Power variation and p -variation of sample functions of stochastic processes Rimas Norvaiˇ sa Department of Econometric Analysis Vilnius University 15 November 2014, Guanajuato, Mexico 5th Workshop on Game-Theoretic Probability and Related Topics R. Norvaiˇ sa (VU) Guanajuato, 2014 1 / 11
Power variation of a function Let f be a regulated function on [0 , T ] , i.e. there exist limits f ( t − ) := lim x ↑ t f ( x ) and f ( s +) := lim x ↓ s f ( x ) for each 0 ≤ s < t ≤ T . i ) m ( n ) Let λ = { λ n : n ≥ 1 } be a nested sequence of partitions λ n = ( t n of i =0 [0 , T ] such that ∪ n λ n is dense in [0 , T ] . Let 1 ≤ p < ∞ . We say that f has p -th power λ -variation on [0 , T ] , if there is a regulated function V on [0 , T ] such that V (0) = 0 and for each 0 ≤ s < t ≤ T m ( n ) � | f (( t n i ∧ t ) ∨ s ) − f (( t n i − 1 ∧ t ) ∨ s )) | p , V ( t ) − V ( s ) = lim n →∞ i =1 V ( t ) − V ( t − ) = | f ( t ) − f ( t − ) | p V ( s +) − V ( s ) = | f ( s +) − f ( s ) | p . and R. Norvaiˇ sa (VU) Guanajuato, 2014 2 / 11
p -variation of a function Let f be a function on [0 , T ] (must be regulated if it has bounded p -variation defined next). Let 1 ≤ p < ∞ . The p -variation of f is the quantity v p ( f, [0 , T ]) defined to be � n � | f ( t i ) − f ( t i − 1 ) | p : ( t i ) n � sup i =0 is a partition of [0 , T ] , i =1 which may be finite or infinite. If v p ( f, [0 , T ]) < ∞ then one says that f has bounded p -variation. The p -variation index of f is the quantity υ ( f, [0 , T ]) defined to be inf { p ≥ 1: v p ( f, [0 , T ]) < ∞} . if the set is non-empty and defined to be + ∞ otherwise. R. Norvaiˇ sa (VU) Guanajuato, 2014 3 / 11
Example: Wiener process Let W = { W ( t ): t ∈ [0 , T ] } be a standard Wiener process. Due to results of N. Wiener (1923) and P. L´ evy (1940): v p ( W, [0 , T ]) < + ∞ a.s. iff p > 2 , and v 2 ( W, [0 , T ]) = + ∞ a.s. Thus the p -variation index υ ( W, [0 , T ]) = 2 a.s. More precise information can be obtained in terms of φ -variation, defined as p -variation except that the power function x �→ x p , x ≥ 0 , is replaced by a function φ . S. J. Taylor (1972): v ψ 1 ( W, [0 , T ]) < + ∞ a. s., where ψ 1 ( x ) := x 2 /LL (1 /x ) , 0 < x ≤ e − e . Also, v ψ ( W ) = + ∞ a.s. for any ψ such that ψ 1 ( x ) = o ( ψ ( x )) as x ↓ 0 . R. Norvaiˇ sa (VU) Guanajuato, 2014 4 / 11
Example: Wiener process Let W = { W ( t ): t ∈ [0 , T ] } be a standard Wiener process. Due to results of N. Wiener (1923) and P. L´ evy (1940): v p ( W, [0 , T ]) < + ∞ a.s. iff p > 2 , and v 2 ( W, [0 , T ]) = + ∞ a.s. Thus the p -variation index υ ( W, [0 , T ]) = 2 a.s. More precise information can be obtained in terms of φ -variation, defined as p -variation except that the power function x �→ x p , x ≥ 0 , is replaced by a function φ . S. J. Taylor (1972): v ψ 1 ( W, [0 , T ]) < + ∞ a. s., where ψ 1 ( x ) := x 2 /LL (1 /x ) , 0 < x ≤ e − e . Also, v ψ ( W ) = + ∞ a.s. for any ψ such that ψ 1 ( x ) = o ( ψ ( x )) as x ↓ 0 . R. Norvaiˇ sa (VU) Guanajuato, 2014 4 / 11
