Discrete rough paths and limit theorems Samy Tindel Purdue University Durham Symposium – 2017 Joint work with Yanghui Liu Samy T. (Purdue) Rough paths and limit theorems Durham 2017 1 / 27
Outline Preliminaries on Breuer-Major type theorems 1 General framework 2 Applications 3 Breuer-Major with controlled weights Limit theorems for numerical schemes Samy T. (Purdue) Rough paths and limit theorems Durham 2017 2 / 27
Outline Preliminaries on Breuer-Major type theorems 1 General framework 2 Applications 3 Breuer-Major with controlled weights Limit theorems for numerical schemes Samy T. (Purdue) Rough paths and limit theorems Durham 2017 3 / 27
Definition of fBm Definition 1. A 1-d fBm is a continuous process B = { B t ; t ≥ 0 } such that B 0 = 0 and for ν ∈ (0 , 1): B is a centered Gaussian process 2 ( | s | 2 ν + | t | 2 ν − | t − s | 2 ν ) E [ B t B s ] = 1 m -dimensional fBm: B = ( B 1 , . . . , B m ), with B i independent 1-d fBm Variance of increments: E [ | B j t − B j s | 2 ] = | t − s | 2 ν Samy T. (Purdue) Rough paths and limit theorems Durham 2017 4 / 27
Examples of fBm paths ν = 0 . 35 ν = 0 . 5 ν = 0 . 7 Samy T. (Purdue) Rough paths and limit theorems Durham 2017 5 / 27
Some notation Uniform partition of [0 , 1]: For n ≥ 1 we set t k = k n Increment of a function: For f : [0 , 1] → R d , we write δ f st = f t − f s Hermite polynomial of order q : defined as 2 d q t 2 dt q e − t 2 H q ( t ) = ( − 1) q e 2 Samy T. (Purdue) Rough paths and limit theorems Durham 2017 6 / 27
Hermite rank Definition 2. Consider γ = N (0 , 1). f ∈ L 2 ( γ ) such that f is centered. Then there exist: d ≥ 1 A sequence { c q ; q ≥ d } such that f admits the expansion: ∞ � f = c q H q . q = d The parameter d is called Hermite rank of f . Samy T. (Purdue) Rough paths and limit theorems Durham 2017 7 / 27
Breuer-Major’s theorem for fBm increments Theorem 3. Let f ∈ L 2 ( γ ) with rank d ≥ 1 B a 1-d fBm with Hurst parameter ν < 1 2 For 0 ≤ s ≤ t ≤ 1 and n ≥ 1, we set: st = n − 1 f ( n ν δ B t k t k +1 ) h n � 2 s ≤ t k < t Then the following convergence holds true: f . d . d . h n − − − → σ d , f W as n → ∞ Samy T. (Purdue) Rough paths and limit theorems Durham 2017 8 / 27
Breuer-Major with weights (1) Motivation for the introduction of weights: Analysis of numerical schemes Parameter estimation based on quadratic variations Convergence of Riemann sums in rough contexts Weighted sums (or discrete integrals): For a function g , we set J t s ( g ( B ); h n ) � g ( B t k ) h n = t k t k +1 s ≤ t k < t n − 1 � g ( B t k ) f ( n ν δ B t k t k +1 ) = 2 s ≤ t k < t Samy T. (Purdue) Rough paths and limit theorems Durham 2017 9 / 27
Breuer-Major with weights (2) Recall: s ( g ( B ); h n ) = n − 1 J t � g ( B t k ) f ( n ν δ B t k t k +1 ) 2 s ≤ t k < t Expected limit result: For W as in Breuer-Major, � t n →∞ J t s ( g ( B ); h n ) = σ d , f lim s g ( B u ) dW u (1) Unexpected phenomenon: The limits of J t s ( g ( B ); h n ) can be quite different from (1) Samy T. (Purdue) Rough paths and limit theorems Durham 2017 10 / 27
Breuer-Major with weights (3) Theorem 4. For d ≥ 1 and g smooth enough we set s ( g ( B ); h n , d ) = n − 1 V n , d st ( g ) = J t g ( B t k ) H d ( n ν δ B t k t k +1 ) � 2 s ≤ t k < t Then the following limits hold true: 1 If d > 2 ν then 1 � t ( d ) V n , d st ( g ) − → c d ,ν s g ( B u ) dW u 1 If d = 2 ν then 2 � t � t ( d ) V n , d s f ( d ) ( B u ) du st ( g ) − → c 1 , d ,ν s g ( B u ) dW u + c 2 , d ,ν 1 If 1 ≤ d < 2 ν then 3 � t n − ( 1 P 2 − ν d ) V n , d s f ( d ) ( B u ) du st ( g ) − → c d Samy T. (Purdue) Rough paths and limit theorems Durham 2017 11 / 27
Breuer-Major with weights (3) Remarks on Theorem 4: Obtained in a series of papers by Corcuera, Nualart, Nourdin, Podolskij, Réveillac, Swanson, Tudor Extensions to p -variations, Itô formulas in law Limitations of Theorem 4: One integrates w.r.t h n , d , in a fixed chaos Results available only for 1-d fBm Weights of the form y = g ( B ) only Aim of our contribution: Generalize in all those directions Samy T. (Purdue) Rough paths and limit theorems Durham 2017 12 / 27
Outline Preliminaries on Breuer-Major type theorems 1 General framework 2 Applications 3 Breuer-Major with controlled weights Limit theorems for numerical schemes Samy T. (Purdue) Rough paths and limit theorems Durham 2017 13 / 27
