Introduction Different ideas for simulation A new scheme Numerical simulation of BSDEs with drivers of quadratic growth Adrien Richou IRMAR, Université de Rennes 1 Roscoff - 2010 Adrien Richou Numerical simulation of quadratic BSDEs
Introduction Different ideas for simulation A new scheme Introduction 1 (Markovian) BSDEs Simulation Quadratic BSDEs Different ideas for simulation 2 A new scheme 3 A time-dependent estimate of Z Convergence of the scheme Adrien Richou Numerical simulation of quadratic BSDEs
Introduction (Markovian) BSDEs Different ideas for simulation Simulation A new scheme Quadratic BSDEs Let (Ω , F , P ) be a probability space, ( W t ) t ∈ R + be a Brownian motion in R d , ( F t ) t ∈ R + be his augmented natural filtration, T be a nonnegative real number. We consider an SDE � t � t X t = x + b ( s , X s ) ds + σ ( s , X s ) dW s , 0 0 with standard assumptions on b and σ , and a Markovian BSDE � T � T Y t = g ( X T ) + f ( s , X s , Y s , Z s ) ds − Z s dW s . t t Adrien Richou Numerical simulation of quadratic BSDEs
Introduction (Markovian) BSDEs Different ideas for simulation Simulation A new scheme Quadratic BSDEs Let (Ω , F , P ) be a probability space, ( W t ) t ∈ R + be a Brownian motion in R d , ( F t ) t ∈ R + be his augmented natural filtration, T be a nonnegative real number. We consider an SDE � t � t X t = x + b ( s , X s ) ds + σ ( s , X s ) dW s , 0 0 with standard assumptions on b and σ , and a Markovian BSDE � T � T Y t = g ( X T ) + f ( s , X s , Y s , Z s ) ds − Z s dW s . t t Definition A solution to this BSDE is a pair of processes ( Y t , Z t ) 0 � t � T such that : ( Y , Z ) is a predicable process with values in R × R 1 × d , 1 P − a . s . t �→ Y t is continuous and 2 � T 0 | f ( r , X r , Y r , Z r ) | + � Z r � 2 dr < ∞ Adrien Richou Numerical simulation of quadratic BSDEs
Introduction (Markovian) BSDEs Different ideas for simulation Simulation A new scheme Quadratic BSDEs Theorem (Pardoux-Peng 1990) Let us assume that f is a Lipschitz function with respect to y and z � T � | g ( X T ) | 2 + � 0 | f ( r , X r , 0 , 0 ) | 2 dr and E < ∞ . Then the previous equation has a unique solution ( Y , Z ) such that � � T � | Y t | 2 � � | Z t | 2 dt < ∞ , < ∞ . E sup E 0 ≤ t ≤ T 0 Adrien Richou Numerical simulation of quadratic BSDEs
Introduction (Markovian) BSDEs Different ideas for simulation Simulation A new scheme Quadratic BSDEs Time discretization We consider a time discretization of the BSDE. We denote the time step by h = T / n and ( t k = kh ) 0 � k � n stands for the discretization times. For X we take the Euler scheme : X n = x 0 X n X n t k + hb ( t k , X n t k ) + σ ( t k , X n = t k )( W t k + 1 − W t k ) , 0 � k � n . t k + 1 For ( Y , Z ) we use the classical dynamic programming equation Y n g ( X n = t n ) t n 1 Z n h E t k [ Y n = t k + 1 ( W t k + 1 − W t k )] , 0 � k � n − 1 , t k Y n E t k [ Y n t k + 1 ] + h E t k [ f ( t k , X n t k , Y n t k + 1 , Z n = t k )] , 0 � k � n − 1 , t k where E t k stands for the conditional expectation given F t k . Adrien Richou Numerical simulation of quadratic BSDEs
Introduction (Markovian) BSDEs Different ideas for simulation Simulation A new scheme Quadratic BSDEs Remarks on simulation the dynamic programming equation is obtained by minimizing the difference � t k + 1 , Z ) − Y − Z ( W t k + 1 − W t k )) 2 � ( Y n t k + 1 + h E t k f ( t k , X n t k , Y n E over F t k -measurable squared integrable random variables ( Y , Z ) . After time discretization, we need to use a spatial discretization in order to compute conditional expectation. We suppose that g and f are Lipschitz functions with respect to x , y , z and t . If we define the error � t k + 1 n − 1 � 2 + E � 2 dt � � � Y n � � Z n � � e ( n ) = sup E t k − Y t k t k − Z t 0 � k � n t k k = 0 then e ( n ) = O ( 1 / n ) . Adrien Richou Numerical simulation of quadratic BSDEs
Introduction (Markovian) BSDEs Different ideas for simulation Simulation A new scheme Quadratic BSDEs References for simulation See, for exemple : B. Bouchard, N. Touzi [2004], J. Zhang [2005], E. Gobet, J.P . Lemor, X. Warin [2005], F . Delarue, S. Menozzi [2006]. Adrien Richou Numerical simulation of quadratic BSDEs
