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Workshop on Stochastic Analysis and Finance ’09 Application of the lent particle method to Poisson driven sde’s Laurent DENIS Universit´ e d’Evry-Val-d’Essonne and Chaire “Risque de Cr´ edit” City University of Hong-Kong, June 29-July 3, 2009 Based on a joint work with N. Bouleau. Laurent DENIS Application of the lent particle method to Poisson driven sde’s

Introduction We are given: • ( X , X , ν, d , γ ): a local symmetric Dirichlet structure which admits a carr´ e du champ operator i.e. ( X , X , ν ) is a measured space, ν is σ -finite and the bilinear form e [ f , g ] = 1 � γ [ f , g ] d ν, 2 is a local Dirichlet form with domain d ⊂ L 2 ( ν ) and carr´ e du champ operator γ . • N : a Poisson random measure on [0 , + ∞ [ × X with intensity dt × ν ( du ) defined on the probability space (Ω 1 , A 1 , P 1 ) where Ω 1 is the configuration space , A 1 the σ -field generated by N and P 1 the law of N . Laurent DENIS Application of the lent particle method to Poisson driven sde’s

Example: the finite dimensional case Let r ∈ N ∗ , ( X , X ) = ( R r , B ( R r )) and ν = kdx where k is non-negative and Borelian. We are given ξ = ( ξ ij ) 1 ≤ i , j ≤ r an R r × r -valued and symmetric Borel function. We assume that there exist an open set O ⊂ R r and a function ψ continuous on O and null on R r \ O such that 1. k > 0 on O ν -a.e. and is locally bounded on O 2. ξ is locally bounded and locally elliptic on O . 3. k ≥ ψ > 0 ν -a.e. on O . 4. for all i , j ∈ { 1 , · · · , r } , ξ i , j ψ belongs to H 1 loc ( O ). Laurent DENIS Application of the lent particle method to Poisson driven sde’s

We denote by H the subspace of functions f ∈ L 2 ( ν ) ∩ L 1 ( ν ) such that the restriction of f to O belongs to C ∞ c ( O ). Then, the bilinear form defined by r � � ∀ f , g ∈ H , e ( f , g ) = ξ i , j ( x ) ∂ i f ( x ) ∂ j g ( x ) ψ ( x ) dx O i , j =1 is closable in L 2 ( ν ). Its closure, ( d , e ), is a local Dirichlet form on L 2 ( ν ) which admits a carr´ e du champ γ . r ξ i , j ( x ) ∂ i f ( x ) ∂ j f ( x ) ψ ( x ) � ∀ f ∈ d , γ ( f )( x ) = k ( x ) . i , j =1 Moreover, it satisfies property (EID) i.e. for any d and for any R d -valued function U whose components are in the domain of the form U ∗ [(det γ [ U , U t ]) · ν ] ≪ λ d where det denotes the determinant and λ d the Lebesgue measure on ( R d , B ( R d )). Laurent DENIS Application of the lent particle method to Poisson driven sde’s

Choice for the gradient • ( R , R , ρ ): an auxiliary probability space s.t. L 2 ( R , R , ρ ) is infinite. • D : a version of the gradient on d with values in the space L 2 0 ( R , R , ρ ) = { g ∈ L 2 ( R , R , ρ ); � R g ( r ) ρ ( dr ) = 0 } . We denote it by ♭ . • N ⊙ ρ the extended marked Poisson measure : it is a random Poisson measure on [0 , + ∞ [ × X × R with compensator dt × ν × ρ defined on the product probability space: (Ω 1 , A 1 , P 1 ) × ( R N , R ⊗ N , P ⊗ N ). Laurent DENIS Application of the lent particle method to Poisson driven sde’s

