4.7 Brownian Bridge(part 2)

報告人：李振綱

OutlineOutline

4.7.4 Multidimensional Distribution of the Brownian Bridge4.7.4 Multidimensional Distribution of the Brownian Bridge

4.7.5 Brownian Bridge as a Conditioned Brownian Motion4.7.5 Brownian Bridge as a Conditioned Brownian Motion

4.7.4 Multidimensional Distribution of the Brownian 4.7.4 Multidimensional Distribution of the Brownian BridgeBridge

We fix and and let denote the Brownian bridge from a to b on . We also fix . We compute the joint density of .

We recall that the Brownian bridge from a to b has the mean function

(P.176)and covariance function

(P.176) When , we may write this as

To simplify notation, we set so that .

a b ( )a bX t

0,T0 1 20 nt t t t T

1( ), , ( )a b a bnX t X t

( ) ( )( )a b b a t T t a bt

m t aT T T

s t

( )( , ) , 0

st s T tc s t s s t T

T T

( , )st

c s t s tT

j jT t 0 T

We define random variable

Because are jointly normal, so that are jointly normal. We compute , and .

1

1

( ) ( )a b a bj j

jj j

X t X tZ

1( ), , ( )a b a bnX t X t

1( ), , ( )nZ t Z tjEZ ( )jVar Z ( , )i jCov Z Z

Brownian Bridge as Gaussian Process (4.7.2)

-1 -1 -1 -11 -1 -1 1

-1 -1 -1 -1

-1 -1 -1 -1

1 1 1 1( , ) ( , ) - ( , ) - ( , ) ( , )

( ) ( ) ( ) ( ) - - 0.

i j i j i j i j i ji j i j i j i j

i j i j i j i j

i j i j i j i j

Cov Z Z c t t c t t c t t c t t

t T t t T t t T t t T t

T T T T

-1 1 -11

1 -1 -1

1

-1

( ) ( )1 1( ) ( ) ( )

( ) .

j j j j j ja b a bj j j

j j j j j j

j j

j j

bt bt bt T t bt T ta aE Z EX t EX t

T T T T T

b t t

1 12 21 1

1 1 12 21 1

-1 -1 1 -1

-1 -1

1 2 1( ) ( ( )) ( ( ), ( )) ( ( ))

1 2 1 ( , ) ( , ) ( , )

2 ( ) 2 (

a b a b a b a bj j j j j

j jj j

j j j j j jj jj j

j j j j j j

j j j

Var Z Var X t Cov X t X t Var X t

c t t c t t c t t

t t t t T t t

T T T

-1 1

-1 -1

) ( ).j j j j j

j j j j

T t t T t t t

T

X,Y are independent Cov(X,Y)=0

( )j jT t a bt

T T

1 1( )j jT t a bt

T T

1 1 11 1

( )( , ) j j j j j j

j j j

t t t T t tc t t t

T T T

If X, Y are normal distribution

課本符號有錯

i j

So we conclude that the normal random variable are independent, and we can write down their joint density, which is

we make the change of variables

1

2

1

-1

( ),..., ( ) 111 1

-1-1

2

1

-1

11

-1

( )

1 1( ,..., ) exp

22

( )

1 exp -

2

n

j jj

nj j

Z t Z t nj jj j j

j jj j

j jj

nj j

j jj

j j

b t tz

f z zt tt t

b t tz

t t

1 1

-1

1.

2

n

j j j

j j

t t

1

1

, 1,..., ,j jj

j j

x xz j n

1 , , nZ Z

mean

VarianceVariance

接下來把 z的部份做變數變換

Where , to find joint density for . We work first on the sum in the exponent to see the effect of this change of variables. We have

Now

2

1

-1

11

-1

2

-1 1 1

1 1 1 -1

2 2 2 2-1 1 1 1 1 1

2 2 2 2 21 11 1 1

( )

( )

( ) 2 2 ( ) 2

j jj

nj j

j jj

j j

nj j j j j j

j j j j j j j

j j j j j j j j j j j j

j j j jj j j j j j

b t tz

t t

x x b t t

t t

x x b t t x x x b t t x

t t

1

21 1

2 2 2-1 1 1 1 1

1 1 1 1 1 1 1

2 21 1 1

1 1 1

( )

( ) 2 2 2=

( ) ( )

- - 21 1-

nj j

j j j

nj j j j j j j j j j

j j j j j j j j j j j j j

j j j j j j

j j j j j j

b t t

x x b t t x x x b x b

t t t t t t

x x

t t t t

1 12

1 1 11 1 1

1 12 .

n n nj j j j

j j jj j j j j j

x x x xb b

t t

1 1 1( ) ( )j j j j j jT t T t t t

0x a 1( ), , ( )a b a bnX t X t

So this last expression is equal to

To change s density, we also need to account for the Jacobian of the change of variables. In this case, we have

and all other partial derivatives are zero. This leads to the Jacobian matrix

2 2 2 21 1 1 12

1 1 1 11 1 1 1

2 2 2n1 2

j=1 1

21

2 1 1 2

( ) 1 1= ( ) 2 ( )

