2BSDEs with Continuous Coefficients Dylan POSSAMAI Ecole - PowerPoint PPT Presentation

Outline Introduction Continuous 2BSDE with monotonicity condition Continuous 2BSDEs with linear growth Conclusion 2BSDEs with Continuous Coefficients Dylan POSSAMAI Ecole Polytechnique Paris New advances in Backward SDEs for financial engineering applications Tamerza, October 27, 2010 Dylan Possamai 2BSDEs with Continuous Coefficients

Outline Introduction Continuous 2BSDE with monotonicity condition Continuous 2BSDEs with linear growth Conclusion 1 Introduction 2 Continuous 2BSDE with monotonicity condition Preliminaries Uniqueness Approximation and Existence of a solution Limitations 3 Continuous 2BSDEs with linear growth Weak Compactness New Hypotheses 4 Conclusion Dylan Possamai 2BSDEs with Continuous Coefficients

Outline Introduction Continuous 2BSDE with monotonicity condition Continuous 2BSDEs with linear growth Conclusion Introduction Motivated by applications in financial mathematics and probabilistic numerical schemes for PDEs, Soner, Touzi and Zhang introduced recently the notion of second order backward stochastic differential equations (2BSDEs for short) [10], which are connected to the larger class of fully non-linear PDEs. They provided a complete theory of existence and uniqueness for 2BSDEs under uniform Lipschitz conditions similar to those of Pardoux and Peng, so our aim here is twofold we want to relax the Lipschitz assumptions on the driver to a linear growth framework as in Lepeltier and San Martin [6] or Matoussi [7]. we want to highlight the major difficulties and differences from the classical BSDE case. Dylan Possamai 2BSDEs with Continuous Coefficients

Outline Preliminaries Introduction Uniqueness Continuous 2BSDE with monotonicity condition Approximation and Existence of a solution Continuous 2BSDEs with linear growth Limitations Conclusion Plan 2 Continuous 2BSDE with monotonicity condition Preliminaries Uniqueness Approximation and Existence of a solution Limitations Dylan Possamai 2BSDEs with Continuous Coefficients

Outline Preliminaries Introduction Uniqueness Continuous 2BSDE with monotonicity condition Approximation and Existence of a solution Continuous 2BSDEs with linear growth Limitations Conclusion The local martingale measures � � ω ∈ C ([0 , 1] , R d ) : ω 0 = 0 Let Ω := be the canonical space equipped with the uniform norm � ω � ∞ := sup 0 ≤ t ≤ T | ω t | , B the canonical process, P 0 the Wiener measure, F := {F t } 0 ≤ t ≤ T the � � filtration generated by B , and F + := F + 0 ≤ t ≤ T the right limit of t F . We first recall the notations introduced Soner, Touzi and Zhang. P is a local martingale measure if the canonical process B is a local martingale under P . By F¨ ollmer [5], there exists an F -progressively � t measurable process, denoted as 0 B s dB s , which coincides with the Itˆ o’s integral, P − a . s . for all local martingale measure P . This provides a pathwise definition of � t � � 1 � B � t := B t B T B s dB T t − 2 s and � a t := lim sup � B � t − � B � t − ǫ . ǫ 0 ǫ ց 0 Dylan Possamai 2BSDEs with Continuous Coefficients

Outline Preliminaries Introduction Uniqueness Continuous 2BSDE with monotonicity condition Approximation and Existence of a solution Continuous 2BSDEs with linear growth Limitations Conclusion The local martingale measures Let P W denote the set of all local martingale measures P such that a takes values in S > 0 � B � t is absolutely continuous in t and � d , P − a . s . We concentrate on the subclass P s ⊂ P W consisting of all probability measures � t P α := P 0 ◦ ( X α ) − 1 where X α α 1 / 2 t := dB s , t ∈ [0 , 1] , P 0 − a . s . s 0 for some F -progressively measurable process α taking values in S > 0 � T d with 0 | α t | dt < + ∞ , P 0 − a . s . Dylan Possamai 2BSDEs with Continuous Coefficients

Outline Preliminaries Introduction Uniqueness Continuous 2BSDE with monotonicity condition Approximation and Existence of a solution Continuous 2BSDEs with linear growth Limitations Conclusion The non-linear generator We consider a map H t ( ω, y , z , γ ) : [0 , T ] × Ω × R × R d × D H → R , where D H ⊂ R d × d is a given subset containing 0. Define the corresponding conjugate of H w.r.t. γ by � 1 � for a ∈ S > 0 F t ( ω, y , z , a ) := sup 2 Tr ( a γ ) − H t ( ω, y , z , γ ) d , γ ∈ D H � a t ) and � F 0 t := � F t ( y , z ) := F t ( y , z , � F t (0 , 0) . We fix a constant κ ∈ (1 , 2] and restrict to P κ H ⊂ P S a P ∈ S > 0 a P ≤ � a ≤ ¯ a P , dt × d P − as for some a P , ¯ d   � �� T �� 2 � � κ � � κ E H , P E P  ess sup P �� F 0  < + ∞ sup ds � t s P ∈P κ 0 ≤ t ≤ T 0 H Dylan Possamai 2BSDEs with Continuous Coefficients

