<{t}_{1}<\dots <{t}_{n}<t$, then
$$\begin{array}{c}\mathbb{P}\left[W\left(t\right)\ge x\phantom{\rule{0.3em}{0ex}}|\phantom{\rule{0.3em}{0ex}}W\left({t}_{0}\right)={x}_{0},W\left({t}_{1}\right)={x}_{1},\dots W\left({t}_{n}\right)={x}_{n}\right]\\ =\mathbb{P}\left[W\left(t\right)\ge x\phantom{\rule{0.3em}{0ex}}|\phantom{\rule{0.3em}{0ex}}W\left({t}_{n}\right)={x}_{n}\right].\end{array}$$This is a statement of the Markov property of the Wiener process.
Recall that the sum of independent random variables that are respectively normally distributed with mean ${\mu}_{1}$ and ${\mu}_{2}$ and variances ${\sigma}_{1}^{2}$ and ${\sigma}_{2}^{2}$ is a normally distributed random variable with mean ${\mu}_{1}+{\mu}_{2}$ and variance ${\sigma}_{1}^{2}+{\sigma}_{2}^{2}$, (see Moment Generating Functions..) Therefore for increments $W\left({t}_{3}\right)-W\left({t}_{2}\right)$ and $W\left({t}_{2}\right)-W\left({t}_{1}\right)$ the sum $W\left({t}_{3}\right)-W\left({t}_{2}\right)+W\left({t}_{2}\right)-W\left({t}_{1}\right)=W\left({t}_{3}\right)-W\left({t}_{1}\right)$ is normally distributed with mean $0$ and variance ${t}_{3}-{t}_{1}$ as we expect. Property 2 of the deﬁnition is consistent with properties of normal random variables.
Let
denote the probability density for a $N\left(0,t\right)$ random variable. Then to derive the joint density of the event
with ${t}_{1}<{t}_{2}<\dots <{t}_{n}$, it is equivalent to know the joint probability density of the equivalent event
Then by part 2, we immediately get the expression for the joint probability density function:
A plot of security prices over time and a plot of one-dimensional Brownian motion versus time has at least a superﬁcial resemblance.
If we were to use Brownian motion to model security prices (ignoring for the moment that security prices are better modeled with the more sophisticated geometric Brownian motion rather than simple Brownian motion) we would need to verify that security prices have the 4 deﬁning properties of Brownian motion.
Another good reason for still using the assumption of normality for the increments is that the normal distribution is easy to work with. The normal probability density uses simple functions familiar from calculus, the normal cumulative probability distribution is tabulated, the moment-generating function of the normal distribution is easy to use, and the sum of independent normal distributions is again normal. A substitution of another distribution is possible but the resulting stochastic process models are diﬃcult to analyze, beyond the scope of this model.
However, the assumption of a normal distribution ignores the small possibility that negative stock prices could result from a large negative change. This is not reasonable. (The log normal distribution from geometric Brownian motion that avoids this possibility is a better model).
Moreover, the assumption of a constant variance on diﬀerent intervals of the same length is not a good assumption since stock volatility itself seems to be volatile. That is, the variance of a stock price changes and need not be proportional to the length of the time interval.
At least as a ﬁrst assumption, we will try to use Brownian motion as a model of stock price movements. Remember the mathematical modeling proverb quoted earlier: All mathematical models are wrong, some mathematical models are useful. The Brownian motion model of stock prices is at least moderately useful.
According to the deﬁning property 1 of Brownian motion, we know that if $s<t$, then the conditional density of $X\left(t\right)$ given $X\left(s\right)=B$ is that of a normal random variable with mean $B$ and variance $t-s$. That is,
This gives the probability of Brownian motion being in the neighborhood of $x$ at time $t$, $t-s$ time units into the future, given that Brownian motion is at $B$ at time $s$, the present.
However the conditional density of $X\left(s\right)$ given that $X\left(t\right)=B$, $s<t$ is also of interest. Notice that this is a much diﬀerent question, since $s$ is “in the middle” between $0$ where $X\left(0\right)=0$ and $t$ where $X\left(t\right)=B$. That is, we seek the probability of being in the neighborhood of $x$ at time $s$, $t-s$ time units in the past from the present value $X\left(t\right)=B$.
