Least-square regression Monte Carlo for approximating BSDEs and semilinear PDEs Plamen Turkedjiev BP International Plc. 20th July 2017 Plamen Turkedjiev (BP) Least-squares regression 20th July 2017 1 / 67
Forward-Backward Stochastic Di ff erential Equations (FBSDEs) Plamen Turkedjiev (BP) Least-squares regression 20th July 2017 2 / 67
Continuous time framework Definitions and relations in continuous time ( X, Y, Z ) are predictable R d ⇥ R ⇥ R q -valued processes Z t Z t X t = X 0 + b ( s, X s ) dt + � ( s, X s ) dW s , 0 0 Z T Z T Y t = Φ ( X T ) + f ( s, X s , Y s , Z s ) ds � Z s dW s . t t Feymann-Kac relation (Pardoux-Peng-92): ( Y t , Z t ) = ( Y ( t, X t ) , Z ( t, X t )) where ( Y ( t, x ) , Z ( t, x )) deterministic and solve Y ( t, x ) = u ( t, x ) and Z ( t, x ) = r u ( t, x ) � ( t, x ) for @ t u ( t, x ) + L ( t, x ) u ( t, x ) = f ( t, x, u, r x u � ) , u ( T, x ) = Φ ( x ) , L ( t, x ) g ( x ) = h b ( t, x ) , r x g ( x ) i + 1 2 trace ( �� > ( t, x ) Hess ( g )( x )) . CE Z loss Plamen Turkedjiev (BP) Least-squares regression 20th July 2017 3 / 67
First steps First steps to discrete time approximation Plamen Turkedjiev (BP) Least-squares regression 20th July 2017 4 / 67
First steps Goals Goals of numerical method (1) approximate the stochastic process ˜ X ⇡ X ; (2) compute approximations of Y ( t, x ) and Z ( t, x ) minimizing the loss function Z T | � ( t, ˜ | ( t, ˜ X t ) � Y ( t, X t ) | 2 ] + E [ X t ) � Z ( t, X t ) | 2 dt ]; l ( � , ) := E [ sup 0 t T 0 (3) tune the approximation algorithm to minimize the computational cost. In this talk, we are not concerned with approximating X ; we drop the notation ˜ X hereafter. The loss function is not tractable and we must make an approximation. Plamen Turkedjiev (BP) Least-squares regression 20th July 2017 5 / 67
First steps Finite time grid Finite time grid approximation Let ⇡ = { 0 = t 0 < . . . < t n = T } and define the loss function Z t i +1 X t 2 ⇡ E [ | � ( t, X t ) � Y ( t, X t ) | 2 ]+ | ( t, X t ) � Z ( s, X s ) | 2 ds ] . l ⇡ ( � , ) := max E [ t i i • Clearly l ⇡ ( · ) is an approximation of l ( · ) . • The choice of ⇡ will a ff ect the e ffi ciency of the approximation. • The regularity and boundedness of Φ , f , b , and � will influence the e ffi ciency of the approximation. Plamen Turkedjiev (BP) Least-squares regression 20th July 2017 6 / 67
First steps Finite time grid Conditional expectation formulation BSDE : By taking conditional expectations in Z T � � � Y t = E Φ ( X T ) + f ( s, X s , Y s , Z s ) ds a.s. � F t � t "� 2 # Z T � � � = arg inf Ψ t 2 A ( t ) E � Φ ( X T ) + f ( s, X s , Y s , Z s ) ds � Ψ t � � � t where A ( t ) = L 2 ( F t ; R ) . Markov property: replace A ( t ) by � A t = { : R d ! R � E [ | ( X t ) | 2 ] < 1 } , � "� 2 # Z T � � � Y t = arg inf ( t, · ) 2 A t E � Φ ( X T ) + f ( s, X s , Y s , Z s ) ds � ( t, X t ) � � � t D1 Plamen Turkedjiev (BP) Least-squares regression 20th July 2017 7 / 67
