General setup and standard results Multi-dimensional linear case Backward Stochastic Differential Equations with Infinite Time Horizon Holger Metzler PhD advisor: Prof. G. Tessitore Universit` a di Milano-Bicocca Spring School “Stochastic Control in Finance” Roscoff, March 2010 Holger Metzler PhD advisor: Prof. G. Tessitore BSDEs with Infinite Time Horizon
General setup and standard results Multi-dimensional linear case Outline General setup and standard results 1 The multi-dimensional nonlinear case The one-dimensional nonlinear case Multi-dimensional linear case 2 Holger Metzler PhD advisor: Prof. G. Tessitore BSDEs with Infinite Time Horizon
General setup and standard results The multi-dimensional nonlinear case Multi-dimensional linear case The one-dimensional nonlinear case Outline General setup and standard results 1 The multi-dimensional nonlinear case The one-dimensional nonlinear case Multi-dimensional linear case 2 Holger Metzler PhD advisor: Prof. G. Tessitore BSDEs with Infinite Time Horizon
General setup and standard results The multi-dimensional nonlinear case Multi-dimensional linear case The one-dimensional nonlinear case General setup Throughout this talk, we are given a complete probability space (Ω , F , P ), carrying a standard d -dimensional Brownian motion ( W t ) t ≥ 0 , the filtration ( ˜ F t ) generated by W , the filtration ( F t ), which is ( ˜ F t ) augmented by all P -null sets. = ⇒ ( F t ) satisfies the usual conditions Adapted processes are always assumed to be ( F t )-adapted. Holger Metzler PhD advisor: Prof. G. Tessitore BSDEs with Infinite Time Horizon
General setup and standard results The multi-dimensional nonlinear case Multi-dimensional linear case The one-dimensional nonlinear case General setup Throughout this talk, we are given a complete probability space (Ω , F , P ), carrying a standard d -dimensional Brownian motion ( W t ) t ≥ 0 , the filtration ( ˜ F t ) generated by W , the filtration ( F t ), which is ( ˜ F t ) augmented by all P -null sets. = ⇒ ( F t ) satisfies the usual conditions Adapted processes are always assumed to be ( F t )-adapted. We denote by M 2 ,̺ ( E ) the Hilbert space of processes X with: X is progressively measurable, with values in the Euclidean space E , � ∞ � e ̺ s � X s � 2 � E E ds < ∞ . 0 Holger Metzler PhD advisor: Prof. G. Tessitore BSDEs with Infinite Time Horizon
General setup and standard results The multi-dimensional nonlinear case Multi-dimensional linear case The one-dimensional nonlinear case Consider the BSDE with infinite time horizon − dY t = ψ ( t , Y t , Z t ) dt − Z t dW t , t ∈ [0 , T ] , T ≥ 0 . (1) ψ : Ω × R + × R n × L ( R d , R n ) → R n is such that ψ ( · , y , z ) is a progressively measurable process. A solution is a couple of progressively measurable processes ( Y , Z ) with values in R n × L ( R d , R n ), such that, for all t ≤ T with t , T ≥ 0, T T � � Y t = Y T + ψ ( s , Y s , Z s ) ds − Z s dW s . t t Holger Metzler PhD advisor: Prof. G. Tessitore BSDEs with Infinite Time Horizon
General setup and standard results The multi-dimensional nonlinear case Multi-dimensional linear case The one-dimensional nonlinear case Assumption (A1) (A1 ) There exist C ≥ 0, γ ≥ 0 and µ ∈ R , such that (1) ψ is uniformly lipschitz, i.e. | ψ ( t , y , z ) − ψ ( t , y ′ , z ′ ) | ≤ C | y − y ′ | + γ � z − z ′ � ; (2) ψ is monotone in y : � y − y ′ , ψ ( t , y , z ) − ψ ( t , y ′ , z ) � ≤ − µ | y − y ′ | 2 ; (3) There exists ̺ ∈ R , such that ̺ > γ 2 − 2 µ , and ∞ � e ̺ s | ψ ( s , 0 , 0) | 2 ds ≤ C . E 0 Holger Metzler PhD advisor: Prof. G. Tessitore BSDEs with Infinite Time Horizon
General setup and standard results The multi-dimensional nonlinear case Multi-dimensional linear case The one-dimensional nonlinear case Set λ := γ 2 2 − µ . This implies ̺ > 2 λ . Darling and Pardoux (1997) established the following result. Theorem If (A1) holds then BSDE (1) has a unique solution ( Y , Z ) in M 2 , 2 λ ( R n × L ( R d , R n )) . The solution actually belongs to M 2 ,̺ ( R n × L ( R d , R n )) . The major restriction is the structural condition in part (3) of (A1): We want to solve the equation for arbitrary bounded ψ ( · , 0 , 0). So we need µ > 1 2 γ 2 . This condition is not natural in applications and, hence, is very unpleasant. Holger Metzler PhD advisor: Prof. G. Tessitore BSDEs with Infinite Time Horizon