Creation and annihilation operators ε + ( t , u ) ( w 1 ) = w 1 1 { ( t , u ) ∈ supp w 1 } + ( w 1 + ε ( t , u ) } ) 1 { ( t , u ) / ∈ supp w 1 } ε − ( t , u ) ( w 1 ) = w 1 1 { ( t , u ) / ∈ supp w 1 } + ( w 1 − ε ( t , u ) } ) 1 { ( t , u ) ∈ supp w 1 } . In a natural way, we extend these operators to the functionals by ε + H ( w 1 , t , u ) = H ( ε + ε − H ( w 1 , t , u ) = H ( ε − ( t , u ) w 1 , t , u ) ( t , u ) w 1 , t , u ) . we denote by P N the measure P N = P 1 ( dw ) N w ( dt , du ). Laurent DENIS Application of the lent particle method to Poisson driven sde’s

Upper Dirichlet structure From this, as explained in the previous talk, we are able to construct a Dirichlet form ( D , E ) on L 2 (Ω 1 ) which admits a gradient operator that we denote by ♯ and given by the following formula: � + ∞ � F ♯ = ε − (( ε + F ) ♭ ) dN ⊙ ρ ∈ L 2 ( P 1 × ˆ ∀ F ∈ D , P ) . (1) 0 X × R Moreover, we have for all F ∈ D � + ∞ � E ( F ♯ ) 2 = Γ[ F ] = ˆ ε − ( γ [ ε + F ]) dN , (2) 0 X and ( D , E ) satisfies (EID). Laurent DENIS Application of the lent particle method to Poisson driven sde’s

The SDE we consider We consider another probability space (Ω 2 , A 2 , P 2 ) on which an R n -valued semimartingale Z = ( Z 1 , · · · , Z n ) is defined, n ∈ N ∗ . Assumption on Z : There exists a positive constant C such that for any square integrable R n -valued predictable process h : � t � t h s dZ s ) 2 ] ≤ C 2 E [ | h s | 2 ds ] . ∀ t ≥ 0 , E [( (3) 0 0 We shall work on the product probability space: (Ω , A , P ) = (Ω 1 × Ω 2 , A 1 ⊗ A 2 , P 1 × P 2 ) . Let d ∈ N ∗ , we consider the following SDE : � t � t � c ( s , X s − , u )˜ X t = x + N ( ds , du ) + σ ( s , X s − ) dZ s (4) 0 0 X where x ∈ R d , c : R + × R d × X → R d and σ : R + × R d → R d × n satisfy the set of hypotheses below denoted (R). Laurent DENIS Application of the lent particle method to Poisson driven sde’s

Hypotheses (R) For simplicity, we fix a finite terminal time T > 0. 1. There exists η ∈ L 2 ( X , ν ) such that: a) for all t ∈ [0 , T ] and u ∈ X , c ( t , · , u ) is differentiable with continuous derivative and ∀ u ∈ X , t ∈ [0 , T ] , x ∈ R d | D x c ( t , x , u ) | ≤ η ( u ) , sup b) ∀ ( t , u ) ∈ [0 , T ] × U , | c ( t , 0 , u ) | ≤ η ( u ), c) for all t ∈ [0 , T ] and x ∈ R d , c ( t , x , · ) ∈ d and t ∈ [0 , T ] , x ∈ R d γ [ c ( t , x , · )]( u ) ≤ η ( u ) , sup d) for all t ∈ [0 , T ], all x ∈ R d and u ∈ X , the matrix I + D x c ( t , x , u ) is invertible and � � ( I + D x c ( t , x , u )) − 1 � sup � ≤ η ( u ) . � � t ∈ [0 , T ] , x ∈ R d Laurent DENIS Application of the lent particle method to Poisson driven sde’s

2. For all t ∈ [0 , T ] , σ ( t , · ) is differentiable with continuous derivative and t ∈ [0 , T ] , x ∈ R d | D x σ ( t , x ) | < + ∞ . sup 3. As a consequence of hypotheses 1. and 2. above, it is well known that equation (4) admits a unique solution X such that E [sup t ∈ [0 , T ] | X t | 2 ] < + ∞ . We suppose that for all t ∈ [0 , T ], the matrix ( I + � n j =1 D x σ · , j ( t , X t − )∆ Z j t ) is invertible and its inverse is bounded by a deterministic constant uniformly with respect to t ∈ [0 , T ]. Laurent DENIS Application of the lent particle method to Poisson driven sde’s