Introduction (Markovian) BSDEs Different ideas for simulation Simulation A new scheme Quadratic BSDEs Quadratic BSDEs What happened if f has a quadratic growth with respect to z ? when g is bounded : existence and uniqueness results have been proved by M. Kobylanski [2000]. when g is unbounded : an existence result has been proved by P . Briand and Y. Hu [2006], partial uniqueness results has been proved by P . Briand and Y. Hu [2008], F . Delbaen, Y. Hu and A. R. [2010]. Such BSDEs have applications in finance : this class arises, for example, in the context of utility optimization problems with exponential utility functions (see e.g. Y. Hu, P . Imkeller and M. Müller [2005]). Adrien Richou Numerical simulation of quadratic BSDEs
Introduction (Markovian) BSDEs Different ideas for simulation Simulation A new scheme Quadratic BSDEs BMO tool Definition � t For a brownian martingale Φ t = 0 φ s dW s , t ∈ [ 0 , T ] , we say that Φ is a BMO martingale if � 1 / 2 �� T � � φ 2 � Φ � BMO = sup E s ds � F τ < + ∞ , � � τ ∈ [ 0 , T ] τ where the supremum is taken over all stopping times in [ 0 , T ] . Adrien Richou Numerical simulation of quadratic BSDEs
Introduction (Markovian) BSDEs Different ideas for simulation Simulation A new scheme Quadratic BSDEs BMO tool the very important feature of BMO martingales is the following lemma : Lemma Let Φ be a BMO martingale. Then we have : The stochastic exponential 1 �� t � t � φ s dW s − 1 | φ s | 2 ds E (Φ) t = E t = exp , 0 � t � T , 2 0 0 is a uniformly integrable martingale. Thanks to the reverse Hölder inequality, there exists p > 1 2 such that E T ∈ L p . The maximal p with this property can be expressed in terms of the BMO norm of Φ . Adrien Richou Numerical simulation of quadratic BSDEs
Introduction (Markovian) BSDEs Different ideas for simulation Simulation A new scheme Quadratic BSDEs Theorem (Briand, Confortola (2008), Ankirchner and al. (2007)) We suppose that M f ( 1 + | y | + | z | 2 ) , | f ( t , x , y , z ) | � | f ( t , x , y , z ) − f ( t , x ′ , y ′ , z ′ ) | K f , x | x − x ′ | + K f , y | y − y ′ | � +( K f , z + L f , z ( | z | + | z ′ | )) | z − z ′ | , | g ( x ) | M g . � The SDE-BSDE system has a unique solution ( X , Y , Z ) such that E [ sup t ∈ [ 0 , T ] | X | 2 ] < + ∞ , Y is a bounded measurable � T 0 | Z s | 2 ds ] < + ∞ . The martingale Z ∗ W process and E [ belongs to the space of BMO martingales and � Z ∗ W � BMO only depends on T, M g and M f . Moreover, there exists r > 1 such that E ( Z ∗ W ) ∈ L r . Adrien Richou Numerical simulation of quadratic BSDEs
Introduction (Markovian) BSDEs Different ideas for simulation Simulation A new scheme Quadratic BSDEs Proposition (Briand, Confortola (2008), Ankirchner and al. (2007)) If we denote ( Y i , Z i ) the solution of a BSDE with a terminal condition g i and a driver f i , then we have � T � 2 ds ] � 2 ] + E [ � � Y 1 t − Y 2 � � � Z 1 s − Z 2 � E [ sup t s t ∈ [ 0 , T ] 0 1 / q � 2 q �� T | g 1 ( X T ) − g 2 ( X T ) | 2 q + � ds � ( f 1 − f 2 )( s , X s , Y 2 � s , Z 2 � � E s ) . 0 where 1 / r + 1 / q = 1 . Adrien Richou Numerical simulation of quadratic BSDEs
Introduction (Markovian) BSDEs Different ideas for simulation Simulation A new scheme Quadratic BSDEs Goal The aim of our work is to give a time discretization scheme for quadratic BSDEs, and to obtain a “good” convergence rate for this scheme. Adrien Richou Numerical simulation of quadratic BSDEs
Introduction Different ideas for simulation A new scheme The exponential transformation When the generator has the specific form f ( t , x , y , z ) = l ( t , x , y ) + a ( t , z ) + γ 2 | z | 2 , with a and l Lipschitz functions and a homogeneous with respect to z , it is possible to use an exponential transform (also known as the Cole-Hopf transformation) : ( e γ Y , γ e γ Y Z ) is the solution of a BSDE with a driver of linear growth. See P . Imkeller, G. dos Reis and J. Zhang [2010]. Adrien Richou Numerical simulation of quadratic BSDEs
Introduction Different ideas for simulation A new scheme g Lipschitz Proposition If g is a Lipschitz function with a Lipschitz constant K g and σ does not depend on x, then, ∀ t ∈ [ 0 , T ] , | Z t | � C ( 1 + K g ) . In this situation the driver becomes a Lipschitz function with respect to z , and so we are allowed to use the classical discrete time approximation. Adrien Richou Numerical simulation of quadratic BSDEs
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