Example: Wiener process Let W = { W ( t ): t ∈ [0 , T ] } be a standard Wiener process. Due to results of N. Wiener (1923) and P. L´ evy (1940): v p ( W, [0 , T ]) < + ∞ a.s. iff p > 2 , and v 2 ( W, [0 , T ]) = + ∞ a.s. Thus the p -variation index υ ( W, [0 , T ]) = 2 a.s. More precise information can be obtained in terms of φ -variation, defined as p -variation except that the power function x �→ x p , x ≥ 0 , is replaced by a function φ . S. J. Taylor (1972): v ψ 1 ( W, [0 , T ]) < + ∞ a. s., where ψ 1 ( x ) := x 2 /LL (1 /x ) , 0 < x ≤ e − e . Also, v ψ ( W ) = + ∞ a.s. for any ψ such that ψ 1 ( x ) = o ( ψ ( x )) as x ↓ 0 . R. Norvaiˇ sa (VU) Guanajuato, 2014 4 / 11
Example: fractional Brownian motion Let B H = { B H ( t ): t ∈ [0 , T ] } be a fractional Brownian motion with the Hurst index H ∈ (0 , 1) , i.e. a Gaussian stochastic process with mean zero and the covariance function EB H ( t ) B H ( s ) = 1 t 2 H + s 2 H − | t − s | 2 H � � s, t ∈ [0 , T ] . 2 i ) m ( n ) Let λ n = ( t n i =0 , n ∈ N , be a sequence of partitions of [0 , T ] such that � 1 ∧ (2 H ) log n → 0 as n → ∞ . Then a.s. max i ( t n i − t n � i − 1 ) m ( n ) � 1 /H = E | η | 1 /H T, � � B H ( t n i ) − B H ( t n � � lim i − 1 ) n →∞ i =1 where η is a standard normal random variable. Thus, almost every sample function of B H has 1 /H power λ -variation t �→ c H t , t ∈ [0 , T ] . Also, a.s. v 1 /H ( B H , [0 , T ]) = + ∞ and υ ( B H , [0 , T ]) = 1 /H . R. Norvaiˇ sa (VU) Guanajuato, 2014 5 / 11
Weighted power variation for a Gaussian process Let X = { X ( t ): t ∈ [0 , T ] } be a mean zero Gaussian process s.t. there is a real valued function ρ defined on [0 , T ] and ”equivalent” to h �→ ( E [ X ( s + h ) − X ( s )] 2 ) 1 / 2 near zero uniformly in s ∈ [ ǫ, T ) for each ǫ > 0 . If X has stationary increments, then one can take ρ ( h ) = ( E [ X ( s + h ) − X ( s )] 2 ) 1 / 2 . Under suitable hypotheses on the covariance of X and for a suitable set of positive r we proved that a.s. m n | X ( t n i ) − X ( t n i − 1 ) | r � ( t n i − t n i − 1 ) = E | η | r T, lim (1) [ ρ ( t n i − t n i − 1 )] r n →∞ i =1 i ) m n where η is a standard normal random variable, and (( t n i =0 ) is a sequence of partitions of [0 , T ] such that the mesh max i ( t n i − t n i − 1 ) tends to zero as n → ∞ sufficiently fast. R. Norvaiˇ sa (VU) Guanajuato, 2014 6 / 11
Weighted power variation for a Gaussian process Let X = { X ( t ): t ∈ [0 , T ] } be a mean zero Gaussian process s.t. there is a real valued function ρ defined on [0 , T ] and ”equivalent” to h �→ ( E [ X ( s + h ) − X ( s )] 2 ) 1 / 2 near zero uniformly in s ∈ [ ǫ, T ) for each ǫ > 0 . If X has stationary increments, then one can take ρ ( h ) = ( E [ X ( s + h ) − X ( s )] 2 ) 1 / 2 . Under suitable hypotheses on the covariance of X and for a suitable set of positive r we proved that a.s. m n | X ( t n i ) − X ( t n i − 1 ) | r � ( t n i − t n i − 1 ) = E | η | r T, lim (1) [ ρ ( t n i − t n i − 1 )] r n →∞ i =1 i ) m n where η is a standard normal random variable, and (( t n i =0 ) is a sequence of partitions of [0 , T ] such that the mesh max i ( t n i − t n i − 1 ) tends to zero as n → ∞ sufficiently fast. R. Norvaiˇ sa (VU) Guanajuato, 2014 6 / 11