Outline Preliminaries Introduction Uniqueness Continuous 2BSDE with monotonicity condition Approximation and Existence of a solution Continuous 2BSDEs with linear growth Limitations Conclusion The non-linear generator We assume (i) The domain D F t ( y , z ) = D F t is independent of ( ω, y , z ). (ii) For fixed ( y , z , γ ), F is F -progressively measurable in D F t . (iii) We have the following uniform Lipschitz-type property � � � � � � � � �� F t ( y , z ) − � a 1 / 2 ′ , t ) , ′ ) ′ ) � , P κ ∀ ( y , z , z F t ( y , z � ≤ C �� ( z − z H − q . s . t (iv) F is uniformly continuous in ω for the || · || ∞ norm. (v) F is continuous in y and has the following growth property | F t ( ω, y , 0 , a ) | ≤ | F t ( ω, 0 , 0 , a ) | + C (1+ | y | ) , P κ ∃ C > 0 s.t. H − q . s . (vi) We have the following monotonicity condition ∃ µ > 0 s.t. ( y 1 − y 2 )( F t ( ω, y 1 , z , γ ) − F t ( ω, y 2 , z , γ )) ≤ µ | y 1 − y 2 | 2 . Dylan Possamai 2BSDEs with Continuous Coefficients

Outline Preliminaries Introduction Uniqueness Continuous 2BSDE with monotonicity condition Approximation and Existence of a solution Continuous 2BSDEs with linear growth Limitations Conclusion The non-linear generator Let us comment on these assumptions Assumptions ( i ) and ( iv ) are taken from [10] and are needed to deal with the technicalities induced by the quasi-sure framework. Assumptions ( ii ) and ( iii ) are quite standard in the classical BSDE litterature. Assumptions ( v ) and ( vi ) where introduced by Pardoux in [8] in a more general setting (namely with a general growth condition in y ) and are also quite commonplace in the litterature (see e.g. Briand et al. [1], [2]). Dylan Possamai 2BSDEs with Continuous Coefficients

Outline Preliminaries Introduction Uniqueness Continuous 2BSDE with monotonicity condition Approximation and Existence of a solution Continuous 2BSDEs with linear growth Limitations Conclusion The spaces and norms For p ≥ 1, L p ,κ denotes the space of all F T -measurable scalar r.v. H ξ with E P [ | ξ | p ] < + ∞ . � ξ � p := sup L p ,κ H P ∈P κ H H p ,κ denotes the space of all F + -progressively measurable H R d -valued processes Z with ��� T 2 � � p � Z � p a 1 / 2 E P Z t | 2 dt := sup | � < + ∞ . H p ,κ t H P ∈P κ 0 H D p ,κ denotes the space of all F + -progressively measurable H R -valued processes Y with � � ag paths, and � Y � p P κ E P | Y t | p H − q . s . c` adl` := sup sup < + ∞ . H p ,κ H P ∈P κ 0 ≤ t ≤ T H Dylan Possamai 2BSDEs with Continuous Coefficients

Outline Preliminaries Introduction Uniqueness Continuous 2BSDE with monotonicity condition Approximation and Existence of a solution Continuous 2BSDEs with linear growth Limitations Conclusion The spaces and norms For each ξ ∈ L 1 ,κ H , P ∈ P κ H and t ∈ [0 , T ] denote ′ E H , P P E P [ ξ ] := ess sup t [ ξ ] , t P ′ ∈P κ H ( t + , P ) � � ′ ∈ P κ ′ = P on F + where P κ H ( t + , P ) := P H : P . t Then we define for each p ≥ κ , � � L p ,κ ξ ∈ L p ,κ := : � ξ � L p ,κ < + ∞ , H H H � � P � � p E H , P where � ξ � p E P [ | ξ | κ ] κ := sup ess sup . L p ,κ t H P ∈P κ 0 ≤ t ≤ T H Dylan Possamai 2BSDEs with Continuous Coefficients

Outline Preliminaries Introduction Uniqueness Continuous 2BSDE with monotonicity condition Approximation and Existence of a solution Continuous 2BSDEs with linear growth Limitations Conclusion The spaces and norms Finally, we denote by UC b (Ω) the collection of all bounded and uniformly continuous maps ξ : Ω → R with respect to the �·� ∞ -norm, and we let L p ,κ := the closure of UC b (Ω) under the norm �·� L p ,κ H . H Dylan Possamai 2BSDEs with Continuous Coefficients

Outline Preliminaries Introduction Uniqueness Continuous 2BSDE with monotonicity condition Approximation and Existence of a solution Continuous 2BSDEs with linear growth Limitations Conclusion Formulation Definition For ξ ∈ L 2 ,κ H , we say ( Y , Z ) ∈ D 2 ,κ H × H 2 ,κ is a solution to the H 2BSDE if : • Y T = ξ P κ H − qs . H , the process K P has non-decreasing paths P − as • ∀ P ∈ P κ � t � t � K P t := Y 0 − Y t − F s ( Y s , Z s ) ds + Z s dB s , 0 ≤ t ≤ T . 0 0 � � K P , P ∈ P κ • The family satisfies the minimum condition H � � ′ ′ K P ess inf P E P K P , 0 ≤ t ≤ T , P − as , ∀ P ∈ P κ t = H . t T P ′ ∈P H ( t + , P ) Dylan Possamai 2BSDEs with Continuous Coefficients