n4 - brownian motion

7/25/2019 N4 - Brownian Motion

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THE UNIVERSITY OF HONG KONG 201516

DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE

STAT3603 Probability Modelling

Brownian motion

The following is adopted from Chapter 3 of the lecture notes on stochastic processes by

David Nualart at https://www.math.ku.edu/~nualart/StochasticCalculus.pdf .

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3 Stochastic Calculus

3.1 Brownian motion

In 1827 Robert Brown observed the complex and erratic motion of grains of pollen suspended in aliquid. It was later discovered that such irregular motion comes from the extremely large numberof collisions of the suspended pollen grains with the molecules of the liquid. In the 20s Norbert

Wiener presented a mathematical model for this motion based on the theory of stochastic processes.The position of a particle at each time t0 is a three dimensional random vector Bt.

The mathematical definition of a Brownian motion is the following:

Definition 19 A stochastic process{Bt, t 0} is called a Brownian motion if it satisfies thefollowing conditions:

i) B0= 0

ii) For all0t1


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Therefore, by Kolmogorovs theorem there exists a Gaussian process with zero mean and co-variance function X(s, t) = min(s, t). On the other hand, Kolmogorovs continuity criterionallows to choose a version of this process with continuous trajectories. Indeed, the incrementBtBs has the normal distribution N(0, t s), and this implies that for any natural numberk we have

E(BtBs)

2k=(2k)!

2k

k!

(t

s)k. (8)

So, choosingk= 2, it is enough because

E

(Bt Bs)4

= 3(t s)2.

4) In the definition of the Brownian motion we have assumed that the probability space (, F, P)is arbitrary. The mapping

C([0, ), R) B()

induces a probability measure PB = P B1 , called the Wiener measure, on the space ofcontinuous functions C = C([0, ),

R

) equipped with its Borel -fieldBC . Then we cantake as canonical probability space for the Brownian motion the space (C, BC, PB). In thiscanonical space, the random variables are the evaluation maps: Xt() =(t).

3.1.1 Regularity of the trajectories

From the Kolmogorovs continuity criterium and using (8) we get that for all >0 there exists arandom variable G,T such that

|Bt Bs| G,T|t s| 12,

for alls, t[0, T]. That is, the trajectories of the Brownian motion are Holder continuous of order12 for all >0. Intuitively, this means that

Bt= Bt+t Bt (t)12 .

This approximation is exact in mean: E

(Bt)2

= t.

3.1.2 Quadratic variation

Fix a time interval [0, t] and consider a subdivision of this interval

0 =t0< t1


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whereasn

k=1

(Bk)2 n

t

n=t.

These properties can be formalized as follows. First, we will show thatn

k=1(Bk)2 converges in

mean square to the length of the interval as the norm of the subdivision tends to zero:

E

nk=1

(Bk)2 t

2 = E n

k=1

(Bk)

2 tk2

=n

k=1

E

(Bk)

2 tk2

=n

k=1

3 (tk)

2 2 (tk)2 + (tk)2

= 2

nk=1

(tk)2 2t||||0 0.

On the other hand, the total variation, defined by

V= sup

nk=1

|Bk|

is infinite with probability one. In fact, using the continuity of the trajectories of the Brownianmotion, we have

nk=1

(Bk)2 sup

k|Bk|

nk=1

|Bk|

V supk

|Bk|||0 0 (9)

ifV 0 the process a1/2Bat, t0

is a Brownian motion. In fact, this process verifies easily properties (i) to (iv).

3.1.4 Stochastic Processes Related to Brownian Motion

1.- Brownian bridge: Consider the process

Xt= B

ttB

1,

t[0, 1]. It is a centered normal process with autocovariance functionE(XtXs) = min(s, t) st,

which verifies X0= 0, X1= 0.

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2.- Brownian motion with drift: Consider the process

Xt = Bt+ t,

t0, where >0 and Rare constants. It is a Gaussian process with

E(Xt) = t,

X(s, t) = 2 min(s, t).

3.- Geometric Brownian motion: It is the stochastic process proposed by Black, Scholes andMerton as model for the curve of prices of financial assets. By definition this process is givenby

Xt= eBt+t,

t 0, where > 0 and R are constants. That is, this process is the exponential of aBrownian motion with drift. This process is not Gaussian, and the probability distributionofXt is lognormal.

