Stochastic Processes MATH5835, P. Del Moral UNSW, School of - PowerPoint PPT Presentation

Stochastic Processes MATH5835, P. Del Moral UNSW, School of Mathematics & Statistics Lectures Notes, No. 6 Consultations (RC 5112): Wednesday 3.30 pm � 4.30 pm & Thursday 3.30 pm � 4.30 pm 1/26

Reminder + Information References in the slides ◮ Material for research projects � Moodle ( Stochastic Processes and Applications ∋ variety of applications) ◮ Important results ⊂ Assessment/Final exam = LOGO = 2/26

Citations of the day The essence of mathematics is not to make simple things complicated, but to make complicated things simple. S. Gudder 3/26

Citations of the day The essence of mathematics is not to make simple things complicated, but to make complicated things simple. S. Gudder 3/26

2 citations - US National Academy of Science states If one disqualifies the Pythagorean Theorem from contention, it is hard to think of a mathematical result which is better known and more widely applied in the world today than ”It¯ o’s Lemma” . This result holds the same position in stochastic analysis that Newton’s fundamental theorem holds in classical analysis. That is, it is the sine qua non of the subject. ⇓ Everything related to ”this lemma” = It¯ o-Doeblin formula is .... 4/26

2 citations - US National Academy of Science states If one disqualifies the Pythagorean Theorem from contention, it is hard to think of a mathematical result which is better known and more widely applied in the world today than ”It¯ o’s Lemma” . This result holds the same position in stochastic analysis that Newton’s fundamental theorem holds in classical analysis. That is, it is the sine qua non of the subject. ⇓ Everything related to ”this lemma” = It¯ o-Doeblin formula is .... 4/26

Plan of the lecture ◮ Reminder + multi-dimensional Brownian motions ◮ Diffusion processes: ◮ Ito Calculus ◮ Infinitesimal generators ◮ Evolution semigroups ◮ Fokker Planck equation ◮ Some applications: ◮ Ornstein-Ulhenbeck processe ◮ Geometric Brownian motion ◮ Mathematical finance (Black-Scholes-Merton formula) 5/26

Reminder - Brownian motion W t Simple random walk model √ W t + dt := W t + ǫ t dt with ǫ t := ± 1 Proba 1 / 2 or ǫ t ∼ N ( 0 , 1 ) (1) 6/26

Reminder - Brownian motion W t Simple random walk model √ W t + dt := W t + ǫ t dt with ǫ t := ± 1 Proba 1 / 2 or ǫ t ∼ N ( 0 , 1 ) (1) � Only ”3 simple ingredients”:  E ( ... )=1   � �� �   ǫ 2 dW t × dW t = t dt ” = ” dt (1) ⇒ dt × dt = 0  √    dt × dW t = dt × ± dt = 0 ⊕ Randomness encapsulated in F t = σ ( W s : s ≤ t ) ⋆ . 6/26

2d-Brownian motion ( abbreviated 2d-BM ) W t = ( W (1) , W (2) ) ∼ 2 independent ( ǫ (1) t , ǫ (2) t ) t t ⇓ √ W ( i ) t + dt − W ( i ) = ǫ ( i ) ∀ i , j ∈ { 1 , 2 } dt (2) t t 7/26

2d-Brownian motion ( abbreviated 2d-BM ) W t = ( W (1) , W (2) ) ∼ 2 independent ( ǫ (1) t , ǫ (2) t ) t t ⇓ √ W ( i ) t + dt − W ( i ) = ǫ ( i ) ∀ i , j ∈ { 1 , 2 } dt (2) t t ⇓ ( W (1) t + dt − W (1) ) × ( W (2) t + dt − W (2) ) = ǫ (1) t ǫ (2) dt t t t � �� � E ( ... )=0 ⇓ 7/26

2d-Brownian motion ( abbreviated 2d-BM ) W t = ( W (1) , W (2) ) ∼ 2 independent ( ǫ (1) t , ǫ (2) t ) t t ⇓ √ W ( i ) t + dt − W ( i ) = ǫ ( i ) ∀ i , j ∈ { 1 , 2 } dt (2) t t ⇓ ( W (1) t + dt − W (1) ) × ( W (2) t + dt − W (2) ) = ǫ (1) t ǫ (2) dt t t t � �� � E ( ... )=0 ⇓ � New ingredient: dW ( i ) × dW ( j ) (2) ⇒ ∀ i , j ∈ { 1 , 2 } = 1 i = j dt t t ⊕ Randomness encapsulated in F t = σ ( W s : s ≤ t ) ⋆ . [YouTube video] 7/26

