Discretization of Stochastic Differential Systems With Singular - PowerPoint PPT Presentation

Discretization of Stochastic Differential Systems With Singular Coefficients Part I Denis Talay INRIA Sophia Antipolis, France TOSCA Project-team ICERM - Brown – November 2012

Outline Introduction Monte Carlo Methods For Linear PDEs Discretization of Stochastic Hamiltonian Dissipative Systems Stochastic Lagrangian Models for Turbulent Flows Conclusion

Outline Introduction Monte Carlo Methods For Linear PDEs Discretization of Stochastic Hamiltonian Dissipative Systems Stochastic Lagrangian Models for Turbulent Flows Conclusion

Why is Probability useful? THE WORLD IS COMPLEX ◮ The physical model is badly calibrated (e.g., MEG or electrical neuronal activity: few sensors), ◮ The physical law is not completely known (e.g., turbulence, meteorology,. . . ), ◮ There is no physical law (e.g., finance). THE PARTIAL DIFFERENTIAL EQUATIONS ARE COMPLEX ◮ Mathematical analysis (existence, uniqueness, smoothness), ◮ Probabilistic analysis of deterministic numerical methods (cf. Kushner, or domain decompositions, or artificial boundary conditions), ◮ Probabilistic numerical methods for high dimensional problems and/or equations in domains with possibly complex geometries and/or small viscosities (high Reynolds numbers),. . . ).

SUMMARY: ◮ Probability theory (in particuler, stochastic integration theory) is used to solve problems which, by nature, are deterministic or ’stochastic’, ◮ Probabilistic models and numerical methods are used when deterministic ones are unefficient. ◮ In all cases, one seeks a statistical information on the model: classical numerical analysis needs to be deeply adapted. Remarks: ◮ For physicists, Stochastic PDEs often are PDEs with random coefficients, ◮ Stochastic collocation methods are not stochastic.

Outline Introduction Monte Carlo Methods For Linear PDEs Discretization of Stochastic Hamiltonian Dissipative Systems Stochastic Lagrangian Models for Turbulent Flows Conclusion

General parabolic PDEs Let b : R d → R d and σ j : R d → R d , ( 1 ≤ j ≤ r ) . Consider the elliptic operator d d � � b i ( x ) ∂ i ψ ( x ) + 1 a i L ψ ( x ) := j ( x ) ∂ ij ψ ( x ) , 2 i = 1 i , j = 1 where a ( x ) := σ ( x ) σ ( x ) t , and the evolution problem  ∂ u  = Lu ( t , x ) , t > 0 , x ∈ R d , ∂ t ( t , x )  = f ( x ) , x ∈ R d . u ( 0 , x )

The Euler scheme for SDEs Let ( G j p ) be i.i.d. N ( 0 , 1 ) and h > 0 be the discretization step.  ¯ X h 0 ( x ) = x ,   � �  ¯ = ¯ ¯ X h X h X h ( p + 1 ) h ( x ) ph ( x ) + b ph ( x ) h � � √  + � r   ¯ h G j X h j = 1 σ j ph ( x ) p + 1 . ◮ Easy to simulate (even for Lévy driven SDEs). ◮ Discretizes the stochastic differential equation � t � t X t ( x ) = x + b ( X s ( x )) ds + σ ( X s ( x )) dW s . 0 0

Moments of the Euler scheme � ¯ � E { ¯ ( p + 1 ) h ( x ) − ¯ X h X h X h ph } = E b ph ( x ) h , � h 2 � E { (¯ ( p + 1 ) h ( x ) − ¯ ph ) · (¯ ( p + 1 ) h ( x ) − ¯ ph ) t } = E a (¯ X h X h X h X h X h ph ) h + O .

Moments of the Euler scheme � ¯ � E { ¯ ( p + 1 ) h ( x ) − ¯ X h X h X h ph } = E b ph ( x ) h , � h 2 � E { (¯ ( p + 1 ) h ( x ) − ¯ ph ) · (¯ ( p + 1 ) h ( x ) − ¯ ph ) t } = E a (¯ X h X h X h X h X h ph ) h + O .

Probabilistic interpretation of parabolic PDEs E f (¯ X h T ( x )) − u ( T , x ) T / h − 1 � � � �� � � T − ( p + 1 ) h , ¯ T − ph , ¯ X h X h − u = E u ( p + 1 ) h ( x ) ph ( x ) p = 0 T / h − 1 � � � � � �� T − ( p + 1 ) h , ¯ T − ph , ¯ X h X h − u = E u ph ( x ) ph ( x ) p = 0 T / h − 1 T / h − 1 � � � � �� � h 2 � T − ( p + 1 ) h , ¯ X h + h E Lu ph ( x ) + O p = 0 p = 0 � �� T / h − 1 � � � � − ∂ u T − ph , ¯ T − ph , ¯ X h X h = h E Lu ph ( x ) ph ( x ) + O ( h ) ∂ t p = 0 = O ( h ) , since ∂ u ∂ t ( t , x ) = Lu ( t , x ) .

Probabilistic interpretation of parabolic PDEs E f (¯ X h T ( x )) − u ( T , x ) T / h − 1 � � � �� � � T − ( p + 1 ) h , ¯ T − ph , ¯ X h X h − u = E u ( p + 1 ) h ( x ) ph ( x ) p = 0 T / h − 1 � � � � � �� T − ( p + 1 ) h , ¯ T − ph , ¯ X h X h − u = E u ph ( x ) ph ( x ) p = 0 T / h − 1 T / h − 1 � � � � �� � h 2 � T − ( p + 1 ) h , ¯ X h + h E Lu ph ( x ) + O p = 0 p = 0 � �� T / h − 1 � � � � − ∂ u T − ph , ¯ T − ph , ¯ X h X h = h E Lu ph ( x ) ph ( x ) + O ( h ) ∂ t p = 0 = O ( h ) , since ∂ u ∂ t ( t , x ) = Lu ( t , x ) .