Theorem 1. The conditional distribution of $X\left(s\right)$, given $X\left(t\right)=B$, $s<t$, is normal with mean $Bs\u2215t$ and variance $\left(s\u2215t\right)\left(t-s\right)$.
$$\begin{array}{c}\mathbb{P}\left[X\left(s\right)\in \left(x,x+\Delta x\right)\phantom{\rule{0.3em}{0ex}}|\phantom{\rule{0.3em}{0ex}}X\left(t\right)=B\right]\\ \approx \frac{1}{\sqrt{2\pi \left(s\u2215t\right)\left(t-s\right)}}exp\left(\frac{-{\left(x-Bs\u2215t\right)}^{2}}{2\left(t-s\right)}\right)\Delta x.\end{array}$$
Proof. The conditional density is
$$\begin{array}{llll}\hfill {f}_{s\phantom{\rule{0.3em}{0ex}}|\phantom{\rule{0.3em}{0ex}}t}\left(x\phantom{\rule{0.3em}{0ex}}|\phantom{\rule{0.3em}{0ex}}B\right)& =\left({f}_{s}\left(x\right){f}_{t-s}\left(B-x\right)\right)\u2215{f}_{t}\left(B\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & ={K}_{1}exp\left(\frac{-{x}^{2}}{2s}-\frac{{\left(B-x\right)}^{2}}{2\left(t-s\right)}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & ={K}_{2}exp\left(-{x}^{2}\left(\frac{1}{2s}+\frac{1}{2\left(t-s\right)}\right)+\frac{Bx}{t-s}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & ={K}_{2}exp\left(\frac{-t}{2s\left(t-s\right)}\cdot {x}^{2}-\frac{2sBx}{t}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & ={K}_{3}exp\left(\frac{-t{\left(x-Bs\u2215t\right)}^{2}}{2s\left(t-s\right)}\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$where ${K}_{1}$, ${K}_{2}$, and ${K}_{3}$ are constants that do not depend on $x$. For example, ${K}_{1}$ is the product of $1\u2215\sqrt{2\pi s}$ from the ${f}_{s}\left(x\right)$ term, and $1\u2215\sqrt{2\pi \left(t-s\right)}$ from the ${f}_{t-s}\left(B-x\right)$ term, times the $1\u2215{f}_{t}\left(B\right)$ term in the denominator. The ${K}_{2}$ term multiplies in an $exp\left(-{B}^{2}\u2215\left(2\left(t-s\right)\right)\right)$ term. The ${K}_{3}$ term comes from the adjustment in the exponential to account for completing the square. We know that the result is a conditional density, so the ${K}_{3}$ factor must be the correct normalizing factor, and we recognize from the form that the result is a normal distribution with mean $Bs\u2215t$ and variance $\left(s\u2215t\right)\left(t-s\right)$. □
Corollary 1. The conditional density of $X\left(t\right)$ for ${t}_{1}<t<{t}_{2}$ given $X\left({t}_{1}\right)=A$ and $X\left({t}_{2}\right)=B$ is a normal density with mean
and variance
Proof. $X\left(t\right)$ subject to the conditions $X\left({t}_{1}\right)=A$ and $X\left({t}_{2}\right)=B$ has the same density as the random variable $A+X\left(t-{t}_{1}\right)$, under the condition $X\left({t}_{2}-{t}_{1}\right)=B-A$ by condition 2 of the deﬁnition of Brownian motion. Then apply the theorem with $s=t-{t}_{1}$ and $t={t}_{2}-{t}_{1}$. □
The material in this section is drawn from A First Course in Stochastic Processes by S. Karlin, and H. Taylor, Academic Press, 1975, pages 343–345 and Introduction to Probability Models by S. Ross, pages 413-416.
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Simulate a sample path of the Wiener process as follows, see [1]. Divide the interval $\left[0,T\right]$ into a grid $0={t}_{0}<{t}_{1}<\dots <{t}_{N-1}<{t}_{N}=T$ with ${t}_{i+1}-{t}_{i}=\Delta t$. Set $i=1$ and $W\left(0\right)=W\left({t}_{0}\right)=0$ and iterate the following algorithm.
This method of approximation is valid only on the points of the grid. In between any two points ${t}_{i}$ and ${t}_{i-1}$, the Wiener process is approximated by linear interpolation.
Octave script for Wiener process.
Perl PDL script for Winer process.
Scientiﬁc Python script for Wiener process.
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the covariance of $X$ and $Y$. If $X$ and $Y$ are independent, then $Cov\left(X,Y\right)=0$. A positive value of $Cov\left(X,Y\right)$ indicates that $Y$ tends to increases as $X$ does, while a negative value indicates that $Y$ tends to decrease when $X$ increases. Thus, $Cov\left(X,Y\right)$ is an indication of the mutual dependence of $X$ and $Y$. Show that
satisﬁes the partial diﬀerential equation for heat ﬂow (the heat equation)
Discuss how changes in the parameters of the simulation change the Wiener process.
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[1] Stefano M. Iacus. Simulation and Inference for Stochastic Diﬀerential Equations. Number XVIII in Springer Series in Statistics. Springer-Verlag, 2008.
[2] S. Karlin and H. Taylor. A Second Course in Stochastic Processes. Academic Press, 1981.
[3] Sheldon M. Ross. Introduction to Probability Models. Elsevier, 8th edition, 2003.
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