Large Deviations for Multi-valued Stochastic Differential Equations - PowerPoint PPT Presentation

(i) X 0 = x 0 and X ( t ) ∈ D ( A ) ∀ t ; a.s., (ii) K = { K ( t ) , F t ; t ∈ R + } is of finite variation and K (0) = 0 a.s. ; (iii) dX ( t ) = b ( X ( t )) dt + σ ( X ( t )) dw ( t ) − dK ( t ) , t ∈ R + , a.s.; (iv) for any continuous processes ( α, β ) satisfying ( α ( t ) , β ( t )) ∈ Gr ( A ) , ∀ t ∈ R + , the measure � X ( t ) − α ( t ) , dK ( t ) − β ( t ) dt � � 0 (Formally, this amounts to saying � X ( t ) − α ( t ) , K ′ ( t ) dt − β ( t ) dt � � 0 or � X ( t ) − α ( t ) , K ′ ( t ) − β ( t ) � � 0 )

Existence and Uniqueness

Existence and Uniqueness Theorem (C´ epa, 1995) If σ and b are Lipschitz, then � d X ( t ) ∈ b ( X ( t )) d t + σ ( X ( t )) d W ( t ) − A ( X ( t )) d t, X (0) = x ∈ D ( A ) , ε ∈ (0 , 1] .

Existence and Uniqueness Theorem (C´ epa, 1995) If σ and b are Lipschitz, then � d X ( t ) ∈ b ( X ( t )) d t + σ ( X ( t )) d W ( t ) − A ( X ( t )) d t, X (0) = x ∈ D ( A ) , ε ∈ (0 , 1] . admits a unique solution, ( X, K ) .

Existence and Uniqueness Theorem (C´ epa, 1995) If σ and b are Lipschitz, then � d X ( t ) ∈ b ( X ( t )) d t + σ ( X ( t )) d W ( t ) − A ( X ( t )) d t, X (0) = x ∈ D ( A ) , ε ∈ (0 , 1] . admits a unique solution, ( X, K ) . The unique solution of

Existence and Uniqueness Theorem (C´ epa, 1995) If σ and b are Lipschitz, then � d X ( t ) ∈ b ( X ( t )) d t + σ ( X ( t )) d W ( t ) − A ( X ( t )) d t, X (0) = x ∈ D ( A ) , ε ∈ (0 , 1] . admits a unique solution, ( X, K ) . The unique solution of � d X ε ( t ) ∈ b ( X ε ( t )) d t + √ εσ ( X ε ( t )) d W ( t ) − A ( X ε ( t )) d t, X ε (0) = x ∈ D ( A ) , ε ∈ (0 , 1] .

Existence and Uniqueness Theorem (C´ epa, 1995) If σ and b are Lipschitz, then � d X ( t ) ∈ b ( X ( t )) d t + σ ( X ( t )) d W ( t ) − A ( X ( t )) d t, X (0) = x ∈ D ( A ) , ε ∈ (0 , 1] . admits a unique solution, ( X, K ) . The unique solution of � d X ε ( t ) ∈ b ( X ε ( t )) d t + √ εσ ( X ε ( t )) d W ( t ) − A ( X ε ( t )) d t, X ε (0) = x ∈ D ( A ) , ε ∈ (0 , 1] . will be denoted by

Existence and Uniqueness Theorem (C´ epa, 1995) If σ and b are Lipschitz, then � d X ( t ) ∈ b ( X ( t )) d t + σ ( X ( t )) d W ( t ) − A ( X ( t )) d t, X (0) = x ∈ D ( A ) , ε ∈ (0 , 1] . admits a unique solution, ( X, K ) . The unique solution of � d X ε ( t ) ∈ b ( X ε ( t )) d t + √ εσ ( X ε ( t )) d W ( t ) − A ( X ε ( t )) d t, X ε (0) = x ∈ D ( A ) , ε ∈ (0 , 1] . will be denoted by ( X ε , K ε )

normally reflected SDE as a special case of MSDE

normally reflected SDE as a special case of MSDE Suppose

normally reflected SDE as a special case of MSDE Suppose O : a closed convex subset of R m ,

normally reflected SDE as a special case of MSDE Suppose O : a closed convex subset of R m , I O : the indicator function of O ,

normally reflected SDE as a special case of MSDE Suppose O : a closed convex subset of R m , I O : the indicator function of O , i.e, � I O ( x ) =

normally reflected SDE as a special case of MSDE Suppose O : a closed convex subset of R m , I O : the indicator function of O , i.e, � 0 , if x ∈ O , I O ( x ) =

normally reflected SDE as a special case of MSDE Suppose O : a closed convex subset of R m , I O : the indicator function of O , i.e, � 0 , if x ∈ O , I O ( x ) = + ∞ , if x / ∈ O .