Rough path Notation: We consider ν ∈ (0 , 1), Hölder continuity exponent ℓ = ⌊ 1 ν ⌋ , order of the rough path p > 1, integrability order R m , state space for a process x S 2 ≡ simplex in [0 , 1] 2 = { ( s , t ); 0 ≤ s ≤ t ≤ 1 } Rough path: Collection x = { x i ; i ≤ ℓ } such that x i = { x i st ∈ ( R m ) ⊗ i ; ( s , t ) ∈ S 2 } x i st = � s ≤ s 1 < ··· < s i ≤ t dx s 1 ⊗ · · · ⊗ dx s i (to be defined rigorously) We have | x i uv | L p | x i | p ,ν ≡ | v − u | ν i < ∞ sup ( u , v ) ∈S 2 Samy T. (Purdue) Rough paths and limit theorems Durham 2017 14 / 27
Controlled processes (incomplete definition) Definition 5. Let: ℓ = ⌊ 1 ν ⌋ x a ( L p , ν, ℓ )-rough path A family y = ( y , y (1) , . . . , y ( ℓ − 1) ) of processes We say that y is a process controlled by x if ℓ − 1 y ( i ) � s x i | r st | L p � | t − s | νℓ . δ y st = st + r st , and i =1 Remark: Typical examples of controlled process → solutions of differential equations driven by x , or g ( x ) ֒ Samy T. (Purdue) Rough paths and limit theorems Durham 2017 15 / 27
Abstract transfer theorem: setting Objects under consideration: Let α limiting regularity exponent. Typically α = 1 2 or α = 1 x rough path of order ℓ h n such that uniformly in n : s ( x i ; h n ) | L 2 ≤ K ( t − s ) α + ν i |J t (2) y controlled process of order ℓ ( ω i , i ∈ I ) family of processes independent of x → Typically ω i t = Brownian motion, or ω i t = t ֒ Samy T. (Purdue) Rough paths and limit theorems Durham 2017 16 / 27
Abstract transfer theorem (1) Recall: h n satisfies: s ( x i ; h n ) | L 2 ≤ K ( t − s ) α + ν i |J t Illustration: General transfer
Abstract transfer theorem (1) Recall: h n satisfies: s ( x i ; h n ) | L 2 ≤ K ( t − s ) α + ν i |J t Illustration: n →∞ Breuer-Major: h n → ω 0 − − − y controlled process General transfer n →∞ � x i st k h n → ω i − − − t k t k +1 k
Abstract transfer theorem (1) Recall: h n satisfies: s ( x i ; h n ) | L 2 ≤ K ( t − s ) α + ν i |J t Illustration: n →∞ Breuer-Major: h n → ω 0 − − − y t k h n � t k t k +1 k y controlled process n → ∞ General transfer � y ( i ) d ω i � n →∞ � x i st k h n → ω i − − − t k t k +1 i k Samy T. (Purdue) Rough paths and limit theorems Durham 2017 17 / 27
Abstract transfer theorem Theorem 6. We assume that (2) holds and: As n → ∞ : 1 � � → { x , ω i ; 0 ≤ i ≤ ℓ − 1 } . f . d . d . x , J ( x i ; h n ) ; 0 ≤ i ≤ ℓ − 1 − − − � y d ω i . One additional technical condition on 2 Then the following convergence holds true as n → ∞ : ℓ − 1 � f.d.d., stable y ( i ) d ω i . J ( y ; h n ) � − − − − − − → i =0 Samy T. (Purdue) Rough paths and limit theorems Durham 2017 18 / 27
Outline Preliminaries on Breuer-Major type theorems 1 General framework 2 Applications 3 Breuer-Major with controlled weights Limit theorems for numerical schemes Samy T. (Purdue) Rough paths and limit theorems Durham 2017 19 / 27
Outline Preliminaries on Breuer-Major type theorems 1 General framework 2 Applications 3 Breuer-Major with controlled weights Limit theorems for numerical schemes Samy T. (Purdue) Rough paths and limit theorems Durham 2017 20 / 27
Notation Setting: We consider A 1-d fractional Brownian motion B Hurst parameter: ν < 1 2 st = ( δ B st ) i 1-d rough path: B i i ! y controlled process f smooth enough with Hermite rank d W Wiener process independent of B Quantity under consideration: s ( y ; h n , d ) = n − 1 J t � y t k f ( n ν δ B t k t k +1 ) 2 s ≤ t k < t Samy T. (Purdue) Rough paths and limit theorems Durham 2017 21 / 27
Breuer-Major with controlled weights Theorem 7. For f smooth with Hermite rank d and y controlled we set s ( y ; h n , d ) = n − 1 J t y t k f ( n ν δ B t k t k +1 ) � 2 s ≤ t k < t Then the following limits hold true: 1 If d > 2 ν then 1 � t ( d ) J t s ( y ; h n , d ) − → c d ,ν s y u dW u 1 If d = 2 ν then 2 � t � t ( d ) J t s ( y ; h n , d ) s y ( d ) − → c 1 , d ,ν s y u dW u + c 2 , d ,ν du u 1 If 1 ≤ d < 2 ν then 3 � t n − ( 1 P 2 − ν d ) J t s ( y ; h n , d ) s y ( d ) − → c d du u Samy T. (Purdue) Rough paths and limit theorems Durham 2017 22 / 27
Breuer-Major with controlled weights (2) Improvements of Theorem 7: One integrates w.r.t a general f ( n ν δ B t k t k +1 ) ֒ → with f smooth enough Results can be generalized to d-dim situations General controlled weights y Other applications: Itô formulas in law, convergence of Riemann sums Asymptotic behavior of p -variations Samy T. (Purdue) Rough paths and limit theorems Durham 2017 23 / 27
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