2d- BM W t = ( W 1 t , W 2 t ) df ( W t ) = f ( W t + dW t ) − f ( W t ) ∂ x 1 ( W t ) dW (1) ∂ x 2 ( W t ) dW (2) ∂ f + ∂ f = t t √ ∂ 2 f ∂ 2 f ∂ x 1 ( W t ) dW (1) dW (1) ∂ x 2 ( W t ) dW (2) dW (2) + 1 + 1 + ” O ( dt dt )” 2 t t 2 t t � �� � � �� � = dt = dt 8/26

2d- BM W t = ( W 1 t , W 2 t ) df ( W t ) = f ( W t + dW t ) − f ( W t ) ∂ x 1 ( W t ) dW (1) ∂ x 2 ( W t ) dW (2) ∂ f + ∂ f = t t √ ∂ 2 f ∂ 2 f ∂ x 1 ( W t ) dW (1) dW (1) ∂ x 2 ( W t ) dW (2) dW (2) + 1 + 1 + ” O ( dt dt )” 2 t t 2 t t � �� � � �� � = dt = dt ⇓ 2d - Ito-Doeblin formula df ( W t ) = L ( f )( W t ) dt + dM t ( f ) � �� � = dM (1) ( f )+ dM (2) with ( f ) t t L ( f ) = ∆( f ) ∂ 2 f ∂ 2 f 1 + 1 t ( f ) = ∂ f dM ( i ) ( W t ) dW ( i ) := & t ∂ x 2 ∂ x 2 2 2 ∂ x i 1 2 8/26

2d- BM & Heat eq. � � W (1) ∈ dx , W (2) P ∈ dy = p t ( x , y ) dxdy t t ⇓ [Ito-Doeblin formula] � f ( x , y ) ∂ d E ( f ( W t )) = ∂ t p t ( x , y ) dxdy dt 9/26

2d- BM & Heat eq. � � W (1) ∈ dx , W (2) P ∈ dy = p t ( x , y ) dxdy t t ⇓ [Ito-Doeblin formula] � f ( x , y ) ∂ d E ( f ( W t )) = ∂ t p t ( x , y ) dxdy dt � ∂ 2 � � ∂ x 2 + ∂ 2 1 = f ( x , y ) p t ( x , y ) dxdy dt 2 ∂ y 2 9/26

2d- BM & Heat eq. � � W (1) ∈ dx , W (2) P ∈ dy = p t ( x , y ) dxdy t t ⇓ [Ito-Doeblin formula] � f ( x , y ) ∂ d E ( f ( W t )) = ∂ t p t ( x , y ) dxdy dt � ∂ 2 � � ∂ x 2 + ∂ 2 1 = f ( x , y ) p t ( x , y ) dxdy dt 2 ∂ y 2 � ∂ 2 � � ∂ x 2 + ∂ 2 f ( x , y ) 1 = p t ( x , y ) dxdy dt ∂ y 2 2 9/26

2d- BM & Heat eq. � � W (1) ∈ dx , W (2) P ∈ dy = p t ( x , y ) dxdy t t ⇓ [Ito-Doeblin formula] � f ( x , y ) ∂ d E ( f ( W t )) = ∂ t p t ( x , y ) dxdy dt � ∂ 2 � � ∂ x 2 + ∂ 2 1 = f ( x , y ) p t ( x , y ) dxdy dt 2 ∂ y 2 � ∂ 2 � � ∂ x 2 + ∂ 2 f ( x , y ) 1 = p t ( x , y ) dxdy dt ∂ y 2 2 ⇓ 2d - Heat equation � ∂ 2 � ∂ x 2 + ∂ 2 ∂ t p t ( x , y ) = 1 ∂ p t ( x , y ) ∂ y 2 2 9/26

1d - diffusion processes Simple Markov chain model on R ( b t , σ t smooth + bounded) X t + dt − X t := b t ( X t ) dt + σ t ( X t ) ( W t + dt − W t ) 10/26

1d - diffusion processes Simple Markov chain model on R ( b t , σ t smooth + bounded) X t + dt − X t := b t ( X t ) dt + σ t ( X t ) ( W t + dt − W t ) ⇓ One dimensional diffusion model dX t = b t ( X t ) dt + σ t ( X t ) dW t ⊕ Randomness ⊕ information encapsulated in F t = σ ( W s : s ≤ t ) = σ ( X s , s ≤ t ) ⋆⋆ . 10/26

