THE LENT PARTICLE FORMULA Nicolas BOULEAU, Laurent DENIS, Paris. - PDF document

THE LENT PARTICLE FORMULA Nicolas BOULEAU, Laurent DENIS, Paris. Workshop on Stochastic Analysis and Finance, Hong-Kong, June-July 2009 This is part of a joint work with Laurent Denis, concerning the approach of the regularity of Poisson functionals by Dirichlet forms methods cf. [6] et [7]. 1

Analysis in the spirit of Malliavin calculus vs Dirichlet forms approach a) The arguments hold under only Lipschitz hypotheses, b) A general criterion exists, (EID) the Energy Image Den- sity property, (proved on the Wiener space for the Ornstein- Uhlenbeck form but still a conjecture in general since 1986 cf Bouleau-Hirsch [5]) c) Dirichlet forms are easy to construct in the infinite dimensio- nal frameworks encountered in probability theory and this yields a theory of errors propagation through the stochastic calculus, especially for finance and physics cf Bouleau [2], but also for numerical analysis of pde and spde cf Scotti [18]. Extensions of Malliavin calculus to the case of Poisson measures and SDE’s with jumps • either dealing with local operators acting on the size of the jumps (Bichteler-Gravereaux-Jacod [1] Coquio[9] Ma-Röckner[13] etc.) • or based on the Fock space representation of the Poisson space and finite difference operators (Nualart-Vives[15] Picard[16] Ishikawa- Kunita[12] etc.). 2

The lent particle method To calculate the Malliavin matrix, add a particle to the system, compute the gradient of the functional on this particle, take back the particle before integrating by the Poisson measure. Let ( X, X , ν, d , γ ) be a local symmetric Dirichlet structure which admits a carré du champ operator i.e. ( X, X , ν ) is a measured space called the bottom space , ν is σ -finite and the bilinear form e [ f, g ] = 1 � γ [ f, g ] dν, is a local Dirichlet form 2 with domain d ⊂ L 2 ( ν ) and carré du champ γ (cf Bouleau- Hirsch [5]). Consider a Poisson random measure on this state space. A Dirichlet structure may be constructed on the Poisson space, called the upper space , that we denote (Ω , A , P , D , Γ) . The main result is the formula : For all F ∈ D � ε − ( γ [ ε + F ]) dN. Γ[ F ] = X in which ε + and ε − are the creation and annihilation operators. 3

Example. Let Y t be a centered Lévy process with Lévy measure σ in- tegrating x 2 . We assume that σ is such that a local Dirichlet structure may be constructed on R \{ 0 } with carré du champ γ [ f ] = x 2 f ′ 2 ( x ) . We define a gradient ♭ associated with γ by choosing ξ such � 1 � 1 0 ξ 2 ( r ) dr = 1 and putting that 0 ξ ( r ) dr = 0 and f ♭ = xf ′ ( x ) ξ ( r ) . N is the Poisson random measure associated with Y with in- � t 1 [0 ,t ] ( s ) h ( s ) x ˜ � tensity dt × σ such that 0 h ( s ) dY s = N ( dsdx ) We study the regularity of � t V = ϕ ( Y s − ) dY s 0 where ϕ is Lipschitz and C 1 . 4

1 o . First step. We add a particle ( α, x ) i.e. a jump to Y at time α with size x what gives � t ε + V − V = ϕ ( Y α − ) x + ] α ( ϕ ( Y s − + x ) − ϕ ( Y s − )) dY s 2 o . V ♭ = 0 since V does not depend on x , and � t � � ( ε + V ) ♭ = ] α ϕ ′ ( Y s − + x ) xdY s ϕ ( Y α − ) x + ξ ( r ) because x ♭ = xξ ( r ) . 3 o . We compute � t γ [ ε + V ] = ( ε + V ) ♭ 2 dr = ( ϕ ( Y α − ) x + ] α ϕ ′ ( Y s − + x ) xdY s ) 2 � 4 o . We take back the particle and compute Γ[ V ] = ε − γ [ ε + V ] dN . � � t � 2 � � x 2 N ( dαdx ) ϕ ′ ( Y s − ) dY s Γ[ V ] = ϕ ( Y α − ) + ] α � t � ∆ Y 2 ϕ ′ ( Y s − ) dY s + ϕ ( Y α − )) 2 . = α ( ] α α � t . 5

Second example same hypotheses on Y (which imply 1 + ∆ Y s � = 0 a.s.) We want to study the existence of density for the pair ( Y t , E ( Y ) t ) where E ( Y ) is the Doléans exponential of Y . E ( Y ) t = e Y t � (1 + ∆ Y s ) e − ∆ Y s . s � t 1 0 / we add aparticle ( α, y ) i.e. a jump to Y at time α � t with size y : (1+∆ Y s ) e − ∆ Y s (1+ y ) e − y = E ( Y ) t (1+ y ) . ( α,y ) ( E ( Y ) t ) = e Y t + y � ε + s � t 2 0 / we compute γ [ ε + E ( Y ) t ]( y ) = ( E ( Y ) t ) 2 y 2 . 3 0 / we take back the particle : E ( Y ) t (1 + y ) − 1 � 2 y 2 ε − γ [ ε + E ( Y ) t ] = � we integrate in N and that gives the upper squared field opera- tor : E ( Y ) t (1 + y ) − 1 � 2 y 2 N ( dαdy ) � � Γ[ E ( Y ) t ] = [0 ,t ] × R E ( Y ) t (1 + ∆ Y α ) − 1 � 2 ∆ Y 2 � = � α . α � t 6

