Homogenization of the L evy operators with asymmetric L evy - PDF document

Homogenization of the L´ evy operators with asymmetric L´ evy measures Mariko ARISAWA Wolfgang Pauli Institute Faculty of Maths., Univ. Vienna E-mail: mariko.arisawa@univie.ac.at

Periodic homogenisation problem: (P) ∂u ε ∂t + F ( x, u ε , ∇ u ε , ∇ 2 u ε ) − c ( x � ε ) R N [ u ε ( x + z ) − u ε ( x ) − 1 | z | < 1 �∇ u ε ( x ) , z � ] q ( z ) dz = 0 x ∈ Ω , x ∈ Ω c , u ε ( x, t ) = φ ( x ) t > 0 , u ε ( x, 0) = u 0 x ∈ Ω . • Is there a unique limit : ∃ lim ε ↓ 0 u ε ( x ) = u ( x ) ? • Characterize u by an effective PIDE: ∂u ∂t + I ( x, u, ∇ u, ∇ 2 u, I [ u ]) = 0 t > 0 .

Type I (Pure Jump Process). • Linear problem: Ex. F = 0 . • First-order nonlinear problem: Ex. F = a ( x ) | p | . Type II (Jump-Diffusion process). F : uniformly elliptic, i.e. ∃ θ > 0 ∀ Q ′ ≥ O, F ( x, u, p, Q + Q ′ ) <F ( x, u, p, Q ) − θTrQ ′ Ex. F ( x, u, ∇ u, ∇ 2 u ) = − Tr ( ∇ 2 u ) = − ∆ u .

Method . In the case of PDE (elliptic, parabolic), the effective PDE is obtained by • Formal asymptotic expansion • Cell problem (ergodic problem of PDE) • Averaging principle in the underlying stochas- tic process • Rigorous justification : for nonlinear PDEs Perturbed test function method by using viscosity solutions References. A. Bensoussain, J.-L. Lions, and G. Papanicolaou, P.-L. Lions, G. Papanico- laou, and S.R.S. Varadhan, L.C. Evans, etc.

M. A. Homogenizations of PIDE with L´ evy op- erators, submitted. M.A. Some remarks on the homogenizations of L´ evy operators with asymmetric densities, in preparation.

Pure jump case . � − R N [ u ( x + z ) − u ( x ) − 1 | z | < 1 �∇ u ( x ) , z � ] q ( z ) dz � � | z | < 1 | z | 2 q ( z ) dz + s.t. | z | > 1 1 q ( z ) dz < ∞ . Homogenisations � Regularization effect (averaging principle) � Singular L´ evy density Examples. α • Symmetric α -stable process ( − ∆ 2 , 0 < α < 2) 1 q ( z ) dz = | z | N + α ( dz ) . Remark. As α → 2, the operator tends to − ∆.

• α -Stable process ( N = 1, 0 < α < 2) 1 q ( z ) dz = c 1 | z | 1+ α ( dz ) z < 0 , 1 = c 2 | z | 1+ α ( dz ) z > 0 where c 1 , c 2 ≥ 0, and at least one c i � = 0. • CGMY model ( N = 1, C > 0, G ≥ 0, M ≥ 0, 0 < Y < 2) 1 q ( z ) dz = C ( I z< 0 e − G | z | + I z> 0 e − M | z | ) | z | 1+ Y ( dz ) . • Asymmetric singularity ( N = 1, 0 < α 1 < α 2 < 2) 1 q ( z ) dz = c 1 | z | 1+ α 1 ( dz ) z < 0 , 1 = c 2 | z | 1+ α 2 ( dz ) z > 0

• Nonlinear operator ( N = 2, 0 < α 1 , α 2 < 2) � max {− R [ u ( x 1 + z 1 , x 2 ) − u ( x 1 , x 2 ) 1 − 1 | z 1 | < 1 �∇ x 1 u ( x ) , z 1 � ] | z 1 | 1+ α 1 dz 1 , � − R [ u ( x 1 , x 2 + z 2 ) − u ( x 1 , x 2 ) 1 − 1 | z 2 | < 1 �∇ x 2 u ( x ) , z 2 � ] | z 2 | 1+ α 2 dz 2 }

Jump diffusion case . Homogenisations � Regularization effect (averaging principle) � Effect of the diffusion of ” − ∆ ” Examples . (Bdd L´ evy measures can be added.) • Discrete L´ evy measure q ( dz ) = c Σ d j =1 p j δ a j ( dz ) , p j ≥ 0, Σ d j =1 p j = 1, c > 0: frequency of the jump, a i : jump lengths. • Gaussian distribution 2 πv exp( −| z − m | 2 1 √ q ( z ) dz = c ) dz 2 v

c > 0: frequency of the jump; jump distri- bution: the normal distribution. • Variance gamma process ( c , c 1 , c 2 > 0) q ( z ) dz = c ( I z< 0 e − c 1 | z | + I z> 0 e − c 2 | z | ) 1 | z | dz.