normally reflected SDE as a special case of MSDE Suppose O : a closed convex subset of R m , I O : the indicator function of O , i.e, � 0 , if x ∈ O , I O ( x ) = + ∞ , if x / ∈ O . The subdifferential of I O is given by

normally reflected SDE as a special case of MSDE Suppose O : a closed convex subset of R m , I O : the indicator function of O , i.e, � 0 , if x ∈ O , I O ( x ) = + ∞ , if x / ∈ O . The subdifferential of I O is given by { y ∈ R m : � y, x − z � R m � 0 , ∀ z ∈ O } ∂I O ( x ) =

normally reflected SDE as a special case of MSDE Suppose O : a closed convex subset of R m , I O : the indicator function of O , i.e, � 0 , if x ∈ O , I O ( x ) = + ∞ , if x / ∈ O . The subdifferential of I O is given by { y ∈ R m : � y, x − z � R m � 0 , ∀ z ∈ O } ∂I O ( x ) =   = 

normally reflected SDE as a special case of MSDE Suppose O : a closed convex subset of R m , I O : the indicator function of O , i.e, � 0 , if x ∈ O , I O ( x ) = + ∞ , if x / ∈ O . The subdifferential of I O is given by { y ∈ R m : � y, x − z � R m � 0 , ∀ z ∈ O } ∂I O ( x ) =  ∅ , if x / ∈ O ,  = 

normally reflected SDE as a special case of MSDE Suppose O : a closed convex subset of R m , I O : the indicator function of O , i.e, � 0 , if x ∈ O , I O ( x ) = + ∞ , if x / ∈ O . The subdifferential of I O is given by { y ∈ R m : � y, x − z � R m � 0 , ∀ z ∈ O } ∂I O ( x ) =  ∅ , if x / ∈ O ,  { 0 } , x ∈ Int( O ) , = if 

normally reflected SDE as a special case of MSDE Suppose O : a closed convex subset of R m , I O : the indicator function of O , i.e, � 0 , if x ∈ O , I O ( x ) = + ∞ , if x / ∈ O . The subdifferential of I O is given by { y ∈ R m : � y, x − z � R m � 0 , ∀ z ∈ O } ∂I O ( x ) =  ∅ , if x / ∈ O ,  { 0 } , x ∈ Int( O ) , = if Λ x , if x ∈ ∂ O , 

normally reflected SDE as a special case of MSDE Suppose O : a closed convex subset of R m , I O : the indicator function of O , i.e, � 0 , if x ∈ O , I O ( x ) = + ∞ , if x / ∈ O . The subdifferential of I O is given by { y ∈ R m : � y, x − z � R m � 0 , ∀ z ∈ O } ∂I O ( x ) =  ∅ , if x / ∈ O ,  { 0 } , x ∈ Int( O ) , = if Λ x , if x ∈ ∂ O ,  where

normally reflected SDE as a special case of MSDE Suppose O : a closed convex subset of R m , I O : the indicator function of O , i.e, � 0 , if x ∈ O , I O ( x ) = + ∞ , if x / ∈ O . The subdifferential of I O is given by { y ∈ R m : � y, x − z � R m � 0 , ∀ z ∈ O } ∂I O ( x ) =  ∅ , if x / ∈ O ,  { 0 } , x ∈ Int( O ) , = if Λ x , if x ∈ ∂ O ,  where Int( O ) is the interior of O

normally reflected SDE as a special case of MSDE Suppose O : a closed convex subset of R m , I O : the indicator function of O , i.e, � 0 , if x ∈ O , I O ( x ) = + ∞ , if x / ∈ O . The subdifferential of I O is given by { y ∈ R m : � y, x − z � R m � 0 , ∀ z ∈ O } ∂I O ( x ) =  ∅ , if x / ∈ O ,  { 0 } , x ∈ Int( O ) , = if Λ x , if x ∈ ∂ O ,  where Int( O ) is the interior of O Λ x is the exterior normal cone at x .