Derivation of the equation H D : the set of real valued processes ( H t ) t ∈ [0 , T ] , which belong to L 2 ([0 , T ]; D ) i.e. such that � T � T � H � 2 | H t | 2 dt ] + H D = E [ E ( H t ) dt < + ∞ . 0 0 Proposition The equation (4) admits a unique solution X in H d D . Moreover, the gradient of X satisfies: � t � X ♯ D x c ( s , X s − , u ) · X ♯ s − ˜ = N ( ds , du ) t 0 U � t � c ♭ ( s , X s − , u , r ) N ⊙ ρ ( ds , du , dr ) + 0 X × R � t D x σ ( s , X s − ) · X ♯ + s − dZ s 0 Laurent DENIS Application of the lent particle method to Poisson driven sde’s

Derivative of the flow generated by X Let us define the R d × d -valued processes U and K by n � D x σ ., j ( s , X s − ) dZ j dU s = s . j =1 � t � t � D x c ( s , X s − , u ) K s − ˜ K t = I + N ( ds , du ) + dU s K s − 0 X 0 Under our hypotheses, for all t ≥ 0, the matrix K t is invertible and K t = ( K t ) − 1 satisfies: it inverse ¯ � t � ¯ K s − ( I + D x c ( s , X s − , u )) − 1 D x c ( s , X s − , u )˜ ¯ = I − N ( ds , du ) K t 0 X � t ¯ � K s − (∆ U s ) 2 ( I + ∆ U s ) − 1 ¯ − K s − dU s + 0 s ≤ t � t K s d < U c , U c > s . ¯ + 0 Laurent DENIS Application of the lent particle method to Poisson driven sde’s

Obtaining the carr´ e du champ matrix Theorem For all t ∈ [0 , T ] , � t � K s γ [ c ( s , X s − , · )] ¯ ¯ K ∗ s N ( ds , du ) K ∗ Γ[ X t ] = t . K t 0 X Proof : Let ( α, u ) ∈ [0 , T ] × X . We put X ( α, u ) = ε + ( α, u ) X t . t � α � X ( α, u ) c ( s , X ( α, u ) , u ′ )˜ N ( ds , du ′ ) = x + t s − 0 X � α σ ( s , X ( α, u ) ) dZ s + c ( α, X ( α, u ) + , u ) s − α − 0 � � � c ( s , X ( α, u ) σ ( s , X ( α, u ) , u ′ )˜ N ( ds , du ′ ) + + ) dZ s . s − s − ] α, t ] X ] α, t ] Laurent DENIS Application of the lent particle method to Poisson driven sde’s

Let us remark that X ( α, u ) = X t if t < α so that, taking the t gradient with respect to the variable u , we obtain: ( X ( α, u ) ( c ( α, X ( α, u ) ) ♭ , u )) ♭ = t α − � � D x c ( s , X ( α, u ) , u ′ ) · ( X ( α, u ) ) ♭ ˜ N ( ds , du ′ ) + s − s − ] α, t ] X � D x σ ( s , X ( α, u ) ) · ( X ( α, u ) ) ♭ dZ s . + s − s − ] α, t ] Let us now introduce the process K ( α, u ) = ε + ( α, u ) ( K t ) which t satisfies the following SDE: � t � t � K ( α, u ) D x c ( s , X ( α, u ) , u ′ ) K ( α, u ) dU ( α, u ) K ( α, u ) ˜ N ( ds , du ′ )+ = I + s t s − s − s − 0 X 0 K ( α, u ) = ( K ( α, u ) and its inverse ¯ ) − 1 . Then, using the flow property, t t we have: ) ♭ = K ( α, u ) ∀ t ≥ 0 , ( X ( α, u ) K ( α, u ) ¯ ( c ( α, X α − , u )) ♭ . t t α Laurent DENIS Application of the lent particle method to Poisson driven sde’s