Standard Brownian Motion / Wiener Process: An Introduction

in this short video let's take an

introductory look at Wiener processes

now Wiener processes don't have a

dedicated chapter or reading in the frm

curriculum but we do find applications

of Wiener processes in frm exam part 2

when we come to interest rate models so

to understand

Wiener processes in a pretty

introductory kind of way let's take a

step back let's first understand what a

stochastic process is in general think

of a stochastic process to be a variable

whose value changes over time in an

uncertain way so to denote this

stochastic process we denote it as X

subscript T where X denotes my random

variable and T denotes the time at which

a particular value of the random

variable has been observed now this X

can either be a discrete variable or it

can be a continuous variable this D it

can evolve or pass in a discrete way or

the time can pass in a continuous way

depending on this choice of discrete

versus continuous I can categorize

stochastic processes into four camps

discrete variable discrete time discrete

variable continuous time continuous

variable discrete time and continuous

variable continuous time when it comes

to now Wiener processes I will denote

them as W subscript T and they fall in

this category of continuous variable

continuous time so let's do this let's

now build our analysis of the Wiener

process in as introductory away as

possible and let's then in the end get

down to the properties of this Vener

process the only background knowledge

that you would need to work out these

properties is the set of properties of

the normal distribution okay so now

let's do this let's start with a very

convenient starting point let me assume

that at time T equal to zero my W it's

at a value zero now this tells me that

whichever path I simulate for my W all

these parts they have a common starting

point and that starting point is zero at

time T equal to zero now let's do this

let's try and write down the logic which

this W will follow when it comes to

changes in the W over a very tiny time

interval now I have told you that W is

continuous variable continuous time for

a moment let's assume that time is not

continuous but rather discrete let me

start at time T equal to zero and for

this complete time interval which starts

from T equal to zero and ends at time T

equal to capital T think of this capital

T to be the horizon which I have in mind

let me discretize this entire time

interval into a number of time steps

each of these time steps let us assume

it's a duration or length delta T and

let me assume that there are n such time

steps so n times delta T gives me my

capital T now let's assume that at any

point in time which is lowercase T the

value or the level of my Wiener process

is WT and the level at time T plus delta

T is w subscript T plus delta T okay the

change in my Wiener process over this

time interval from T to T plus delta T

let me denote it as Delta W subscript T

so subscript T means it's a period which

begins at time T now comes the logic how

do I write down the change in my Wiener

process over this time interval I just

chose any time interval from 0 to time T

ok to write down that change in my

Wiener process I have to keep two things

in mind the first thing is that the

change in my Wiener process is by no

means deterministic the change is random

and hence when I write down the logic

behind this Delta

WT I need to have a random component in

it that's number one number two the

change in my wiener process over a time

interval which is of a length or

duration delta T I need to also take

this delta T into account smaller is my

delta T smaller intuitively speaking

should be the increment or change in my

W okay so therefore let me write down

this based on both of these components

component one was a random component

component two was some kind of a

dependence or a scaling that happens

taking into account the delta T so let

me write down my delta WT as epsilon T

which is my random component that times

square root of delta T that's my

dependence with respect to the time step

delta T okay this epsilon T is a

standard normal variable so it's

normally distributed at 0 mean and

variance of 1 and this epsilon T is

serially uncorrelated that means the

epsilon which I choose to work out

increment in my WT from time T to time T

plus delta T I call it epsilon T that

epsilon T in no way depends on the

Epsilon's which I chose for the previous

time steps okay so my Epsilon's are

serially uncorrelated I would need this

property going forward I started at time

T equal to 0 at a value 0

that was my property 1 then Delta W

which is over this time interval from 0

to delta T I would call that as Delta W

0 remember in the subscript I always put

the time instant which is the beginning

of the interval so here it is 0 so Delta

W 0 is epsilon 0 the standard normal

variate which i pick at time T equal to

0 scaled by square root of delta T this

then gives me the level of W at the

first time slice let's call it W at

Delta T 2

equal to w at time zero which I know was

