A variational formula for functionals of fBM and applications to - PowerPoint PPT Presentation

A variational formula for functionals of fBM and applications to LDPs Andr´ e de Oliveira Gomes IMECC-UNICAMP, Campinas SP, Brasil Escola Brasileira de Probabilidade XXIII-2019 S˜ ao Carlos, SP Brazil joint work with Pedro Catuogno (IMECC-UNICAMP) 26 July 2019

Overview 1. The baby problem. 2. Features of the fBM. 3. A Quick tour on the weak convergence approach to LDT. 4. A variational formula for bounded measurable functionals of fBM. 5. A sufficient condition for LDPs of SDEs driven by fBM. 7. Donsker-Varadhan LDPs.

The starting point- the baby problem We would like to understand the deviations in an exponentially small scale, when ε → 0, of the solutions of the following SDEs  d  dZ ε, z = V 0 ( Z ε, z � V i ( Z ε, z ) ◦ dX i  ) dt + ε t ,  t t t (1) i =1  Z ε, z = z ,   0 where ( X t ) t ≥ 0 := ( X 1 t , . . . , X d t ) t ≥ 0 is a d -dimensional Gaussian process and V 0 , . . . , V d is a collection of smooth vector fields: R n − → R .

The fractional Brownian Motion The driving signal ( X t ) t ≥ 0 is a fractional Brownian motion (fBM) with Hurst parameter H ∈ (0 , 1), that is, X = B H , where B H is a centered Gaussian process with covariance given by s ] = 1 � | t | 2 H + | s | 2 H − | t − s | 2 H � R H ( s , t ) = E [ B H t B H 2 Remark 2 the process B H is a standard Brownian motion. If H = 1

Features of the fBM 1. Self-similarity: For any a > 0, one has { B H ( at ) | t ≥ 0 } = d { a H B H ( t ) | t ≥ 0 } . It results from the structure of the covariance function. 2. Stationary increments: { B H ( t + h ) − B H ( h ) } = d { B H ( t ) } , for every h > 0. 3. Independent increments:no!! fBMs have independent increments iff H = 1 2 and in this case E [ B H t B H s ] = t ∧ s . When H � = 1 2 the increments are not independent. When H > 1 2 the increments are positively correlated; if H < 1 2 they are negatively correlated. 4. Long range dependence: Let ( X t ) t ≥ 0 be an H -self similar process with stationary increments and non degenerate for all t ≥ 0 with E [ | X 1 | 2 ] < ∞ . Write ξ n = X n +1 − X n and r ( n ) = E [ ξ (0) ξ ( n )], for all n ≥ 0. For 1 2 < H < 1 we have � n | r n | = ∞ and this property is called long range dependence .

Features of the fBM 5 Markovian pp: A Gaussian process with covariance R is Markovian iif R ( s , u ) = R ( s , t ) R ( s , u ) , s ≤ t ≤ u . R ( t , t ) The fBM is Markovian iif H = 1 2 . 6. β -H¨ older continuity: fBM admits a modification which is H¨ older continuous of order β iif β ∈ (0 , H ). The value of the Hurst parameter decides the regularity of the sample paths. 7. Differentiability: fBM is a.s. nowhere differentiable. 8. p-variation: fBM has bounded p -variation when p > 1 H and unbounded p-variation when p < 1 H . 9. It is not a semimartingale. If B H t = A H t + M H t for all t ≥ 0, by Doob-Meyer, when H < 1 2 we have [ M H ] t = ∞ and | A H t | TV = ∞ if H > 1 2 . Therefore no stochastic calculus. NO ITO!!!

Features of the fBM � t u s dB H 10. How to define s ? 0 i) When H > 1 2 one uses Young’s integral. � � 1 3 , 1 ii) When H ∈ one uses RPtheory (Coutin, Hairer, 2 Baudoin, Gubinelli...) iii) Nualart’s antecipative calculus via the divergence operator (Skorohod integral) We choose Rough Paths theory.

LDP: the weak convergence approach Let ( X ε ) ε> 0 be a family of r.vs defined on (Ω , F , P ) with values in a complete separable metric space E (Polish). • A function I : E − → [0 , ∞ ] is called a good rate function if I is lower semicontinuous and if the sublevel sets { x ∈ E | I ( x ) ≤ c } are compact c ≥ 0. • The family ( X ε ) ε> 0 is said to satisfy a large deviations principle on E with the good rate function I if ε ln P ( X ε ∈ F ) ≤ − inf lim sup x ∈ F I ( x ) ε → 0 ε → 0 ε ln P ( X ε ∈ G ) ≥ − inf lim inf x ∈ G I ( x ) , for every F ∈ B ( E ) closed and G ∈ B ( E ) open.

LDPs via the weak convergence approach • Laplace’s method: for any h ∈ C b ([0 , 1]) one has � 1 1 e − h ( x ) dx = − min lim n ln x ∈ [0 , 1] h ( x ) . n →∞ 0 • ( X ε ) ε> 0 a family of E -valued r.vs. is said to satisfy the Laplace-Varadhan principle with the good rate function I if � ε h ( X ε ) � e − 1 lim sup ε ln E ≤ − inf x ∈ E { I ( x ) + h ( x ) } , ε → 0 � ε h ( X ε ) � e − 1 lim inf ε → 0 ε ln E ≥ − inf x ∈ E { h ( x ) + I ( x ) } , for every h ∈ C b ( E ). • Since E is Polish LDP ⇔ LVP.