3.1.5 Simulation of the Brownian Motion

Brownian motion can be regarded as the limit of a symmetric random walk. Indeed, fix a timeinterval [0, T]. Considern independent are identically distributed random variables 1, . . . , n withzero mean and variance Tn . Define the partial sums

Rk =1+ + k, k= 1, . . . , n .

By the Central Limit Theorem the sequence Rn converges, as n tends to infinity, to the normaldistribution N(0, T).

Consider the continuous stochastic process Sn(t) defined by linear interpolation from the values

Sn(kT

n ) =Rk k= 0, . . . , n .

Then, a functional version of the Central Limit Theorem, known as Donsker Invariance Principle,says that the sequence of stochastic processes Sn(t) converges in law to the Brownian motion on[0, T]. This means that for any continuous and bounded function : C([0, T]) R, we have

E((Sn))E((B)),

as ntends to infinity.The trajectories of the Brownian motion can also be simulated by means of Fourier series

with random coefficients. Suppose that

{en, n

0

} is an orthonormal basis of the Hilbert space

L2([0, T]). Suppose that{Zn, n 0} are independent random variables with law N(0, 1). Then,the random series

n=0

Zn

t0

en(s)ds

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converges uniformly on [0, T], for almost all , to a Brownian motion{Bt, t[0, T]}, that is,

sup0tT

Nn=0

Zn

t0

en(s)ds Bt a.s. 0.

This convergence also holds in mean square. Notice that

E

Nn=0

Zn

t0

en(r)dr

Nn=0

Zn

s0

en(r)dr

=

Nn=0

t0

en(r)dr

s0

en(r)dr

=Nn=0

1[0,t], en

L2([0,T])

1[0,s], en

L2([0,T])

N 1[0,t], 1[0,s]L2([0,T])= s t.In particular, if we take the basis formed by trigonometric functions, en(t) =

1

cos(nt/2),

for n

1, and e0(t) =

1

2, on the interval [0, 2], we obtain the Paley-Wiener representation of

Brownian motion:

Bt = Z0t2

+ 2

n=1

Znsin(nt/2)

n , t[0, 2].

In order to use this formula to get a simulation of Brownian motion, we have to choose somenumber Mof trigonometric functions and a number Nof discretization points:

Z0tj2

+ 2

Mn=1

Znsin(ntj/2)

n ,

wheretj = 2jN ,j= 0, 1, . . . , N .

3.2 Martingales Related with Brownian Motion

Consider a Brownian motion{Bt, t 0} defined on a probability space (, F, P). For any timet, we define the -fieldFt generated by the random variables{Bs, st} and the events inF ofprobability zero. That is,Ft is the smallest -field that contains the sets of the form

{BsA} N,where 0st,A is a Borel subset ofR, andN Fis such that P(N) = 0. Notice thatFs Ftifs t, that is ,{Ft, t0} is a non-decreasing family of-fields. We say that{Ft, t0} is afiltration in the probability space (, F, P).

We say that a stochastic process{

ut, t

0

} is adapted (to the filtration

Ft) if for all t the

random variable ut isFt-measurable.The inclusion of the events of probability zero in each -fieldFt has the following important

consequences:

a) Any version of an adapted process is adapted.

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b) The family of-fields is right-continuous: For all t0s>tFs =Ft.

IfBt is a Brownian motion andFt is the filtration generated by Bt, then, the processesBt

B2t texp(aBt a

2t

2 )

are martingales. In fact, Bt is a martingale because

E(Bt Bs|Fs) =E(Bt Bs) = 0.For B2t t, we can write, using the properties of the conditional expectation, for s < t

E(B2t |Fs) = E((Bt Bs+ Bs)2 |Fs)= E((Bt

Bs )

2

|Fs) + 2E((Bt

Bs ) Bs

|Fs)

+E(B2s |Fs)= E(Bt Bs )2 + 2BsE((Bt Bs ) |Fs) + B2s= t s + B2s .

Finally, for exp(aBt a2t2 ) we have

E(eaBta2t2|Fs) = eaBsE(ea(BtBs)

a2t2|Fs)

= eaBsE(ea(BtBs)a2t2 )

= eaBsea2(ts)

2 a2t

2 =eaBsa2s2 .

