Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients Large deviation principle for SDE’s with discontinuous coefficients D. Sobolieva 1 1 Department of Probability Theory Kyiv National Taras Shevchenko University 24.10.2014 Berlin-Padova Meeting for Young Researchers Stochastic Analysis and Applications in Biology, Finance and Physics D.Sobolieva
Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients Outline Introduction 1 LDP for one-dimensional SDE’s without drift 2 LDP for one-dimensional SDE’s with discontinuous coefficients 3 D.Sobolieva
Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients Large deviation principle Let ( X , ρ ) be a complete separable metric space with X -valued random variables X n , n ≥ 1 . We say that family { X n } satisfies large deviation principle (LDP) with rate function I : X → [0 , ∞ ] if, for each opened set A , 1 n log P { X n ∈ A } ≥ − inf lim inf x ∈ A I ( x ) , (1) n →∞ and, for each closed set B , 1 n log P { X n ∈ B } ≤ − inf lim sup x ∈ B I ( x ) . (2) n →∞ If (1) holds and (2) holds for each compact set B only then family { X n } satisfies weak LDP. If each level set { x : I ( x ) ≤ a } , a ≥ 0 , is compact then we call rate function I “good”. D.Sobolieva
Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients Large deviation principle Let ( X , ρ ) be a complete separable metric space with X -valued random variables X n , n ≥ 1 . We say that family { X n } satisfies large deviation principle (LDP) with rate function I : X → [0 , ∞ ] if, for each opened set A , 1 n log P { X n ∈ A } ≥ − inf lim inf x ∈ A I ( x ) , (1) n →∞ and, for each closed set B , 1 n log P { X n ∈ B } ≤ − inf lim sup x ∈ B I ( x ) . (2) n →∞ If (1) holds and (2) holds for each compact set B only then family { X n } satisfies weak LDP. If each level set { x : I ( x ) ≤ a } , a ≥ 0 , is compact then we call rate function I “good”. D.Sobolieva
Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients Large deviation principle Let ( X , ρ ) be a complete separable metric space with X -valued random variables X n , n ≥ 1 . We say that family { X n } satisfies large deviation principle (LDP) with rate function I : X → [0 , ∞ ] if, for each opened set A , 1 n log P { X n ∈ A } ≥ − inf lim inf x ∈ A I ( x ) , (1) n →∞ and, for each closed set B , 1 n log P { X n ∈ B } ≤ − inf lim sup x ∈ B I ( x ) . (2) n →∞ If (1) holds and (2) holds for each compact set B only then family { X n } satisfies weak LDP. If each level set { x : I ( x ) ≤ a } , a ≥ 0 , is compact then we call rate function I “good”. D.Sobolieva
Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients Large deviation principle Let ( X , ρ ) be a complete separable metric space with X -valued random variables X n , n ≥ 1 . We say that family { X n } satisfies large deviation principle (LDP) with rate function I : X → [0 , ∞ ] if, for each opened set A , 1 n log P { X n ∈ A } ≥ − inf lim inf x ∈ A I ( x ) , (1) n →∞ and, for each closed set B , 1 n log P { X n ∈ B } ≤ − inf lim sup x ∈ B I ( x ) . (2) n →∞ If (1) holds and (2) holds for each compact set B only then family { X n } satisfies weak LDP. If each level set { x : I ( x ) ≤ a } , a ≥ 0 , is compact then we call rate function I “good”. D.Sobolieva
Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients Large deviation principle Let ( X , ρ ) be a complete separable metric space with X -valued random variables X n , n ≥ 1 . We say that family { X n } satisfies large deviation principle (LDP) with rate function I : X → [0 , ∞ ] if, for each opened set A , 1 n log P { X n ∈ A } ≥ − inf lim inf x ∈ A I ( x ) , (1) n →∞ and, for each closed set B , 1 n log P { X n ∈ B } ≤ − inf lim sup x ∈ B I ( x ) . (2) n →∞ If (1) holds and (2) holds for each compact set B only then family { X n } satisfies weak LDP. If each level set { x : I ( x ) ≤ a } , a ≥ 0 , is compact then we call rate function I “good”. D.Sobolieva
Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients The model Consider one-dimensional SDE’s 1 dX n t = a ( X n √ nσ ( X n t ) dt + t ) dW t (3) with initial conditions X n 0 = x 0 . Freidlin, Wentzell’79 D.Sobolieva
Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients The model Consider one-dimensional SDE’s 1 dX n t = a ( X n √ nσ ( X n t ) dt + t ) dW t (3) with initial conditions X n 0 = x 0 . Freidlin, Wentzell’79 D.Sobolieva
Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients Case a ≡ 0 . The model Consider ∞ 1 � X = C ([0 , ∞ )) , ρ ( f, g ) = 2 k ( sup | f ( t ) − g ( t ) | ∧ 1) . t ∈ [0 ,k ] k =1 Let us consider SDE’s without drift, which means that coefficient a ≡ 0 . In this case our model becomes of the following form: 1 dY n √ nσ ( Y n t = t ) dW t (4) with initial conditions Y n 0 = y 0 . D.Sobolieva
Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients Case a ≡ 0 . The model Consider ∞ 1 � X = C ([0 , ∞ )) , ρ ( f, g ) = 2 k ( sup | f ( t ) − g ( t ) | ∧ 1) . t ∈ [0 ,k ] k =1 Let us consider SDE’s without drift, which means that coefficient a ≡ 0 . In this case our model becomes of the following form: 1 dY n √ nσ ( Y n t = t ) dW t (4) with initial conditions Y n 0 = y 0 . D.Sobolieva
Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients Case a ≡ 0 . The representation for the solution � 1 The solutions to (4) can be represented as follows: Y n = F � n W . Here function F : C ([0 , ∞ )) → C ([0 , ∞ )) is defined by F t ( f ) = f ( η − 1 ( f )) + y 0 , t ≥ 0 , f ∈ C ([0 , ∞ )) t where the transformation η : C ([0 , ∞ )) → C ([0 , ∞ )) is defined by � t σ − 2 ( y 0 + f ( s )) ds, η t ( f ) = t ≥ 0 , f ∈ C ([0 , ∞ )) 0 D.Sobolieva
Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients Case a ≡ 0 . The representation for the solution � 1 The solutions to (4) can be represented as follows: Y n = F � n W . Here function F : C ([0 , ∞ )) → C ([0 , ∞ )) is defined by F t ( f ) = f ( η − 1 ( f )) + y 0 , t ≥ 0 , f ∈ C ([0 , ∞ )) t where the transformation η : C ([0 , ∞ )) → C ([0 , ∞ )) is defined by � t σ − 2 ( y 0 + f ( s )) ds, η t ( f ) = t ≥ 0 , f ∈ C ([0 , ∞ )) 0 D.Sobolieva
Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients Case a ≡ 0 . The representation for the solution � 1 The solutions to (4) can be represented as follows: Y n = F � n W . Here function F : C ([0 , ∞ )) → C ([0 , ∞ )) is defined by F t ( f ) = f ( η − 1 ( f )) + y 0 , t ≥ 0 , f ∈ C ([0 , ∞ )) t where the transformation η : C ([0 , ∞ )) → C ([0 , ∞ )) is defined by � t σ − 2 ( y 0 + f ( s )) ds, η t ( f ) = t ≥ 0 , f ∈ C ([0 , ∞ )) 0 D.Sobolieva
Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients Contraction principle and its generalizations Contraction principle Let family { X n } satisfy LDP with good rate function I and consider a continuous function F : X → Y , such that Y n = F ( X n ) . Then family { Y n } satisfies LDP with good rate function J ( y ) = inf { I ( x ) : y = F ( x ) } Dembo, Zeitouni’98 D.Sobolieva
Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients Contraction principle and its generalizations Contraction principle Let family { X n } satisfy LDP with good rate function I and consider a continuous function F : X → Y , such that Y n = F ( X n ) . Then family { Y n } satisfies LDP with good rate function J ( y ) = inf { I ( x ) : y = F ( x ) } Dembo, Zeitouni’98 D.Sobolieva
Introduction LDP for one-dimensional SDE’s without drift LDP for one-dimensional SDE’s with discontinuous coefficients Contraction principle and its generalizations Contraction principle Let family { X n } satisfy LDP with good rate function I and consider a continuous function F : X → Y , such that Y n = F ( X n ) . Then family { Y n } satisfies LDP with good rate function J ( y ) = inf { I ( x ) : y = F ( x ) } Dembo, Zeitouni’98 D.Sobolieva
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