Trivariate Density Revisited Steven E. Shreve Carnegie Mellon - PowerPoint PPT Presentation

Trivariate Density Revisited Steven E. Shreve Carnegie Mellon University – Conference in Honor of Ioannis Karatzas Columbia University June 4–8, 2012 1 / 94

Outline Rank-based diffusion Bang-bang control Trivariate density Personal reflections 2 / 94

Rank-based diffusion E. R. Fernholz, T. Ichiba, I. Karatzas & V. Prokaj, Planar diffusions with rank-based characteristics and perturbed Tanaka equation, Probability Theory and Related Fields , to appear. dX 1 = g dt + σ dB 1 dX 2 = − h dt + ρ dB 2 dX 1 = − h dt + ρ dB 1 dX 2 = g dt + σ dB 2 ρ 2 + σ 2 = 1 g + h > 0 B 1 , B 2 independent Br. motions 3 / 94

Remarks Karatzas, et. al. examine: ◮ Weak and strong existence and uniqueness in law; ◮ Properties of X 1 ∨ X 2 and X 1 ∧ X 2 ; ◮ The reversed dynamics of ( X 1 , X 2 ); 4 / 94

Remarks Karatzas, et. al. examine: ◮ Weak and strong existence and uniqueness in law; ◮ Properties of X 1 ∨ X 2 and X 1 ∧ X 2 ; ◮ The reversed dynamics of ( X 1 , X 2 ); ◮ Transition probabilities ( X 1 , X 2 ). 5 / 94

The difference process Define Y ( t ) = X 1 ( t ) − X 2 ( t ) . 6 / 94

The difference process Define Y ( t ) = X 1 ( t ) − X 2 ( t ) . Then dY = l I { Y ≤ 0 } ( g + h ) dt − l I { Y > 0 } ( g + h ) dt + l I { Y ≤ 0 } σ dB 1 − l I { Y ≤ 0 } ρ dB 2 + l I { Y > 0 } ρ dB 1 − l I { Y > 0 } σ dB 2 7 / 94

The difference process Define Y ( t ) = X 1 ( t ) − X 2 ( t ) . Then dY = l I { Y ≤ 0 } ( g + h ) dt − l I { Y > 0 } ( g + h ) dt + l I { Y ≤ 0 } σ dB 1 − l I { Y ≤ 0 } ρ dB 2 + l I { Y > 0 } ρ dB 1 − l I { Y > 0 } σ dB 2 � � = − λ sgn Y ( t ) dt + dW , 8 / 94

The difference process Define Y ( t ) = X 1 ( t ) − X 2 ( t ) . Then dY = l I { Y ≤ 0 } ( g + h ) dt − l I { Y > 0 } ( g + h ) dt + l I { Y ≤ 0 } σ dB 1 − l I { Y ≤ 0 } ρ dB 2 + l I { Y > 0 } ρ dB 1 − l I { Y > 0 } σ dB 2 � � = − λ sgn Y ( t ) dt + dW , where λ = g + h > 0 , 9 / 94

The difference process Define Y ( t ) = X 1 ( t ) − X 2 ( t ) . Then dY = l I { Y ≤ 0 } ( g + h ) dt − l I { Y > 0 } ( g + h ) dt + l I { Y ≤ 0 } σ dB 1 − l I { Y ≤ 0 } ρ dB 2 + l I { Y > 0 } ρ dB 1 − l I { Y > 0 } σ dB 2 � � = − λ sgn Y ( t ) dt + dW , where λ = g + h > 0 , � � dW = I { Y ≤ 0 } dB 1 − l l I { Y > 0 } dB 2 σ � �� � dW 1 � � + ρ l I { Y > 0 } dB 1 − l I { Y ≤ 0 } dB 2 . � �� � dW 2 10 / 94

The sum process Define Z ( t ) = X 1 ( t ) + X 2 ( t ) . 11 / 94

The sum process Define Z ( t ) = X 1 ( t ) + X 2 ( t ) . Then dZ = ( g − h ) dt + l I { Y ≤ 0 } σ dB 1 + l I { Y ≤ 0 } ρ dB 2 + l I { Y > 0 } ρ dB 1 + l I { Y > 0 } σ dB 2 12 / 94