Ex.: Brownian fluid flow models Fluid particle ( X 0 = 0) : √ dX t = [ v + U ′ ( X t )] dt + 2 D dW t ◮ Fluid velocity flow v. ◮ Diffusion coefficient = D. ◮ Energy well = U ( x ) Example U ( x ) = k x 2 / 2; k = spring constant � Wolfram -[Brownian-Fluid-model-(v,D).cdf ⊕ Brownian-Motion-2D-TheFokker-Planck-Equation.cdf ] 11/26

1d - diffusion processes - Ito formula dX t = b t ( X t ) dt + σ t ( X t ) dW t � �� � � �� � drift term diffusion term 12/26

1d - diffusion processes - Ito formula dX t = b t ( X t ) dt + σ t ( X t ) dW t � �� � � �� � drift term diffusion term ⇓ df ( t , X t ) = f ( t + dt , X t + dX t ) − f ( t , X t ) ∂ 2 f ∂ f ∂ t ( t , X t ) dt + ∂ f ∂ x ( t , X t ) dX t + 1 = ∂ x 2 ( t , X t ) dX t dX t 2 12/26

1d - diffusion processes - Ito formula dX t = b t ( X t ) dt + σ t ( X t ) dW t � �� � � �� � drift term diffusion term ⇓ df ( t , X t ) = f ( t + dt , X t + dX t ) − f ( t , X t ) ∂ 2 f ∂ f ∂ t ( t , X t ) dt + ∂ f ∂ x ( t , X t ) dX t + 1 = ∂ x 2 ( t , X t ) dX t dX t 2 � ∂ f b t ( X t ) ∂ f = ∂ t ( t , X t ) + ∂ x ( t , X t ) � t ( X t ) ∂ 2 f + 1 dt + ∂ f 2 σ 2 ∂ x 2 ( t , X t ) ∂ x ( t , X t ) σ t ( X t ) dW t � �� � � �� � � ∂ � = ∂ t + L t ( f )( t , X t ) dt + dM t ( f ) 12/26

First key observation df ( t , X t ) = f ( t + dt , X t + dX t ) − f ( t , X t ) � ∂ f b t ( X t ) ∂ f = ∂ t ( t , X t ) + ∂ x ( t , X t ) � t ( X t ) ∂ 2 f + 1 dt + ∂ f 2 σ 2 ∂ x 2 ( t , X t ) ∂ x ( t , X t ) σ t ( X t ) dW t � �� � Martingale increment ⇓ Local description of the predictable increment � ∂ � E ([ f ( t + dt , X t + dX t ) − f ( t , X t )] | F t ) = ∂ t + L t ( f )( t , X t ) dt with the (infinitesimal) generator ∂ 2 f ∂ f ∂ x + 1 2 σ 2 L t ( f ) = b t t ∂ x 2 13/26

Second key observation dM t ( f ) := ∂ f ∂ x ( t , X t ) σ t ( X t ) dW t 14/26

Second key observation dM t ( f ) := ∂ f ∂ x ( t , X t ) σ t ( X t ) dW t ⇓ ∂ f E ( dM t ( f ) | F t ) = ∂ x ( t , X t ) σ t ( X t ) E ([ W t + dt − W t ] | F t ) = 0 � ∂ f � 2 � ( dM t ( f )) 2 | F t � � [ W t + dt − W t ] 2 | F t � = ∂ x ( t , X t ) σ t ( X t ) E E � ∂ f � 2 = ∂ x ( t , X t ) σ t ( X t ) dt 14/26

Second key observation dM t ( f ) := ∂ f ∂ x ( t , X t ) σ t ( X t ) dW t ⇓ ∂ f E ( dM t ( f ) | F t ) = ∂ x ( t , X t ) σ t ( X t ) E ([ W t + dt − W t ] | F t ) = 0 � ∂ f � 2 � ( dM t ( f )) 2 | F t � � [ W t + dt − W t ] 2 | F t � = ∂ x ( t , X t ) σ t ( X t ) E E � ∂ f � 2 = ∂ x ( t , X t ) σ t ( X t ) dt ⇓ M t ( f ) Martingale s.t. M 2 t ( f ) = � M ( f ) � t + martingale with the angle bracket � t � ∂ f � 2 � M ( f ) � t = ∂ x ( s , X s ) σ s ( X s ) ds 0 14/26