a zero plus the change in W which was

epsilon naught times square root of

delta T so my W at Delta T becomes

epsilon naught that times square root in

delta T let's move on to the next time

slice that's a time slice at time two

delta T this can be worked out by first

working out the increment over this time

step the one which runs from delta T to

two delta T that increment I'll call it

as delta w delta T as subscript this is

the standard normal variate which was

picked at time delta T that scaled by

square root of delta T and this becomes

W 2 delta T is equal to w at delta T the

level and delta T plus the new increment

which is epsilon delta T scaled by

square root of delta T basically what

this logic is telling me is that at

every x time step i work out the

increment or the change and I add that

change to the previous level to arrive

at the new level

so basically when I finally reach my

time slice T which is my last time slice

the level of my Wiener process WT would

be equal to the sum of all changes or

increments that happened over all these

time steps so I can write it down as

epsilon naught square root delta T plus

epsilon delta T square root delta T all

the way till the last epsilon which I

picked and that was epsilon at t minus

delta T that times square root of delta

T that's something which you have to

remember from this page ok let's move on

we made a choice the choice was delta w

t is equal to epsilon t scaled by square

root of delta t let's take a look at

what sort of distribution or delta WT

you will have epsilon t a standard

normal distributed i scale that epsilon

t with square root of delta t and

therefore it tells me

that the expected value of Delta WT will

be equal to zero it comes from this guy

the variance of Delta WT will be equal

to root delta T squared because it's a

constant with which this random variable

has been scaled this times the variance

of this guy which I know is one so

variance of my delta WT is equal to

delta T okay now next step let's try and

spend a moment as to why we chose this

as our logic why didn't we choose some

other logic such as Delta WT is equal to

epsilon T times just delta T why did we

take a square root of delta T okay

to understand that let's hypothetically

do two checks my first check is one in

which I make my delta T smaller and

smaller and let it approach a value 0

when that happens then if I choose

square root of delta T in my logic then

square root of delta T it does die down

to a zero when delta T dies down to zero

but the speed at which square root of

delta T dies down to zero is much slower

as compared to the speed at which delta

T dies down to zero therefore when I

make my delta T go down to zero which

definitely I will because I told you

that only for a moment

I made this assumption that time flows

in a discrete way it was just to

highlight various properties of the

Wiener process in the end I need to make

time flow in a continuous way and

continuous flowing time means that any

time increment which I talked about has

to be that tiny that I can approximate

it to be a zero okay so this thing is

bound to happen and if this thing

happens and if this was a delta T

sitting here then Delta WT would have

actually gone down to zero very quickly

when delta WT would have gone down to

zero that quickly

your wiener process would have actually

frozen it would not have evolved over

time okay just take a quick example if

delta T was let's say point zero one

square root of my delta T is actually

point one it's bigger than delta T so

therefore the speed at which the square

root of delta T goes down to zero is

much slower and you do get some changes

or increments in your Wiener process as

your delta T becomes smaller and smaller

so no freezing happens next check what

if delta T was made to be large if delta

T was made to be large square root of

delta T does increase yes but it's

increase is actually at a much much

slower pace as compared to delta T so

based on this we then impart this

behavior on my Wiener process that it

stays more or less contained it does not

really blow up as we keep moving forward

in time so based on these properties

then in the end we have a few properties

that we observe for the various parts of

my Wiener process my Wiener process it

has parts which are continuous these are

parts which can be drawn without lifting

your pen that means continuous parts

parts of the Wiener process are pretty

jagged and that's the behavior which you

observe at any proximity that means even

if you were to zoom in to a path of a

wiener process you don't really arrive

at straight lines you always arrive at

these jagged lines at whatever zoom

level you actually watch this or observe

this path at and because of this guy

it's still finite it doesn't blow up to

infinity unless let's say you were to

let time move to infinity so if you

really want to achieve an infinite value

of W time we really have to go till

infinity for a finite time horizon T

equal to capital T W doesn't go out of

bounds okay so this was a few

a set of properties of W based on the