Partial sum process Let X 1 , X 2 , . . . be real random variables. For each n = 1 , 2 , . . . , let S n be the n -th partial sum process S n ( t ) := X 1 + · · · + X ⌊ tn ⌋ , t ∈ [0 , 1] , Thus for each n = 1 , 2 , . . . and t ∈ [0 , 1] , 0 , if t ∈ [0 , 1 /n ) , if t ∈ [ k n , k +1 X 1 + · · · + X k , n ) , S n ( t ) = k ∈ { 1 , . . . , n − 1 } , X 1 + · · · + X n , if t = 1 . Then for any p ∈ (0 , ∞ ) , m p � � � v p ( S n , [0 , 1]) = max � X k j − 1 +1 + · · · + X k j , � � � j =1 where the maximum is taken over 0 = k 0 < · · · < k m = n and 1 ≤ m ≤ n . R. Norvaiˇ sa (VU) Guanajuato, 2014 7 / 11
Partial sum process Let X 1 , X 2 , . . . be real random variables. For each n = 1 , 2 , . . . , let S n be the n -th partial sum process S n ( t ) := X 1 + · · · + X ⌊ tn ⌋ , t ∈ [0 , 1] , Thus for each n = 1 , 2 , . . . and t ∈ [0 , 1] , 0 , if t ∈ [0 , 1 /n ) , if t ∈ [ k n , k +1 X 1 + · · · + X k , n ) , S n ( t ) = k ∈ { 1 , . . . , n − 1 } , X 1 + · · · + X n , if t = 1 . Then for any p ∈ (0 , ∞ ) , m p � � � v p ( S n , [0 , 1]) = max � X k j − 1 +1 + · · · + X k j , � � � j =1 where the maximum is taken over 0 = k 0 < · · · < k m = n and 1 ≤ m ≤ n . R. Norvaiˇ sa (VU) Guanajuato, 2014 7 / 11
p -variation of partial sum process J. Bretagnolle (1972): given p ∈ (0 , 2) there exists a finite constant C p such that � n n � E | X i | p ≤ � � E | X i | p , Ev p ( S n ) ≤ C p i =1 i =1 provided X 1 , X 2 , . . . are independent, E | X i | p < ∞ and EX i = 0 if p > 1 . Suppose that X 1 , X 2 , . . . are independent identically distributed real random variables, EX 1 = 0 and EX 2 1 = 1 . Let Lx := max { 1 , log x } , x > 0 . J. Qian (1998): boundedness in probability v 2 ( S n ) = O P ( nLLn ) as n → ∞ . Also, O P ( nLLn ) cannot be replaced by o p ( nLLn ) , if in addition E | X 1 | 2+ ǫ < ∞ for some ǫ > 0 . R. Norvaiˇ sa (VU) Guanajuato, 2014 8 / 11
p -variation of partial sum process J. Bretagnolle (1972): given p ∈ (0 , 2) there exists a finite constant C p such that � n n � E | X i | p ≤ � � E | X i | p , Ev p ( S n ) ≤ C p i =1 i =1 provided X 1 , X 2 , . . . are independent, E | X i | p < ∞ and EX i = 0 if p > 1 . Suppose that X 1 , X 2 , . . . are independent identically distributed real random variables, EX 1 = 0 and EX 2 1 = 1 . Let Lx := max { 1 , log x } , x > 0 . J. Qian (1998): boundedness in probability v 2 ( S n ) = O P ( nLLn ) as n → ∞ . Also, O P ( nLLn ) cannot be replaced by o p ( nLLn ) , if in addition E | X 1 | 2+ ǫ < ∞ for some ǫ > 0 . R. Norvaiˇ sa (VU) Guanajuato, 2014 8 / 11
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