Then ∂I O is a multivalued maximal monotone operator.

Then ∂I O is a multivalued maximal monotone operator. In this case, an MSDE is an SDE normally reflected at the boundary of O .

Then ∂I O is a multivalued maximal monotone operator. In this case, an MSDE is an SDE normally reflected at the boundary of O . More generally,

Then ∂I O is a multivalued maximal monotone operator. In this case, an MSDE is an SDE normally reflected at the boundary of O . More generally, the subdifferential of any convex and lower semi-continuous function is a maximal monotone operator.

Then ∂I O is a multivalued maximal monotone operator. In this case, an MSDE is an SDE normally reflected at the boundary of O . More generally, the subdifferential of any convex and lower semi-continuous function is a maximal monotone operator. ∂ϕ ( x ) := { y ∈ R m : � y, z − x � R m � ϕ ( z ) − ϕ ( x ) , ∀ z ∈ R m }

Then ∂I O is a multivalued maximal monotone operator. In this case, an MSDE is an SDE normally reflected at the boundary of O . More generally, the subdifferential of any convex and lower semi-continuous function is a maximal monotone operator. ∂ϕ ( x ) := { y ∈ R m : � y, z − x � R m � ϕ ( z ) − ϕ ( x ) , ∀ z ∈ R m }

Towards a Freidlin-Wentzell theory

Towards a Freidlin-Wentzell theory We now look at the problem of small perturbation:

Towards a Freidlin-Wentzell theory We now look at the problem of small perturbation: � d X ε ( t ) ∈ b ( X ε ( t )) d t + √ εσ ( X ε ( t )) d W ( t ) − A ( X ε ( t )) d t, X ε (0) = x ∈ D ( A ) , ε ∈ (0 , 1] .

Classical theory

Classical theory Recall the classical case:

Classical theory Recall the classical case: A ≡ 0

Classical theory Recall the classical case: A ≡ 0 � d X ε ( t ) = b ( X ε ( t )) d t + √ εσ ( X ε ( t )) d W ( t ) , X ε (0) = x, ε ∈ (0 , 1] .

Classical theory Recall the classical case: A ≡ 0 � d X ε ( t ) = b ( X ε ( t )) d t + √ εσ ( X ε ( t )) d W ( t ) , X ε (0) = x, ε ∈ (0 , 1] . The theory begins in the seminal paper

Classical theory Recall the classical case: A ≡ 0 � d X ε ( t ) = b ( X ε ( t )) d t + √ εσ ( X ε ( t )) d W ( t ) , X ε (0) = x, ε ∈ (0 , 1] . The theory begins in the seminal paper Wentzell-Freidlin: On small random perturbation of dynamical system, 1969

Classical theory Recall the classical case: A ≡ 0 � d X ε ( t ) = b ( X ε ( t )) d t + √ εσ ( X ε ( t )) d W ( t ) , X ε (0) = x, ε ∈ (0 , 1] . The theory begins in the seminal paper Wentzell-Freidlin: On small random perturbation of dynamical system, 1969 There are two standard and well known approaches to this classical problem.

1. Use of Euler Scheme

1. Use of Euler Scheme D.W.Stroock: An introduction to the theory of large deviation, 1984

1. Use of Euler Scheme D.W.Stroock: An introduction to the theory of large deviation, 1984 You look at the Euler Scheme

1. Use of Euler Scheme D.W.Stroock: An introduction to the theory of large deviation, 1984 You look at the Euler Scheme � t � t n ( n − 1 [ sn ])) d s + √ ε X ε b ( X ε σ ( X ε n ( n − 1 [ sn ])) d W ( s ) , n ( t ) = x + 0 0

1. Use of Euler Scheme D.W.Stroock: An introduction to the theory of large deviation, 1984 You look at the Euler Scheme � t � t n ( n − 1 [ sn ])) d s + √ ε X ε b ( X ε σ ( X ε n ( n − 1 [ sn ])) d W ( s ) , n ( t ) = x + 0 0 Then you have to estimate the quantity