As an application of the martingale property of this process we will compute the probabilitydistribution of the arrival time of the Brownian motion to some fixed level.

Example 1 Let Bt be a Brownian motion andFt the filtration generated by Bt. Consider thestopping time

a= inf{t0 :Bt = a},wherea >0. The process Mt= e

Bt2t2 is a martingale such that

E(Mt) =E(M0) = 1.

By the Optional Stopping Theorem we obtain

E(MaN) = 1,

for allN1. Notice that

MaN= exp

BaN2 (a N)

2

ea.

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On the other hand,limN MaN=Ma if a


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such that supi(tnitni1) n 0, then, given two functions f and g on the interval [0, T], the

Riemann Stieltjes integralT0 f dg is defined as

limn

ni=1

f(si1)gi

provided this limit exists and it is independent of the sequences n and n, where gi = g(ti) g(ti1).

The Riemann Stieltjes integralT0 f dg exists iff is continuous and g has bounded variation,

that is,

sup

i

|gi|


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We will define the stochastic integralT0 utdBt for processes u in L

2a,T as the limit in mean

square of the integral of simple processes. By definition a processu in L2a,T is a simple process ifit is of the form

ut =

nj=1

j1(tj1,tj ](t), (11)

where 0 =t0< t1


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wheretj = jTn . The continuity in mean square ofuimplies that

E

T0

ut u(n)t 2 dt

T sup|ts|T/n

E|ut us|2 ,

which converges to zero as ntends to infinity.

2. Suppose now that u is an arbitrary process in the class L2

a,T. Then, we need to show thatthere exists a sequence v (n), of processes in L2a,T, continuous in mean square and such that

limnE

T0

ut v(n)t 2 dt

= 0. (15)

In order to find this sequence we set

v(n)t =n

tt 1

n

usds= n

t0

usds t0

us 1n

ds

,

with the convention u(s) = 0 if s < 0. These processes are continuous in mean square(actually, they have continuous trajectories) and they belong to the class L2a,T. Furthermore,

(15) holds because for each (t, ) we have T0

u(t, ) v(n)(t, )2 dt n 0(this is a consequence of the fact that v

(n)t is defined fromut by means of a convolution with

the kernel n1[ 1n,0]) and we can apply the dominated convergence theorem on the product

space [0, T] because T0

v(n)(t, )

2dt

T0

|u(t, )|2 dt.

Definition 21 The stochastic integral of a processu in theL2a,T is defined as the following limitin mean square T

0utdBt = lim

n

T0

u(n)t dBt, (16)

whereu(n) is an approximating sequence of simple processes that satisfy (14).

Notice that the limit (16) exists because the sequence of random variablesT0 u

(n)t dBtis Cauchy

in the space L2(), due to the isometry property (13):

E T

0

u(n)t dBt

T

0

u(m)t dBt

2

= E T

0u(n)t

u(m)t

2dt

2E T

0

u(n)t ut

2dt

+2E

T0

ut u(m)t

2dt

n,m 0.

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On the other hand, the limit (16) does not depend on the approximating sequence u(n).Properties of the stochastic integral:

1.- Isometry:

E

T0

utdBt

2= E

T0

u2tdt

.

2.- Mean zero:

E

T0

utdBt

= 0.

3.- Linearity: T0

(aut+ bvt) dBt= a

T0

utdBt+ b

T0

vtdBt.

4.- Local property:T0 utdBt = 0 almost surely, on the setG=

T0 u

2tdt= 0

.

Local property holds because on the set Gthe approximating sequence

u(n,m,N)t =

nj=1

m

(j1)Tn

(j1)Tn 1

m

usds

1 (j1)T

n , jT

n

(t)

vanishes.

Example 1 T0

BtdBt =1

2B2T

1

2T.

The process Bt being continuous in mean square, we can choose as approximating sequence

u(n)t =

nj=1

Btj11(tj1,tj ](t),

wheretj = jTn , and we obtain T

0BtdBt = lim

n

nj=1

Btj1

Btj Btj1

= 1

2 limn

nj=1

B2tj B2tj1

1

2 limn

nj=1

Btj Btj1

2=

1

2B2

T1

2T. (17)

Ifxt is a continuously differentiable function such that x0= 0, we know that T0

xtdxt=

T0

xtxtdt=

1

2x2T.

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n4 - brownian motion

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