Outline Outline Itos Equation Itos Equation Fokker Fokker- - - PowerPoint PPT Presentation

SLIDE 1

1

• G. Ahmadi

ME 529 - Stochastics

• G. Ahmadi

ME 529 - Stochastics

Outline Outline

• Ito’s Equation

Ito’s Equation

• Fokker

Fokker-

• Planck Equation for

Planck Equation for Ito’s Equation Ito’s Equation

• General Moment Equation

General Moment Equation

• Example

Example-

• 1

1st

st Order Non

Order Non-

• Linear

Linear

• G. Ahmadi

ME 529 - Stochastics

Ito’s Equation Ito’s Equation

.

with with

( )

n G x g X ⋅ + = t dt d ,

( ) ( )

W x G x g X d t dt t d ⋅ + = , ,

( ) ( )

{ }

( )

τ δ τ

ij j i

D t n t n E 2 = +

{ }

dt D dW dW E

ij j i

2 =

• G. Ahmadi

ME 529 - Stochastics

The joint density function The joint density function f fX

X(x,t

(x,t), with ), with X(t X(t) ) being solution to Ito’s equation, satisfies the being solution to Ito’s equation, satisfies the Fokker Fokker-

• Planck (

Planck (Smoluchowski Smoluchowski) equation ) equation given as : given as :

.

( )

( ) ( ) [ ]

∑∑ ∑

∂ ∂ ∂ + ∂ ∂ − = ∂ ∂

i j ij T j i j j j

f GDG x x f t g x t f

2

, x

SLIDE 2

2

• G. Ahmadi

ME 529 - Stochastics

.

( ) ( ) ( ) { }

x x x − = t E t f δ ,

( ) { } ( ) [ ] ( ) [ ]

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ∂ ∂ − ∂ + − ∂ ∂ = − ∂ ∂ = ∂ ∂

∑ ∑∑

j i j j i j i j j

dX dX X X dX X E dt E t dt t f x X x X x X δ δ δ

2

2 1

Starting with Starting with Taking time derivative and keeping terms up Taking time derivative and keeping terms up to first order in to first order in dt dt (second order in (second order in dW dW), ),

( )

∑

+ =

k k jk j j

dW G dt t , g dX x

Using Using

• G. Ahmadi

ME 529 - Stochastics

.

( )

( )

∑ ∑∑ ∑∑

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ ∂ + ∂ ∂ − = ∂ ∂

i j k k j ik j i j j

D G G f x x f t g x t f

l l l 2

, x

Noting that Noting that dW dWk

k is independent of

is independent of X(t X(t), and ), and it follows that it follows that

( ) ( ) ( ) ( )

⎪ ⎭ ⎪ ⎬ ⎫ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − ∂ ∂ ∂ + ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − ∂ ∂ − = ∂ ∂

∑∑ ∑ ∑ ∑ ∑

i j j j k k ik i j j k k jk j j

dW G dt g dW G dt t , g x x dW G dt t , g x E dt t f

l l l

x x X x x X δ δ

1 2

2 1

{ }

dt D dW dW E

ij j i

2 =

• G. Ahmadi

ME 529 - Stochastics

Given Ito’s Equation Given Ito’s Equation

.

( ) ( )

∑

+ =

j i ij i i

dt dW t G t g dt dX , , x x

( )

dt dW t n

i i

=

( ) { }

= t n E

i

( ) ( )

{ }

( )

2 1 2 1

2 t t D t n t n E

ij j i

− = δ

( ) [ ]

( ) [ ]

∑∑ ∑

⋅ ∂ ∂ ∂ + ∂ ∂ − = ∂ ∂

i j ij T j i i i i

f GDG x x f t g x t f

2

, x

Fokker Fokker-

• Planck Equation

Planck Equation

• G. Ahmadi

ME 529 - Stochastics

Expected value of Expected value of h(X h(X) ) is given as is given as

.