The relative entropy • Let P ( E ) denote the set of probability measures defined on ( E , E ). Given µ ∈ P ( E ) we define R ( . || µ ) : P ( E ) − → [0 , ∞ ] given by �  ln d ν if ν ≪ µ and ln d ν d µ ∈ L 1 ( µ ) d µ ( x ) ν ( dx ) ,  R ( ν || µ ) := E ∞ , otherwise .  • Variational representation of Laplace functionals: Let h ∈ M b ( E ). Let µ ∈ P ( E ). Then � � � � e − h ( z ) µ ( dz ) = − ln inf R ( ν || µ ) + h ( z ) ν ( dz ) ν ∈P ( E ) E E and let ν 0 ∈ P ( E ) such that ν 0 ≪ µ and e − h d ν 0 d µ = E e − h d µ. � Then the infimum above is attained uniquely at ν 0 .

Donsker-Varadhan representation • Let E be a Polish space and µ and ν in P ( E ). One has the representation � � � e f d µ � R ( ν || µ ) = sup fd ν − ln f ∈ C b ( E ) e E � � � � e φ d µ = sup φ d ν − ln . φ ∈ M b ( E ) e E • Laplace functionals and Relative entropies are convex conjugates in the duality of the Fenchel-Legendre transform.

Fractional calculus associated to fBM • Given H ∈ (0 , 1) let H H be the reproducing kernel Hilbert space → R d such associated, which consists on the functions h : [0 , T ] − h ∈ L 2 that have the representation that ˙ � t K H ( t , s )˙ h ( t ) = h ( s ) ds , 0 where K H is the kernel defined by � � t � s � 1 � 1 2 − H � � K H ( t , s ) = c H ( t − s ) H − 1 ( u − s ) H − 3 2 + c H 2 − H 1 − du , 2 u s for some c H > 0. • The scalar product in H H is given by � h 1 , h 2 � H H = � ˙ h 1 , ˙ h 2 � L 2

A bit of Gaussian analysis • For every t ∈ [0 , T ] we denote F B H the σ -field generated by the t random variables B H s , s ∈ [0 , t ] and the P -null sets. • We denote E the set of step functions on [0 , T ]. Let H be the Hilbert space defined as the closure of E wrt to the scalar product � 1 [0 , t ] , 1 [0 , s ] � H := R H ( t , s ) . • The map 1 [0 , t ] �→ B H can be extended to an isometry between t H and the Gaussian space H 1 ( B H ) associated with B H . We will denote this isometry by ϕ �→ B H ( ϕ ). • The covariance kernel can be written as � t ∧ s R H ( t , s ) = K H ( t , r ) K H ( s , r ) dr . 0

A bit of Gaussian analysis • Consider the linear operator K ∗ → L 2 [0 , T ] given by H : E − � T ( ϕ ( r ) − ϕ ( s )) ∂ K H ( K ∗ H ϕ )( s ) = K H ( T , s ) ϕ ( s ) + ∂ r ( r , s ) dr . s • For any pair of step functions ϕ, ψ ∈ E we have � K ∗ H ϕ, K ∗ H ψ � L 2 [0 , T ] = � ϕ, ψ � H . • As a consequence the operator K ∗ H provides an isometry between H and L 2 [0 , T ] . Hence the process W = ( W t ) t ∈ [0 , T ] defined by W t = B H (( K ∗ H ) − 1 1 [0 , t ] ) is a Wiener process wrt to F B H and the process B H has an integral representation of the form � t B H t = K H ( t , s ) dW s , 0 since ( K ∗ H 1 [0 , t ] )( s ) = K H ( t , s ).

Girsanov’s transform • Given an F B H -an adapted process ( u t ) t ∈ [0 , T ] and consider the transformation � t ˜ B H t = B H t + u s ds . 0 • We can write � t � t � t ˜ B H t = B H t + u s ds = K H ( t , s ) dW s + u s ds 0 0 0 � t K H ( t , s ) d ˜ = W s , 0 where � t � � . � � � ˜ K − 1 W t = W t + u s ds ( r ) dr H 0 0

Girsanov transform Theorem Consider the shifted process ˜ B H . defined by the process ( u s ) s ∈ [0 , T ] with integrable paths. Assume that � . H + 1 • It holds that L 2 [0 , T ] P -a.s. 0 u s ds ∈ I 2 0+ • E [ E T ] = 1 where � T � . � T � . ( s ) dW s − 1 � 2 � � � � � K − 1 K − 1 E T = exp − u s ds u s ds ( s ) H H 2 0 0 0 0 B H is an F B H -fBM with Hurst parameter Then the shifted process ˜ P defined by d ¯ H under the new probability ¯ P d P = E T .

A variational formula for bounded measurable functionals of fBM-cf. Dupuis-Ellis’s formula for BM Theorem For any f ∈ M b ( C ([0 , T ]; R d )) � e − f ( B H ) � − ln E � T � � . � . � 1 2 � � � � B H + �� � K − 1 = inf ( r ) dr + f . v ∈A E u s ds u s ds � � H 2 � 0 0 0 where A is the class of all d-dimensional F B H -progressively measurable t processes such that � � T � � . 2 � � �� � K − 1 u s ds < ∞ . E � � H � 0 0

An idea of the proof • We start to see that − ln E [ f ( B H )] is bounded below by the right hand side. • Take v ∈ A b . Then Novikov’s condition (a bit hard) shows that � � t � . � t � . ( s ) dW s − 1 2 � � � � � � � K − 1 K − 1 M t := exp u r dr u r dr ( s ) ds � � H H 2 � � 0 0 0 0 is a martingale wrt to F B H . • Define the measure � A ∈ F B H Q ( A ) := M T d P , T . A � t • Girsanov yields, considering ˜ B H t = B H t − 0 u s ds � ln d Q R ( Q || P ) = d P d Q