Formal asymptotic expansion . Type I (Pure Jump Process). 1. Symmetric α -stable process (1 < α < 2) ∂u ε ∂t + |∇ u ε | − c ( x � ε ) R N [ u ε ( x + z ) − u ε ( x ) 1 | z | N + α dz − g ( x − 1 | z | < 1 �∇ u ε ( x ) , z � ] ε ) = 0 ⇓ u ε ( x, t ) = u ( x, t ) + ε α v ( x ε, t ) + o ( ε α ) ∇ u ε ( x, t ) = ∇ x u ( x, t ) + ε α − 1 ∇ y v ( x ε, t ) . ⇓ ∂u ∂t + |∇ u | − c ( x � ε ) R N [ u ( x + z ) − u ( x ) 1 − 1 | z | < 1 �∇ x u ( x ) , z � ] | z | N + α dz

R N [ v ( x + z − ε α c ( x ) − v ( x � ε ) ε ) ε 1 ∇ y v ( x ε ) , z | z | N + α dz − g ( x � � − 1 | z | < 1 ] ε ) = 0 ε ⇓ ∂u � ∂t + |∇ u | − c ( y ) R N [ u ( x + z ) − u ( x ) 1 � R N [ v ( y + z ′ ) − 1 | z | < 1 �∇ x u ( x ) , z � ] | z | N + α dz − c ( y ) 1 ∇ y v ( y ) , z ′ � � − v ( y ) − 1 | z ′ | < 1 ] | z ′ | N + α dz − g ( y ) = 0 ε ⇓ Ergodic problem (Averaging principle) � R N [ v ( y + z ′ ) −∃ I ( x, u, ∇ u, I ) − c ( y ) I − c ( y ) 1 ∇ y v ( y ) , z ′ � � − v ( y ) − 1 | z ′ | < 1 ] | z ′ | N + α dz − g ( y ) = 0 , ε M.A. Proc.”Stoc. Processes and Applic. to Math. Finance”, World Scientifics, (2007)

Effective integro-differential operators • Uniform sub-ellipticity: ∀ I ′ > 0 I ( x, r, p, I + I ′ ) <I ( x, r, p, I ) − ∃ θI ′ • I ( x, r, p, I + I ′ ) ∈ C (Ω × R × R N × R ). ⇓ Effective integro-differential equations u is the unique solution of ∂u ∂t + I ( x, u, ∇ u, I ) = 0 t > 0 . Remark. The formal argument is justified by the purturbed test fc. method. Remark. Asymmetric α -Stable process c 1 c 2 q ( z ) dz = 1 z< 0 | z | 1+ α dz + 1 z> 0 | z | 1+ α dz

can be treated similarly. 2. Asymmetric singularity ( N = 1, 0 < α 1 < α 2 < 2) c 1 c 2 q ( z ) dz = 1 z < 0 | z | 1+ α 1 dz + 1 z < 0 | z | 1+ α 2 dz, i.e. � 0 ∂u ε ∂t + |∇ u ε | − c ( x ε ) −∞ [ u ε ( x + z ) − u ε ( x ) c 1 − 1 | z | < 1 �∇ u ε ( x ) , z � ] | z | 1+ α 1 dz � ∞ − c ( x 0 [ u ε ( x + z ) − u ε ( x ) ε ) | z | 1+ α 2 dz − g ( x c 2 − 1 | z | < 1 �∇ u ε ( x ) , z � ] ε ) = 0 ⇓ Stronger singularity dominates: u ε ( x, t ) = u ( x, t ) + ε α 2 v ( x ε, t ) + o ( ε α 2 )

3. Nonlinear operator ( N = 2, 0 < α 1 , α 2 < 2) ∂u ε ∂t + c ( x � ε ) max {− R [ u ε ( x 1 + z 1 , x 2 ) − u ε ( x 1 , x 2 ) 1 − 1 | z 1 | < 1 �∇ x 1 u ε ( x ) , z 1 � ] | z 1 | 1+ α 1 dz 1 , � − R [ u ε ( x 1 , x 2 + z 2 ) − u ε ( x 1 , x 2 ) 1 | z 2 | 1+ α 2 dz 2 } − g ( x − 1 | z 2 | < 1 �∇ x 2 u ε ( x ) , z 2 � ] ε ) = 0 . ⇓ Developpements in each directions: u ε ( x 1 , x 2 , t ) = u ( x, t ) + ε α 1 v ( x 1 ε , x 2 , t )+ ε α 2 w ( x 1 , x 2 ε , t )+ o ( ε α )

Theorem. Let us consider the problem (P), which is either Type I or Type II. Let u ε be the solution of (P). Then, there is a unique fonc- tion lim ε ↓ 0 u ε = u exists, which is the unique solution of ∂u ∂t + I ( x, u, ∇ u, ∇ 2 u, I [ u ]) = 0 t > 0 , with the same initial and boundary conditions. Remark. The result is applied to a stochastic volatility model with jumps, in maths finances. (cf. Fouque, Papanicolaou, Sircar.)