( ) { } ( ) ( )

∫ ∫

∞ + ∞ − ∞ + ∞ −

∂ ∂ =

0,

... dx d t f t f h h E dt d x x x x

Taking Time Rate of Change Taking Time Rate of Change

( ) { } ( ) ( ) ( )

∫ ∫

+∞ ∞ − +∞ ∞ −

= , , | , ... x x x x x X d d t f t t f x h h E

SLIDE 3

3

• G. Ahmadi

ME 529 - Stochastics

Eliminating Eliminating ∂ ∂f/ f/∂ ∂t t, ,

.

Integrating by parts leads to the general Integrating by parts leads to the general moment equation: moment equation:

( ) { } ( )

( )

∑∑ ∑

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ∂ ∂ ∂ + ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∂ ∂ =

i j j i ij T i i i

X X h GDG E t g X h E h E dt d

2

, x x ( ) { } ( ) ( ) ( )

( ) ( )

( )

∫ ∫ ∑∑ ∑

∞ + ∞ − ∞ + ∞ −

⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ∂ ∂ ∂ + ∂ ∂ − =

2

, ... x x x x x x d d t f f GDG x x h f g x h h E dt d

i j ij T j i i i i

• G. Ahmadi

ME 529 - Stochastics

Find moment equation corresponding to Find moment equation corresponding to

.

( ) ( )

[ ]

( )

t n t aX t X X + + − =

3

&

( ) ( )

τ δ τ D Rnn 2 =

( )

3

aX X g + − = 1 = G

( ) ( ) ( )

t n t X G t X g X , , + = &

1 1st

st Order System

Order System

( ) { }

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∂ ∂ + ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ∂ ∂ =

2 2 2

X h G DE g x h E X h E dt d

• G. Ahmadi

ME 529 - Stochastics

{ } { } (

)

1

2 2 1

− + =

− −

k k X G DE g X kE m

k k k

&

{ }

k k

X E m =

( ) ( )

2 2

1

− +

− + + − =

k k k k

m k Dk am m k m &

For For

( )

k

X X h =

1 −

= ∂ ∂

k

kx x h

( )

2 2 2

1

−

− = ∂ ∂

k

x k k x h

General Moment Equation General Moment Equation Or Or

• G. Ahmadi

ME 529 - Stochastics

.

k = 1 k = 1

( ) ( ) ( )

t am t m m

3 1 1

+ − = &

( ) ( ) ( )

D t am t m m 2 2

4 2 2

+ + − = &

( ) ( ) ( )

1 5 3 3

6 3 Dm t am t m m + + − = &

k = 2 k = 2 k = 3 k = 3 A closure assumption is now needed, e.g., A closure assumption is now needed, e.g.,

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ + + = + + =

2 2 1 1 4 2 2 1 1 3

m b m b b m m a m a a m

Coefficients may be estimated by Coefficients may be estimated by minimizing mean minimizing mean-

• square error

square error

SLIDE 4

4

• G. Ahmadi

ME 529 - Stochastics

.

Alternative closure scheme Alternative closure scheme assumes assumes X(t X(t) is quasi ) is quasi-

• Gaussian

Gaussian

( )

{ }

( )

{ }⎪

⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ − − − = − − − =

2 2 2 1 4 2 2 2 2 2 1 3 2 1

X b X b b X E e X a X a a X E e

2 1 =

∂ ∂

i

a e

2 2 =

∂ ∂

i

b e 2 , 1 , = i

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ = =

2 2 4 3

3 µ µ µ ( )

{ }

( )

{ }

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − + − = − = − − = − =

4 1 2 2 1 3 1 4 4 1 4 2 1 2 1 3 3 1 3

3 6 4 2 m m m m m m m X E m m m m m X E µ µ

• G. Ahmadi

ME 529 - Stochastics

• G. Ahmadi

ME 529 - Stochastics

• G. Ahmadi

ME 529 - Stochastics

Concluding Remarks Concluding Remarks

• Ito’s Equation

Ito’s Equation

• Fokker

Fokker-

• Planck Equation for

Planck Equation for Ito’s Equation Ito’s Equation

• General Moment Equation

General Moment Equation

• Example

Example-

• 1

1st

st Order Non

Order Non-

• Linear

Linear