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

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 st Order Non � Example � 1 st Example- -1 Order Non- -Linear Linear ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi ( ) d X Ito’s Equation Ito’s Equation The joint density function f f X (x,t), with ), with X(t X(t) ) = + ⋅ The joint density function X (x,t g x , t G n dt 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 : d X g x t dt G x t d W given as : { } ( ) ( ) ( ) + τ = δ τ ∂ ∂ ∂ [ ] with ( ) ( ) with 2 E n t n t 2 D f ( ) ∑ ∑∑ = − + T i j ij g x , t f GDG f ∂ ∂ ∂ ∂ j ij t x x x { } j i j j i j . . = E dW dW 2 D dt i j ij ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 1

( ) { ( ( ) ) } = δ − Starting with Starting with , f x t E x t x ⎧ ∂ ∂ ⎡ ⎛ ⎞ ⎤ ⎪ f ∑ ( ) ( ) ∑ = − δ − ⎜ + ⎟ ⎨ dt E ⎢ X x g x , t dt G dW ⎥ ∂ ⎪ ∂ j jk k ⎝ ⎠ t ⎩ x ⎣ ⎦ j k j ⎫ ⎡ ⎤ Taking time derivative and keeping terms up ∂ ⎛ ⎞ ⎛ ⎞ ⎪ Taking time derivative and keeping terms up 2 1 ∑∑ ( ) ( ) ∑ ∑ + δ − ⎜ + ⎟ ⎜ + ⎟ ⎬ ⎢ ⎥ X x g x , t dt G dW g dt G dW ∂ ∂ l l ⎝ i ik k ⎠ ⎝ j j ⎠ ⎪ 2 x x ⎣ ⎦ ⎭ to first order in dt dt (second order in (second order in dW dW ), ), to first order in l i j 1 j k ∂ ∂ f { ( ) } Noting that dW dW k is independent of X(t X(t), and ), and = δ − Noting that k is independent of X x dt E dt { } ∂ ∂ t t = E dW dW 2 D dt it follows that it follows that [ ( ) ] ⎧ ⎫ i j ij ∂ ∂ δ − ⎪ 2 ⎪ [ ] 1 ∑ ( ) ∑∑ X x = δ − + ⎨ ⎬ X x E dX dX dX ⎪ ∂ j ∂ ∂ i j ⎪ ⎡ ⎤ 2 ∂ ∂ ( ) ∂ ⎛ ⎞ ⎩ X X X ⎭ 2 f ∑ ( ) ∑∑ ∑∑ j j . i j i j . = − + ⎜ ⎟ g x , t f ⎢ f G G D ⎥ ∂ ∂ j ∂ ∂ ik j l k l ⎝ ⎠ ⎣ ⎦ t x x x ( ) ∑ = + i j k l j i j x dX g , t dt G dW Using Using j j jk k k ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi Given Ito’s Equation Given Ito’s Equation Expected value of h(X h(X) ) is given as is given as Expected value of dX ( ) ( ) dW ( ) dW ∑ +∞ +∞ = + = { ( ) } ( ) ( ) ( ) ∫ ∫ i i i g x , t G x , t n t = X ... x , | x , x , x x E h h x f t t f t d d i ij i dt dt dt 0 0 0 0 0 − ∞ − ∞ j { } ( ) ( ) ( ) { ( ) } = = δ − 2 E n t 0 E n t n t D t t i 1 j 2 ij 1 2 Taking Time Rate of Change i Taking Time Rate of Change Fokker- Fokker -Planck Equation Planck Equation ∂ d { ( ) } + ∞ + ∞ ( ) f ( ) ∫ ∫ = [ ] ∂ ∂ ∂ ( ) ... 0 , E h x h x f x t d x dx 2 f [ ( ) ] ∑ ∑∑ ∂ 0 0 − ∞ − ∞ = − + ⋅ dt t T x , g t f . GDG f . ∂ ∂ ∂ ∂ i ij t x x x i i j i i j ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 2