First steps Finite time grid Reformulation of the Y -part of the loss Orthogonality of conditional expectation: Z T f ( s, X s , Y s , Z s ) ds | 2 ] E [ | ( t, X t ) � Φ ( X T ) � t Z T = E [ | ( t, X t ) � Y ( t, X t ) | 2 ] + E [ | Y ( t, X t ) � Φ ( X T ) � f ( s, X s , Y s , Z s ) ds | 2 ] t The Y part of the loss function becomes Z T f ( s, X s , Y s , Z s ) ds | 2 ] . l ⇡ ,y ( t, ) = E [ | ( t, X t ) � Φ ( X T ) � t Plamen Turkedjiev (BP) Least-squares regression 20th July 2017 8 / 67
First steps Finite time grid Z part of the loss BSDE : The optimal discrete Z is also a conditional expectation, Z t i +1 | � ( X t i ) � Z ( s, X s ) | 2 ds ] Z ⇡ ( t i , x ) := arg inf � 2 A t E [ t i Z t i +1 1 = E [ Z s ds | X t i = x ] t i +1 � t i t i W t i +1 � W t i Z T ◆� ✓ � � = E Φ ( X T ) � f ( s, X s , Y s , Z s ) ds � X t i = x � t i +1 � t i t i D1 Plamen Turkedjiev (BP) Least-squares regression 20th July 2017 9 / 67
First steps Finite time grid Z part of the loss As before, we use orthogonality property of the conditional expectation E [ | � ( t i , X t i ) � Z ⇡ ( t i , X t i ) | 2 ] Z T ✓ ◆ + E [ | Z ⇡ ( t i , X t i ) � W t i +1 � W t i | 2 ] Φ ( X T ) + f ( s, X s , Y s , Z s ) ds t i +1 � t i t i Z T ✓ ◆ = E [ | � ( t i , X t i ) � W t i +1 � W t i | 2 ] Φ ( X T ) + f ( s, X s , Y s , Z s ) ds t i +1 � t i t i =: l ⇡ ,z ( t i , � ) . The discrete loss is approximated by X l ⇡ ( , � ) ⇡ max t i 2 ⇡ l ⇡ ,y ( t i , ) + l ⇡ ,z ( t i , � )( t i +1 � t i ) t i 2 ⇡ The loss function is still not tractable because of the integral. Plamen Turkedjiev (BP) Least-squares regression 20th July 2017 10 / 67
Equivalent representations Equivalent continuous time representations Plamen Turkedjiev (BP) Least-squares regression 20th July 2017 11 / 67
Equivalent representations One-step vs. multistep approximation From the tower law, Z T � � � Y ( t i , x ) = E Φ ( X T ) + f ( s, X s , Y s , Z s ) ds � X t i = x � t i Z t i +1 � � � = E Y t i +1 + f ( s, X s , Y s , Z s ) ds � X t i = x . � t i Likewise, W t i +1 � W t i Z T ✓ ◆� � � Z ⇡ ( t i , x ) = E Φ ( X T ) + f ( s, X s , Y s , Z s ) ds � X t i = x � t i +1 � t i t i W t i +1 � W t i Z t i +1 ✓ ◆� � � = E Y t i +1 + f ( s, X s , Y s , Z s ) ds � X t i = x . � t i +1 � t i t i D2 Plamen Turkedjiev (BP) Least-squares regression 20th July 2017 12 / 67
Equivalent representations Decomposition into a system z ) and ( ˜ Y , ˜ Define (ˆ y, ˆ Z ) solving respectively Z T y t = Φ ( X T ) � ˆ z s dW s , ˆ t Z T Z T ˜ y s + ˜ z s + ˜ ˜ Y t = f ( s, X s , ˆ Y s , ˆ Z s ) ds � Z s dW s . t t y t + ˜ z t + ˜ Observe that Y t = ˆ Y t and Z t = ˆ Z t . The representation is beneficial: • The functions ˆ y ( t, X t ) = ˆ y t , ˆ z ( t, X t ) = ˆ z t come from linear equation. • The functions ˜ Y ( t, X t ) = ˜ Y t , ˜ Z ( t, X t ) = ˜ Z t are generally smoother than their Y ( t, x ) , Z ( t, x ) counterparts. ML Plamen Turkedjiev (BP) Least-squares regression 20th July 2017 13 / 67