1. Use of Euler Scheme D.W.Stroock: An introduction to the theory of large deviation, 1984 You look at the Euler Scheme � t � t n ( n − 1 [ sn ])) d s + √ ε X ε b ( X ε σ ( X ε n ( n − 1 [ sn ])) d W ( s ) , n ( t ) = x + 0 0 Then you have to estimate the quantity 0 � t � 1 | X ε ( t, x ) − X ε n | � δ ) P ( max

1. Use of Euler Scheme D.W.Stroock: An introduction to the theory of large deviation, 1984 You look at the Euler Scheme � t � t n ( n − 1 [ sn ])) d s + √ ε X ε b ( X ε σ ( X ε n ( n − 1 [ sn ])) d W ( s ) , n ( t ) = x + 0 0 Then you have to estimate the quantity 0 � t � 1 | X ε ( t, x ) − X ε n | � δ ) P ( max If you have for any δ > 0

1. Use of Euler Scheme D.W.Stroock: An introduction to the theory of large deviation, 1984 You look at the Euler Scheme � t � t n ( n − 1 [ sn ])) d s + √ ε X ε b ( X ε σ ( X ε n ( n − 1 [ sn ])) d W ( s ) , n ( t ) = x + 0 0 Then you have to estimate the quantity 0 � t � 1 | X ε ( t, x ) − X ε n | � δ ) P ( max If you have for any δ > 0 0 � t � 1 | X ε ( t, x ) − X ε n →∞ lim sup lim ε log P ( max n | � δ ) = −∞ , ε ↓ 0

1. Use of Euler Scheme D.W.Stroock: An introduction to the theory of large deviation, 1984 You look at the Euler Scheme � t � t n ( n − 1 [ sn ])) d s + √ ε X ε b ( X ε σ ( X ε n ( n − 1 [ sn ])) d W ( s ) , n ( t ) = x + 0 0 Then you have to estimate the quantity 0 � t � 1 | X ε ( t, x ) − X ε n | � δ ) P ( max If you have for any δ > 0 0 � t � 1 | X ε ( t, x ) − X ε n →∞ lim sup lim ε log P ( max n | � δ ) = −∞ , ε ↓ 0 then you are done.

Unfortunately, this approach does NOT apply to MSDE case

Unfortunately, this approach does NOT apply to MSDE case simply because the Euler scheme cannot be defined for MSDE

Unfortunately, this approach does NOT apply to MSDE case simply because the Euler scheme cannot be defined for MSDE since either A ( X n ( t )) is not defined or may be a set.

2. Use of Freidlin-Wentzell estimate

2. Use of Freidlin-Wentzell estimate (R. Azencott: Grandes deviations et applications, 1980

2. Use of Freidlin-Wentzell estimate (R. Azencott: Grandes deviations et applications, 1980 P. Priouret: Remarques sur les petite perturbations de syst` emes dynamiques, 1982)

2. Use of Freidlin-Wentzell estimate (R. Azencott: Grandes deviations et applications, 1980 P. Priouret: Remarques sur les petite perturbations de syst` emes dynamiques, 1982) Consider an ODE

2. Use of Freidlin-Wentzell estimate (R. Azencott: Grandes deviations et applications, 1980 P. Priouret: Remarques sur les petite perturbations de syst` emes dynamiques, 1982) Consider an ODE � d g ( t ) = b ( g ( t )) d t + σ ( g ( t )) f ′ ( t ) d t, g (0) = x.

2. Use of Freidlin-Wentzell estimate (R. Azencott: Grandes deviations et applications, 1980 P. Priouret: Remarques sur les petite perturbations de syst` emes dynamiques, 1982) Consider an ODE � d g ( t ) = b ( g ( t )) d t + σ ( g ( t )) f ′ ( t ) d t, g (0) = x. where f satisfies

2. Use of Freidlin-Wentzell estimate (R. Azencott: Grandes deviations et applications, 1980 P. Priouret: Remarques sur les petite perturbations de syst` emes dynamiques, 1982) Consider an ODE � d g ( t ) = b ( g ( t )) d t + σ ( g ( t )) f ′ ( t ) d t, g (0) = x. where f satisfies � 1 f ′ ( t ) 2 dt < ∞ . 0

If you can prove

If you can prove ∀ R > 0 ,

If you can prove ∀ R > 0 , ρ > 0 ,

If you can prove ∀ R > 0 , ρ > 0 , ∃ ε 0 > 0 ,