Eliminating ∂ ∂ f/ f/ ∂ ∂ t t, , Find moment equation corresponding to Eliminating Find moment equation corresponding to [ ] ( ) ( ) ( ) ( ) ( ) & = − + + 3 τ = δ τ ∂ X X t aX t n t 2 ⎡ ⎤ R nn D ∑ ( ) ( ) − h x g f ⎢ ⎥ ∂ i d { ( ) } + ∞ + ∞ x ( ) ∫ ∫ ⎢ ⎥ = i i E h x ... f x , t d x d x ( ) ∂ ( ( ) ) 2 0 0 0 ⎢ ( ) ⎥ − ∞ − ∞ ∑∑ = dt = − + + 3 G 1 T h x GDG f g X aX ⎢ ⎥ ∂ ∂ ij ⎣ x x ⎦ i j i j ( ) ( ) ( ) st Order System & = + 1 st X g X , t G X , t n t Integrating by parts leads to the general Order System Integrating by parts leads to the general 1 moment equation: moment equation: ∂ ⎧ ∂ ⎫ ⎧ ⎫ 2 d { ( ) } h h . . = + ⎧ ⎫ 2 ⎧ ∂ ⎫ ∂ ⎨ ⎬ ⎨ ⎬ ⎪ ( ) ⎪ 2 E h X E g DE G d { ( ) } h ( ) h ∑ ∑∑ = + ⎨ ⎬ ⎨ T ⎬ ∂ ∂ E h x E g x , t E GDG ⎩ ⎭ 2 ⎩ ⎭ dt x X ∂ ∂ ∂ i ⎪ ij ⎪ ⎩ ⎭ dt X X X ⎩ ⎭ i i j i i j ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi ( ( ) ( ) ) ∂ ∂ = − + 2 ( ) h h ( ) & For = = − = − − m m t am t For k k 1 2 k = 1 1 k k = 1 h X X kx k k x 1 1 3 ∂ ∂ 2 x x ( ( ) ( ) ) = − + + & 2 2 k = 2 k = 2 m m t am t D { } 2 2 4 m = General Moment Equation General Moment Equation k E X k ( ( ) ( ) ) k = 3 k = 3 = − + + & 3 6 m m t am t Dm { } { } ( ) 3 3 5 1 = − + − − & k 1 2 k 2 1 m kE X g DE G X k k k A closure assumption is now needed, e.g., A closure assumption is now needed, e.g., Or Or . = + + ( ) ( ) ⎧ ⎫ m a a m a m Coefficients may be estimated by Coefficients may be estimated by = − + + − ⎨ 3 0 1 1 2 2 ⎬ & 1 m k m am Dk k m = + + ⎩ m b b m b m ⎭ + − minimizing mean- -square error square error minimizing mean 2 2 4 0 1 1 2 2 k k k k ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 3

{ } ( ) ⎧ ⎫ 2 ⎪ = − − − ⎪ 2 3 2 e E X a a X a X { } ⎪ ⎨ 1 0 1 2 ⎬ ( ) ⎪ = − − − 2 2 4 2 ⎩ ⎭ e E X b b X b X 2 0 1 2 ∂ ∂ 2 2 1 = 2 = = e e 0 , 1 , 2 i 0 0 ∂ ∂ a b i i µ = ⎧ ⎫ 0 Alternative closure scheme Alternative closure scheme 3 ⎨ ⎬ µ = µ 2 ⎩ 3 ⎭ assumes X(t X(t) is quasi ) is quasi- -Gaussian Gaussian assumes 4 2 { } . ( ) ⎧ ⎫ µ = − = − − 3 2 2 E X m m m m m { } ⎨ 3 1 3 1 2 1 ⎬ ( ) µ = − 4 = − + − 2 4 ⎩ 4 6 3 ⎭ E X m m m m m m m 4 1 4 1 3 1 2 1 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 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 st Order Non � Example � Example- -1 1 st Order Non- -Linear Linear ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 4