The sum process Define Z ( t ) = X 1 ( t ) + X 2 ( t ) . Then dZ = ( g − h ) dt + l I { Y ≤ 0 } σ dB 1 + l I { Y ≤ 0 } ρ dB 2 + l I { Y > 0 } ρ dB 1 + l I { Y > 0 } σ dB 2 = ν dt + dV , 13 / 94

The sum process Define Z ( t ) = X 1 ( t ) + X 2 ( t ) . Then dZ = ( g − h ) dt + l I { Y ≤ 0 } σ dB 1 + l I { Y ≤ 0 } ρ dB 2 + l I { Y > 0 } ρ dB 1 + l I { Y > 0 } σ dB 2 = ν dt + dV , where ν = g − h , 14 / 94

The sum process Define Z ( t ) = X 1 ( t ) + X 2 ( t ) . Then dZ = ( g − h ) dt + l I { Y ≤ 0 } σ dB 1 + l I { Y ≤ 0 } ρ dB 2 + l I { Y > 0 } ρ dB 1 + l I { Y > 0 } σ dB 2 = ν dt + dV , where ν = g − h , � � dV = l I { Y ≤ 0 } dB 1 + l I { Y > 0 } dB 2 σ � �� � dV 1 � � + ρ I { Y > 0 } dB 1 + l l I { Y ≤ 0 } dB 2 . � �� � dV 2 15 / 94

Summary X 1 ( t ) + X 2 ( t ) = Z ( t ) = X 1 (0) + X 2 (0) + ν t + V ( t ) , X 1 ( t ) − X 2 ( t ) = Y ( t ) � t � � = X 1 (0) + X 2 (0) − λ sgn Y ( s ) ds + W ( t ) , 0 where V and W are correlated Brownian motions. 16 / 94

Summary X 1 ( t ) + X 2 ( t ) = Z ( t ) = X 1 (0) + X 2 (0) + ν t + V ( t ) , X 1 ( t ) − X 2 ( t ) = Y ( t ) � t � � = X 1 (0) + X 2 (0) − λ sgn Y ( s ) ds + W ( t ) , 0 where V and W are correlated Brownian motions. ◮ To determine transition probabilities for ( X 1 , X 2 ), it suffices to determine transition probabilites for ( Z , Y ). 17 / 94

Summary X 1 ( t ) + X 2 ( t ) = Z ( t ) = X 1 (0) + X 2 (0) + ν t + V ( t ) , X 1 ( t ) − X 2 ( t ) = Y ( t ) � t � � = X 1 (0) + X 2 (0) − λ sgn Y ( s ) ds + W ( t ) , 0 where V and W are correlated Brownian motions. ◮ To determine transition probabilities for ( X 1 , X 2 ), it suffices to determine transition probabilites for ( Z , Y ). ◮ ( Z , W ) is a Gaussian process. ◮ Y is determined by W by � � dY ( t ) = − λ sgn Y ( t ) dt + dW ( t ) . 18 / 94

Summary X 1 ( t ) + X 2 ( t ) = Z ( t ) = X 1 (0) + X 2 (0) + ν t + V ( t ) , X 1 ( t ) − X 2 ( t ) = Y ( t ) � t � � = X 1 (0) + X 2 (0) − λ sgn Y ( s ) ds + W ( t ) , 0 where V and W are correlated Brownian motions. ◮ To determine transition probabilities for ( X 1 , X 2 ), it suffices to determine transition probabilites for ( Z , Y ). ◮ ( Z , W ) is a Gaussian process. ◮ Y is determined by W by � � dY ( t ) = − λ sgn Y ( t ) dt + dW ( t ) . ◮ It suffices to determine the distribution of ( W ( t ) , Y ( t )). 19 / 94

Bang-bang control Minimize � ∞ e − t X 2 t dt , E 0 Subject to � t X t = x + u s ds + W t , a ≤ u t ≤ b , t ≥ 0 . 0 20 / 94

Bang-bang control Minimize � ∞ e − t X 2 t dt , E 0 Subject to � t X t = x + u s ds + W t , a ≤ u t ≤ b , t ≥ 0 . 0 Beneˇ s, Shepp & Witsenhausen (1980): Solution is u t = f ( X t ), where � b , if x < δ, f ( x ) = a , if x ≥ δ, √ √ b 2 + 2 + b ) − 1 − ( a 2 + 2 − a ) − 1 . and δ = ( 21 / 94

