BROWNIAN MOTION

Brownian motion is named after the botanist Robert Brown who used a microscope to study the fertilization mechanism of flowering plants. He first observed the random motion of pollen particles (obtained from the American species Clarkia pulchella) suspended in water, and wrote:

The fovilla or granules fill the whole orbicular disk but do not extend to the projecting angles. They are not spherical but oblong or nearly cylindrical, and the particles have manifest motion. This motion is only visible to my lens which magnifies 370 times. The motion is obscure yet certain (Robert Brown, 12 June 1827; see Ramsbottom, 1932) It appears that Brown considered this motion no more than a curiosity (he believed that the particles werealive) and continuedundistractedwith his botanical research.

The full significance of his observations only became apparent about eighty years later when it was shown, Einstein (1905), that the motion is caused by the collisions that occur between the pollen grains and the water molecules. In 1908 Perrin, see Perrin (1909), was finally able to confirm Einstein’s predictions experimentally. His work was made possible by the development of the ultramicroscope by Zsigmondy and Siedentopf in 1903. He was able to work out from his experimental results and Einstein’s formula the size of the water molecule and a precise value for Avogadro’s number. His work established the physical theory of Brownian motion and ended the skepticism about the existence of atoms and molecules as actual physical entities.

Many of the fundamental properties of Brownian motion were discovered by Levy (1939, 1948), and the first mathematically rigorous treatment was provided by Wiener (1923, 1924). Karatzas and Shreve (1988) is an excellent text book on the theoretical properties of Brownian motion, while Shreveet al. (1997) provides much useful information concerning the use of Brownian processes within finance.

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Brownian motion is also called a random walk, a Wiener process, or sometimes (more poetically) thedrunkards walk.

In formal terms a process Z¼(Zt:t0) is (one-dimensional) Brownian motion if:

(i) Zt is continuous, andZ0¼0 (ii) ZtN(0,t)

(iii) The incrementdZdt¼ZtþdtZt is normally distributed as,dZdtN(0,dt), so E[dZdt]¼0 and Var(dZdt)¼dt. The incrementdZdt is also independent of the history of the process up to timet.

From (iii) we can further state that, since the incrementsdZ_dtare independent of past valuesZ_t, a Brownian process is also a Markovprocess. In addition we shall now show that Brownian process is also a Martingale process.

In a Martingale process P_t,t0, the conditional expectation E(Ptþdt|Ft)¼P_t, whereFtis called thefiltrationgenerated by the process and contains the information learned by observing the process up to timet. Since for Brownian motion we have

EðZtþdtjFtÞ ¼EððZtþdtZtÞ þZtjFtÞ ¼EðZtþdtZtÞ þZt¼EðdZtþdtÞ þZt¼Zt

where we have used the fact thatE[dZtþdt]¼0. SinceE(ZtþdtjFt)¼Zt, the Brownian motionZis a Martingale process.

We will now consider the Brownian increments over the time interval dtin more detail. Over the time intervaldtwe have:

dX_dt¼dZ_dt ð8:1Þ

wheredZdt is a random variable drawn from a normal distribution with mean zero and variance dt, which we denote as dZdtN(0, dt). Equation 8.1 can also be written in the equivalent form:

dXdt¼ ffiffiffiffiffi dt p

ð8:2Þ

whereis a random variable drawn from a standardnormal distribution (that is a normal distribution with zero mean and unit variance), and we use the notation N(0, 1).

Equations 8.1 and 8.2 give the incremental change in the value ofXover the time intervaldtforstandardBrownian motion.

We shall now generalize these equations slightly by introducing the extra (volatil- ity) parameterwhich controls the variance of the process. We now have:

dXdt¼dZdt ð8:3Þ

wheredZdtN(0,dt), anddXdtN(0, ²dt). Equation 8.3 can also be written in the equivalent form:

dXdt¼ ffiffiffiffiffi pdt

i; iNð0;1Þ ð8:4Þ

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or equivalently dX_dt¼ ffiffiffiffiffi pdt

⁰_i; ⁰_iNð0; ²Þ ð8:5Þ

We are now in a position to provide a mathematical description of the movement of the pollen grains in water observed by Robert Brown in 1827. We will start by assuming that the container of water is perfectly level. This will ensure that there is no drift of the pollen grains in any particular direction. Let us denote the position of a particular pollen grain at timetbyXt, and set the position att¼0,X0, to zero. The statistical distribution of the grain’s position, X_T, at some later time t¼T, can be found as shown below.

We divide the time Tinto n equal intervals dt¼T=n. Since the position of the particle changes by the amountdX_i¼ ffiffiffiffiffi

pdt

_i over theith time intervaldt, the final positionX_T is given by:

XT¼Xⁿ

i¼1

ffiffiffiffiffi dt p

i

¼ ffiffiffiffiffi dt p Xⁿ

i¼1

i

Since_iN(0, 1), by the Law of Large numbers, see Appendix F.1, we have that the expected value of positionXT is:

E½XT ¼ ffiffiffiffiffi dt p

E Xⁿ

i¼1

i

" #

¼0 The variance of the positionXT is:

Var½XT ¼Var ffiffiffiffiffi pdtXⁿ

i¼1

i

" #

¼²dt Var Xⁿ

i¼1

i

" #

ð8:6Þ Using the fact thatVar½i ¼1 and that

Var Xⁿ

i¼1

Xi

" #

¼Xⁿ

i¼1

Var X½ ;i

see Appendix F.3, we have:

Var½XT ¼²dtXⁿ

i¼1

Var½i ¼²dtXⁿ

i¼1

1 ð8:7Þ

which gives:

Var½XT ¼²n dt¼T² ð8:8Þ

So, at timeT, the position of the pollen grain,X_T is distributed asX_T N(0, T²).

If the water container is not perfectly level then the pollen grains will exhibit drift in a particular direction. We can modify Equation 8.4 to take this into account as follows:

dXdt¼dtþ ffiffiffiffiffi dt p

; iNð0;1Þ ð8:9Þ

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or equivalently

dXdt¼dtþdZt; dZtNð0;dtÞ ð8:10Þ where we have included theconstantdrift. Proceeding in a similar manner to that for the case ofzero driftBrownian motion we have:

XT¼Xⁿ

i¼1

dtþ ffiffiffiffiffi pdt

i

¼Xⁿ

i¼1

dtþ ffiffiffiffiffi pdtXⁿ

i¼1

i¼Tþ ffiffiffiffiffi pdtXⁿ

i¼1

i

which gives

E½XT ¼E Tþ ffiffiffiffiffi dt p Xⁿ

i¼1

i

" #

¼Tþ ffiffiffiffiffi dt p

E Xⁿ

i¼1

i

" #

¼T

The variance of the positionX_T is:

Var½XT ¼Var Tþ ffiffiffiffiffi dt p Xⁿ

i¼1

i

" #

¼Var ffiffiffiffiffi dt p Xⁿ

i¼1

i

" #

Here we have used the fact (see Appendix F.3) that Var½aþbX ¼b²Var½X, where a¼T, andb¼1. From Equations 8.6 to 8.8 we have:

Var½XT ¼Var ffiffiffiffiffi pdtXⁿ

i¼1

i

" #

¼T²

So, at timeT, the position of the pollen grain,X_T is distributed asX_T N(T,T²).