General setup and standard results The multi-dimensional nonlinear case Multi-dimensional linear case The one-dimensional nonlinear case The one-dimensional case ( n = 1) Significant improvement due to Briand and Hu (1998). Solution exists for all µ > 0, if ψ ( · , 0 , 0) is bounded, i.e. (3’) | ψ ( t , 0 , 0) | ≤ K . µ > 0 means, ψ is dissipative with respect to y . Theorem ( n = 1) Assume parts (1) and (2) of (A1) with µ > 0 , and (3’). Then BSDE (1) has a solution ( Y , Z ) which belongs to M 2 , − 2 µ ( R × R d ) and such that Y is a bounded process. This solution is unique in the class of processes ( Y , Z ) , such that Y is continuous and bounded and Z belongs to M 2 loc ( R d ) . Holger Metzler PhD advisor: Prof. G. Tessitore BSDEs with Infinite Time Horizon
General setup and standard results The multi-dimensional nonlinear case Multi-dimensional linear case The one-dimensional nonlinear case Idea of the proof 1 Consider the equation with finite time horizon [0 , m ]. Call the unique solution ( Y m , Z m ). 2 Establish the a priori bound | Y m ( θ ) | ≤ K µ , for all θ. 3 Use this a priori bound to show that ( Y m , Z m ) m ∈ N is a Cauchy sequence in M 2 , − 2 µ ( R × R d ). The crucial part is to establish the a priori bound. Holger Metzler PhD advisor: Prof. G. Tessitore BSDEs with Infinite Time Horizon
General setup and standard results The multi-dimensional nonlinear case Multi-dimensional linear case The one-dimensional nonlinear case The a priori bound Linearise ψ to ψ ( s , Y m , Z m ) = α m ( s ) Y m ( s ) + β m ( s ) Z m ( s ) + ψ ( s , 0 , 0) with α m ( s ) ≤ − µ and β m bounded. ( Y m , Z m ) solves the equation m � Y m ( t ) = [ α m ( s ) Y m ( s ) + β m ( s ) Z m ( s ) + ψ ( s , 0 , 0)] ds t � m − Z m ( s ) dW s . t Holger Metzler PhD advisor: Prof. G. Tessitore BSDEs with Infinite Time Horizon
General setup and standard results The multi-dimensional nonlinear case Multi-dimensional linear case The one-dimensional nonlinear case Introduce �� t � R m ( t ) := exp α m ( s ) ds , θ � t W m ( t ) := W ( t ) − β m ( s ) ds . 0 Note that R m ( s ) ≤ e − µ ( s − θ ) and ∞ � R m ( s ) ds ≤ 1 µ. θ Holger Metzler PhD advisor: Prof. G. Tessitore BSDEs with Infinite Time Horizon
General setup and standard results The multi-dimensional nonlinear case Multi-dimensional linear case The one-dimensional nonlinear case Apply Itˆ o’s formula to the process R m Y m : m � Y m ( θ ) = R m ( m ) Y m ( m ) + R m ( s ) ψ ( s , 0 , 0) ds θ m � − R m ( s ) Z m ( s ) dW m ( s ) . θ Take into account that Y m ( m ) = 0: m m � � Y m ( θ ) = R m ( s ) ψ ( s , 0 , 0) ds − R m ( s ) Z m ( s ) dW m ( s ) . θ θ Holger Metzler PhD advisor: Prof. G. Tessitore BSDEs with Infinite Time Horizon
General setup and standard results The multi-dimensional nonlinear case Multi-dimensional linear case The one-dimensional nonlinear case Using Girsanov’s theorem, we can consider W m as a Brownian motion with respect to an equivalent measure Q m and hence, we get, Q m -a.s., | Y m ( θ ) | = E Q m [ | Y m ( θ ) | | F θ ] ∞ � ≤ E Q m | ψ ( s , 0 , 0) | R m ( s ) ds | F θ θ ≤ K µ . In the end, this estimate assures also the boundedness of the limit process Y . Holger Metzler PhD advisor: Prof. G. Tessitore BSDEs with Infinite Time Horizon
General setup and standard results The multi-dimensional nonlinear case Multi-dimensional linear case The one-dimensional nonlinear case Problem for n > 1 If Y is a multi-dimensional process ( n > 1), we cannot use this Girsanov trick, because each coordinate needs its own transformation, and these transformations are not consistent among each other. So we are restricted to the case µ > 1 2 γ 2 , whereas the case µ > 0 could have multiple interesting applications, e.g. in stochastic differential games or for homogenisation of PDEs. Holger Metzler PhD advisor: Prof. G. Tessitore BSDEs with Infinite Time Horizon
General setup and standard results Multi-dimensional linear case Outline General setup and standard results 1 The multi-dimensional nonlinear case The one-dimensional nonlinear case Multi-dimensional linear case 2 Holger Metzler PhD advisor: Prof. G. Tessitore BSDEs with Infinite Time Horizon
General setup and standard results Multi-dimensional linear case Multi-dimensional linear case Let us now consider the following equation: d � Γ j Z j − dY t = [ AY t + t + f t ] dt − Z t dW t , t ∈ [0 , T ] , T ≥ 0 . (2) j =1 A , Γ j ∈ R n × n . Z j t denotes the j -th column vector of Z t ∈ R n × d . f t ∈ R n is bounded by K . A is assumed to be dissipative, i.e. there exists µ > 0 such that � y − y ′ , A ( y − y ′ ) � ≤ − µ | y − y ′ | 2 . The coefficients in equation (2) are non-stochastic and, except f t , time-independent. Holger Metzler PhD advisor: Prof. G. Tessitore BSDEs with Infinite Time Horizon
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