Bang-bang control Minimize � ∞ e − t X 2 t dt , E 0 Subject to � t X t = x + u s ds + W t , a ≤ u t ≤ b , t ≥ 0 . 0 Beneˇ s, Shepp & Witsenhausen (1980): Solution is u t = f ( X t ), where � b , if x < δ, f ( x ) = a , if x ≥ δ, √ √ b 2 + 2 + b ) − 1 − ( a 2 + 2 − a ) − 1 . and δ = ( What is the transition density for the controlled process X ? (Beneˇ s, et. al. computed its Laplace transform.) 22 / 94

Transition density by Girsanov Compute the transition density for X , where � t X t = x + f ( X s ) ds + W t . 0 23 / 94

Transition density by Girsanov Compute the transition density for X , where � t X t = x + f ( X s ) ds + W t . 0 Start with a probability measure P X under which X is a Brownian motion. Define W by � t W t = X t − x − f ( X s ) ds . 0 24 / 94

Transition density by Girsanov Compute the transition density for X , where � t X t = x + f ( X s ) ds + W t . 0 Start with a probability measure P X under which X is a Brownian motion. Define W by � t W t = X t − x − f ( X s ) ds . 0 Change to a probability measure P under which W is a Brownian motion. Compute the transition density for X under P . 25 / 94

Transition density by Girsanov Compute the transition density for X , where � t X t = x + f ( X s ) ds + W t . 0 Start with a probability measure P X under which X is a Brownian motion. Define W by � t W t = X t − x − f ( X s ) ds . 0 Change to a probability measure P under which W is a Brownian motion. Compute the transition density for X under P . � � � d P � P { X t ∈ B } = E X l I { X t ∈ B } , � d P X � F t where �� t � t � � d P f ( X s ) dX s − 1 � f 2 ( X s ) ds = exp . � d P X 2 � 0 0 F t 26 / 94

Transition density by Girsanov (continued) To compute P { X t ∈ B } , we need the joint distribution of � t � t f 2 ( X s ) ds . X t , f ( X s ) dX s , 0 0 27 / 94

Transition density by Girsanov (continued) To compute P { X t ∈ B } , we need the joint distribution of � t � t f 2 ( X s ) ds . X t , f ( X s ) dX s , 0 0 Assume without loss of generality that δ = 0. Define � bx , � x if x ≤ 0 , F ( x ) = f ( ξ ) d ξ = ax , if x ≥ 0 . 0 Tanaka’s formula implies � t f ( X s ) dX s + 1 2( a − b ) L X F ( X t ) = t 0 so � t f ( X s ) dX s = F ( X t ) − 1 2( a − b ) L X t . 0 28 / 94

Transition density by Girsanov (continued) To compute P { X t ∈ B } , we need the joint distribution of � t � t f 2 ( X s ) ds . X t , f ( X s ) dX s , 0 0 Assume without loss of generality that δ = 0. Define � bx , � x if x ≤ 0 , F ( x ) = f ( ξ ) d ξ = ax , if x ≥ 0 . 0 Tanaka’s formula implies � t f ( X s ) dX s + 1 2( a − b ) L X F ( X t ) = t 0 so � t f ( X s ) dX s = F ( X t ) − 1 2( a − b ) L X t . 0 We need the joint distribution of � t L X f 2 ( X s ) ds . X t , t , 0 29 / 94

Transition density by Girsanov (continued) Define � t � � Γ + ( t ) = l I (0 , ∞ ) X ( s ) ds , 0 � t � � Γ − ( t ) = l I ( −∞ , 0) X ( s ) ds = t − Γ + ( t ) . 0 30 / 94

Transition density by Girsanov (continued) Define � t � � Γ + ( t ) = l I (0 , ∞ ) X ( s ) ds , 0 � t � � Γ − ( t ) = l I ( −∞ , 0) X ( s ) ds = t − Γ + ( t ) . 0 Then � t f 2 ( X s ) ds = b 2 Γ − ( t ) + a 2 Γ + ( t ) = b 2 t + ( a 2 − b 2 )Γ + ( t ) . 0 31 / 94