By a similar computation the matrix Γ of the pair ( Y t , E ( Y t )) is given by � � E ( Y ) t (1 + ∆ Y α ) − 1 1 � ∆ Y 2 Γ = α . E ( Y ) t (1 + ∆ Y α ) − 1 � 2 E ( Y ) t (1 + ∆ Y α ) − 1 � α � t Hence under hypotheses implying (EID) the density of the pair ( Y t , E ( Y t )) is yielded by the condition �� � � 1 dim L α ∈ JT = 2 E ( Y ) t (1 + ∆ Y α ) − 1 where JT denotes the jump times of Y between 0 and t . Making this in details we obtain Let Y be a Lévy process with infinite Lévy measure with density dominating a positive continuous function � = 0 near 0 , then the pair ( Y t , E ( Y ) t ) possesses a density on R 2 . 7

• The Energy Image Density property (EID). A Dirichlet form on L 2 (Λ) ( Λ σ -finie) with carré du champ γ satisfies (EID) if for any d and all U with values in R d whose components are in the domain of the form U ∗ [(det γ [ U, U t ]) · Λ] ≪ λ d This property is true for th O-U form on the Wiener space, and in several other cases cf. Bouleau-Hirsch. It was conjectured in 1986 that it were always true. It is still a conjecture. Grosso modo here for Poisson measures : as soon as EID is true for the bottom space, EID is true for the upper space. (we use a result of Shiqi Song [19]) 8

Demonstration of the lent particle formula. • The construction ( E, X , m, d , γ ) is a local Dirichlet structure with carré du champ : it is the bottom space , m is σ -finite and the bilinear form e [ f, g ] = 1 � γ [ f, g ] dm, 2 is a local Dirichlet form with domain d ⊂ L 2 ( m ) and with carré du champ γ . For all x ∈ X , { x } is supposed to belong to X , m is diffuse. The associated generator is denoted a , its domain is D ( a ) ⊂ d . We consider a random Poisson measure N , on ( E, X , m ) with intensity m . It is defined on (Ω , A , P ) where Ω is the confi- guration space of countable sums of Dirac masses on E , A is the σ -field generated by N and P is the law of N . (Ω , A , P ) is called the upper space . 9

• Basic formulas and pregenerator. Because of some formulas on functions of the form e iN ( f ) related to the Laplace functional, we consider the space of test functions N ( f ) with f ∈ D ( a ) ∩ L 1 ( m ) et γ [ f ] ∈ L 2 } . D 0 = L{ e i ˜ N ( f p ) in D 0 , we put p λ p e i ˜ and for U = � N ( a [ f p ]) − 1 λ p e i ˜ � N ( f p ) ( i ˜ A 0 [ U ] = 2 N ( γ [ f p ])) . p In order to show that A 0 is uniquely defined and is the generator of a Dirichlet form satisfying the needed properties, - we construct an explicit gradient - we use the Friedrichs’ property 10

• Bottom gradient We suppose the space d separable, then there exists a gra- dient for the bottom space : There is a separable Hilbert space and a linear map D from d into L 2 ( X, m ; H ) such that ∀ u ∈ d , � D [ u ] � 2 H = γ [ u ] , then necessarily - If F : R → R is Lipschitz then ∀ u ∈ d , D [ F ◦ u ] = ( F ′ ◦ u ) Du, - If F is C 1 and Lipschitz from R d into R then D [ F ◦ u ] = � d i =1 ( F ′ i ◦ u ) D [ u i ] ∀ u = ( u 1 , · · · , u d ) ∈ d d . We take for H a space L 2 ( R, R , ρ ) where ( R, R , ρ ) is a probability space s.t. L 2 ( R, R , ρ ) be infinite dimensional. The gradient D is denoted ♭ : ∀ u ∈ d , Du = u ♭ ∈ L 2 ( X × R, X ⊗ R , m ⊗ ρ ) . Without loss of generality, we assume moreover that operator ♭ takes its values in the orthogonal space of 1 in L 2 ( R, R , ρ ) . So that we have � u ♭ dρ = 0 ν - a.e. ∀ u ∈ d , 11

• Gradient for the upper space We introduce the operators ε + and ε − : ∀ x, w ∈ Ω , ε + x ( w ) = w 1 { x ∈ supp w } + ( w + ε x ) 1 { x/ ∈ supp w } . ε − x ( w ) = ( w − ε x ) 1 { x ∈ supp w } + w 1 { x/ ∈ supp w } . So that for all w ∈ Ω , ε + x ( w ) = w et ε − x ( w ) = w − ε x for N w -almost every x ε + x ( w ) = w + ε x et ε − ( w ) = w for m -almost every x Definition. For F ∈ D 0 , we define the pre-gradient � F ♯ = ε − (( ε + F ) ♭ ) dN ⊙ ρ. where N ⊙ ρ is the point process N “marked” by ρ . 12

• Main result Theorem. The formula � ∀ F ∈ D , F ♯ = ε − (( ε + F ) ♭ ) dN ⊙ ρ, E × R extends from D 0 to D , it is justified by the following decomposition : ε − (( . ) ♭ ) ε + − I d ( N ⊙ ρ ) F ♯ ∈ L 2 ( P × ˆ �→ ε + F − F ∈ D ε − (( ε + F ) ♭ ) ∈ L 2 F ∈ D �→ 0 ( P N × ρ ) �→ P ) where each operator is continuous on the range of the preceding one and where L 2 0 ( P N × ρ ) is the closed set of elements G in L 2 ( P N × ρ ) such that � R Gdρ = 0 P N -a.s. Furthermore for all F ∈ D � E ( F ♯ ) 2 = Γ[ F ] = ˆ ε − γ [ ε + F ] dN. E 13