choice which I made for increments in W

now let's do this based on this guy you

know my delta WT and its distribution

properties let's work out the

distribution of W as it stands at time T

okay this particular part of W leads me

to this value of W T if I were to

simulate any other path it might lead me

to this value a new path might lead me

to this value so W T is a random

variable at this time slice it can

arrive at any value on this time slice

now let's take a look at what

distribution would this random variable

follow my previous page it told me that

WT you can think of it to be a number

which you arrived at by accumulating

many many small increments in W each of

these small increments came from an

epsilon multiplied by a square root of

delta T so when I all when I add all

these increments together use the

distributional assumption of each of

these increments

I can arrive at the distribution of WT

the expected value of WT very simple is

expected value of each of these

increments added together each of these

increments has a zero expected value so

the expected value of WT would also be a

zero the variance of WT would be equal

to the variance of each of these

increments remember I had told you the

Epsilon's are serially uncorrelated so

there won't be any covariance terms now

for us to worry about and therefore the

variance of WT as I just said is the sum

of the variances of the individual

increments and what it then tells me is

that it will be equal to square root of

delta T squared that times 1 plus 1 plus

1 n times so it becomes n times delta T

which I know is capital T since my WT is

a linear combination of many many norm

variables put together the distribution

of WT will also be normal and it tells

me that WT follows a normal distribution

with zero mean and variance as T it

tells me that if I were to take two time

slices one at T and let's say one which

is somewhere here this is at some W you

know at some T one for example so it's

WT one so the distribution at capital T

is a normal distribution zero mean and

variance which is governed by T the

variance is much lower at T one so the

mean here is still zero but the variance

is lower okay so the dispersion of this

distribution of WT 1 is much less around

the mean as compared to WT okay now look

at this result I can actually

extrapolate this result by looking at

this result in a slightly different way

when you take the Wiener process from

time T equal to 0 to a time T equal to

capital T the Wiener process changes by

an amount WT minus w0 this guy I know is

a zero this change in my Wiener process

I am saying is normally distributed with

zero mean and with variance t minus zero

this result I can in general write it

for the change in my Wiener process over

any time interval which runs from t1 to

t2 so my change would be given by W at

t2 minus W at t1 and based on this

result I can write down that this change

will be a normally distributed random

variable with 0 mean and variance t2

minus t1 remember it was t minus zero

here so t2 minus t1 last result and that

is I can extend this fact that the WT is

obtained by various increments put

together right from its starting point

to its ending

if I were to pick two time intervals

which are non-overlapping in nature

non-overlapping time intervals would

mean that they don't share any

increments in between them okay

then I can draw this conclusion that if

the two time intervals are 1 which runs

from t1 to t2 and second one which runs

from t2 to t3 then the change in my

Wiener process over these two time

intervals these two changes will be

independent of one another because there

are no common increments included in

these two intervals ok so W t3 minus WT

- the change in the Wiener process that

happens from t2 to t3 this random

variable would be independent of the

random variable which denotes the change

in my Wiener process from t1 to t2

that's W t2 - w t1 ok so what we have

done in this video is to take a look at

the properties of the Wiener process and

what we've done is that we have tried to

let's say you know do some kind of

sketchy proofs as to how we can

substantiate these properties let me

quickly recap the properties for you the

ones which we've covered number one the

Wiener process starts at time T equal to

zero at a value zero number two the

increment or changes in the bener

process you can treat them as a normally

distributed random variable with 0 mean

and variance equal to the length of that

small time interval then we said that

the terminal distribution the

distribution of W at any time slice in

future WT is a normal distribution 0

mean variance equal to the time at which

this terminal distribution is being

plotted its capital T the changes which

happen for my Wiener process over two

time intervals which are non-overlapping

in nature these changes denote two

different random variables

which are independent of one another

okay there is no correlation between

these two changes if they happen over

two in non-overlapping time intervals

okay so this was a quick look at the

Wiener process