( )

n n n nj j j j j j j j

j j j jj j j j j j j j

j j n n

j j n n n

j j

j j

x x x x x x x xb b

t t

x x x xa ab b

t t T t T T t T T t T

x x

t t

2 2n

j=1 1

( ) ( ).n

n

b x b a

T t T

1 1

1 , 1,..., ,

1 , 2,..., ,

j

j j

j

j j

zj n

x

zj n

x

1

1 2

10 0

1 10

10 0

n

J

1

1n

j j

J

Whose determinant is . Multiplying by this

determinant and using the change of variables worked out above, we

obtain the density for ,

11( ),..., ( )

2 2 21

1 1 11 1

-1

2 21 1

11 11

( ,..., )

( ) ( )1 ( ) 1 1exp

2 2( ) 22

( ) ( )1 1exp

2 2( )2 ( )

a b a bn

nX t X t

n nnj j n

j j jj j n j j j

j j

n nj j j n

jj j j j nj j

f x x

x x b x b a

t t T t T t t

x x b x

t t T tt t

2

2 2 21

11 11

1 11

( )

2

( ) ( )1 1 ( )exp

2 2( ) 22 ( )

( - , , )( , , )

( , , )

n nj j n

jjn j j nj j

nn n

j j j jj

b a

T

x x b xT b a

T t t t T t Tt t

p T t x bp t t x x

p T a b

2

(4.7.6)

1 ( ) ( , , ) exp .

22

y xwhere p x y is the transition density for Brownian motion

1

1n

jj

1( ),..., ( ) 1( ,..., )nZ t Z t nf z z

1( ), , ( )a b a bnX t X t

J

Ch3-5 P.108

4.7.5 Brownian Bridge as a Conditioned Brownian 4.7.5 Brownian Bridge as a Conditioned Brownian MotionMotion The joint density (4.7.6) for permits us to give one

more interpretation for Brownian bridge from a to b on . It is a Brownian motion on this time interval, starting at and conditioned to arrive at b at time T (i.e., conditioned on ). Let be given. The joint density of is

This is because is the density for the Brownian motion going from to in the time between and . Similarly, is the density for going from to between time and . The joint density for and is then the product

1( ),..., ( ), ( ) 1 1 11

0

( ,..., , ) ( , , ) ( , , )

, (0)

n

n

W t W t W T n n n j j j jj

f x x b p T t x b p t t x x

where W x a

1 1 2 1 1 2( , , ) ( , , ).p t a x p t t x x

1( ), , ( )a b a bnX t X t

0,T( )W t (0)W a

( )W T b0 1 20 nt t t t T

1( ), , ( ), ( )nW t W t W T

1 0 0 1 1 1( , , ) ( , , )p t t x x p t a x (0)W a 1 1( )W t x 0t

1t t2 1 1 2( , , )p t t x x 1 1( )W t x 2 2( )W t x

1t t

2t t 1( )W t 2( )W t

??

1 2 1 2 1 1 2 1( ), ( ) 1 2 ( )| (0) 1 ( )| ( ) 2 1 ( ) 1 ( )| ( ) 2 1( , ) ( | ) ( | ) ( ) ( | )W t W t W t W W t W t W t W t W tf x x f x a f x x f x f x x

So, the density of conditioned on is thus the quotient

and this is of (4.7.6).

Finally, let us define

to be the maximum value obtained by the Brownian bridge from a to b on . This random variable has the following distribution.

1 11

( - , , )( , , )

( , , )

nn n

j j j jj

p T t x bp t t x x

p T a b

0( ) max ( )a b a b

t TM T X t

1( ), , ( )nW t W t ( )W T b

11( ),..., ( )

( ,..., )a b a bn

nX t X tf x x

0,T

Corollary 4.7.7.

The density of is

Proof : Because the Brownian bridge from 0 to on is a Brownian motion conditioned on , the maximum of on is the maximum of on conditioned on . Therefore, the density of was computed in Corollary 3.7.4 and is

The density of can be obtained by translating from the initial condition to and using (4.7.9). In particular, in (4.7.9) we replace by and replace by . This result in (4.7.8).

2( )( )

( )

2(2 )( ) , max{ , }. (4.7.8)a b

y a y bT

M T

y b af y e y a b

T

0

2 ( )

( )

2(2 )( ) , , 0. (4.7.9)w

m m w

TM T

m wf m e w m m

T

( )a bM T

0,T( )W T w

w0 wX

0,T 0,T

0 ( )wM TW ( )W T w

( )( )a bM T

f y

(0)W a (0) 0W m y a w b a

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