Equivalent representations Adding zero From the conditional expectation Z T � � � Y ( t i , x ) = E Φ ( X T ) + f ( s, X s , Y s , Z s ) ds � X t i = x � t i 2 3 Z T Z T � � = E 4 Φ ( X T ) + f ( s, X s , Y s , Z s ) ds � Z s dW s � X t i = x � 5 t i t i | {z } = Y ( t i , x ) In other words, the integrand has conditional variance zero. More to come... Plamen Turkedjiev (BP) Least-squares regression 20th July 2017 14 / 67
Equivalent representations Adding zero From the conditional expectation W t i +1 � W t i Z T ◆� ✓ � � Z ⇡ ( t i , x ) = E Φ ( X T ) + f ( s, X s , Y s , Z s ) ds � X t i = x � t i +1 � t i t i W t i +1 � W t i = E t i +1 � t i 0 1 Z T Z T � � � B C ⇥ @ Φ ( X T ) + f ( s, X s , Y s , Z s ) ds � Y ( t i , x ) � Z s dW s � X t i = x � A t i t i +1 | {z } Z t i +1 = Y ( t i +1 , X t i +1 ) � Y ( t i , x ) + f ( s, X s , Y s , Z s ) ds t i The integrand has low conditional variance zero. More to come... D3 Plamen Turkedjiev (BP) Least-squares regression 20th July 2017 15 / 67
Equivalent representations Malliavin representation (Hu-Nualart-Song-11) Rather than computing Z ⇡ ( t, x ) , directly use the representation Z T � � � Z ( t, x ) = E D t Φ ( X T ) + r x f ( s, X s , Y s , Z s ) D t X s ds � X t = x � t i Z T � � � + E @ y f ( s, X s , Y s , Z s ) D t Y s ds � X t = x � t Z T � � � + E r z f ( s, X s , Y s , Z s ) D t Z s ds � X t = x � t Z T � � � = E Γ ( t, T ) D t Φ ( X T ) + Γ ( t, s ) r x f ( s, X s , Y s , Z s ) D t X s ds � X t = x � t i with D t X ⌧ = r x X ⌧ ( r x X t ) � 1 � ( t, X t ) and ✓Z s Z s ◆ ( @ y f ⌧ � 1 2 | r z f ⌧ | 2 d ⌧ Γ ( t, s ) = exp r z f ⌧ dW ⌧ + t t Valid under restricted conditions. Plamen Turkedjiev (BP) Least-squares regression 20th July 2017 16 / 67
Equivalent representations Malliavin integration by parts (Ma-Zhang-02)(T.-15) Rather than computing Z ⇡ ( t, x ) , directly use the representation Z T � � � Z ( t, x ) = E Φ ( X T ) M ( t, T ) + f ( s, X s , Y s , Z s ) M ( t, s ) ds � X t = x � t i for random variables Z s 1 � � 1 ( ⌧ , X ⌧ ) D t X ⌧ dW ⌧ ) > . M ( t, s ) := s � t t Valid under restricted conditions. Sometimes M ( t, s ) is available in closed form. E.g. for X t = W t or geometric Brownian motion, M ( t, s ) = W s � W t . s � t D3 Plamen Turkedjiev (BP) Least-squares regression 20th July 2017 17 / 67
Continuous time approximations Continuous time approximations Plamen Turkedjiev (BP) Least-squares regression 20th July 2017 18 / 67
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