Probabilistic interpretation of parabolic PDEs E f (¯ X h T ( x )) − u ( T , x ) T / h − 1 � � � �� � � T − ( p + 1 ) h , ¯ T − ph , ¯ X h X h − u = E u ( p + 1 ) h ( x ) ph ( x ) p = 0 T / h − 1 � � � � � �� T − ( p + 1 ) h , ¯ T − ph , ¯ X h X h − u = E u ph ( x ) ph ( x ) p = 0 T / h − 1 T / h − 1 � � � � �� � h 2 � T − ( p + 1 ) h , ¯ X h + h E Lu ph ( x ) + O p = 0 p = 0 � �� T / h − 1 � � � � − ∂ u T − ph , ¯ T − ph , ¯ X h X h = h E Lu ph ( x ) ph ( x ) + O ( h ) ∂ t p = 0 = O ( h ) , since ∂ u ∂ t ( t , x ) = Lu ( t , x ) .

Probabilistic interpretation of parabolic PDEs E f (¯ X h T ( x )) − u ( T , x ) T / h − 1 � � � �� � � T − ( p + 1 ) h , ¯ T − ph , ¯ X h X h − u = E u ( p + 1 ) h ( x ) ph ( x ) p = 0 T / h − 1 � � � � � �� T − ( p + 1 ) h , ¯ T − ph , ¯ X h X h − u = E u ph ( x ) ph ( x ) p = 0 T / h − 1 T / h − 1 � � � � �� � h 2 � T − ( p + 1 ) h , ¯ X h + h E Lu ph ( x ) + O p = 0 p = 0 � �� T / h − 1 � � � � − ∂ u T − ph , ¯ T − ph , ¯ X h X h = h E Lu ph ( x ) ph ( x ) + O ( h ) ∂ t p = 0 = O ( h ) , since ∂ u ∂ t ( t , x ) = Lu ( t , x ) .

Probabilistic interpretation of parabolic PDEs E f (¯ X h T ( x )) − u ( T , x ) T / h − 1 � � � �� � � T − ( p + 1 ) h , ¯ T − ph , ¯ X h X h − u = E u ( p + 1 ) h ( x ) ph ( x ) p = 0 T / h − 1 � � � � � �� T − ( p + 1 ) h , ¯ T − ph , ¯ X h X h − u = E u ph ( x ) ph ( x ) p = 0 T / h − 1 T / h − 1 � � � � �� � h 2 � T − ( p + 1 ) h , ¯ X h + h E Lu ph ( x ) + O p = 0 p = 0 � �� T / h − 1 � � � � − ∂ u T − ph , ¯ T − ph , ¯ X h X h = h E Lu ph ( x ) ph ( x ) + O ( h ) ∂ t p = 0 = O ( h ) , since ∂ u ∂ t ( t , x ) = Lu ( t , x ) .

Convergence rate Let F ( · ) be a functional on the path space. The global error of a Monte Carlo method is N � � ¯ � E F ( X · ) − 1 = E F ( X · ) − E F (¯ X h , k X h E F · ) · N � �� � k = 1 =: ǫ d ( h ) N � � ¯ � · ) − 1 + E F (¯ X h X h , k F . · N k = 1 � �� � =: ǫ s ( h , N ) The statistical error satisfies C ∃ C > 0 , E | ǫ s ( h ) | ≤ √ for all h . N

Concerning the discretization error : Suppose that f has a polynomial growth at infinity. Under hypoellipticity conditions, or when all the functions of the problem are smooth, one has (T.-Tubaro, Bally-T. etc.) e d ( h ) = C f ( T , x ) h + Q h ( f , T , x ) h 2 , where | C f ( T , x ) | + sup h | Q h ( f , T , x ) | ≤ C ( 1 + � x � Q ) 1 + K ( T ) T q Thus, Romberg extrapolation techniques can be used: � �� � � � N N � � � h 2 � 2 − 1 X h / 2 , k ¯ X h , k ¯ = O E f f . T T N N k = 1 k = 1 Remark: The technique used in the proofs is purely probabilistic (stochastic flows of diffeomorphisms, Malliavin variations calculus).

Dirichlet boundary conditions For  ∂ u   ∂ t ( t , x ) = Lu ( t , x ) , t > 0 , x ∈ D ,  u ( 0 , x ) = f ( x ) , x ∈ D ,    = g ( x ) , x ∈ ∂ D , u ( t , x ) one has u ( t , x ) = E f ( X t ( x )) I t<τ + E g ( X τ ( x )) I t ≥ τ , where τ := ‘first boundary hitting time of ( X t ) ’. The stopped Euler scheme is defined as ¯ X h ph ∧ τ h ( x ) , where τ h := ‘first boundary hitting time of the Euler scheme’. For a convergence rate analysis, see Gobet, Menozzi, etc.

Neumann boundary conditions For  ∂ u   ∂ t ( t , x ) = Lu ( t , x ) , t > 0 , x ∈ D ,  u ( 0 , x ) = f ( x ) , x ∈ D ,    ∇ u ( t , x ) · n ( x ) = 0 , x ∈ ∂ D , one has u ( t , x ) = E f ( X t ( x )) where X := ‘reflected diffusion process’: � t � t � t X t ( x ) = x + b ( X s ( x )) ds + σ ( X s ( x )) dW s + n ( X s ) dL s ( X ) . 0 0 0 Here, ( L t ( X )) is an increasing process, namely the local time of X at the boundary . The reflected Euler scheme is defined in such a way that the simulation of the local time is avoided.