Introduction and setup Results Proof methods A change of variable formula with Itˆ o correction term* Jason Swanson Department of Mathematics University of Central Florida Isaac Newton Institute, May 24, 2010 *Joint work with Chris Burdzy (University of Washington) Jason Swanson A change of variable formula with Itˆ o correction term*
Introduction and setup Heuristics and preliminaries Results Alternating quadratic variation Proof methods Classical Itˆ o formula for Brownian motion: � t � t g ′ ( B ( s )) dB ( s ) + 1 g ′′ ( B ( s )) ds g ( B ( t )) = g ( B ( 0 )) + 2 0 0 Correction term due to E | B ( t + ∆ t ) − B ( t ) | 2 = ∆ t . For a process F with E | F ( t + ∆ t ) − F ( t ) | 4 ≈ ∆ t , we construct an integral such that � t � t g ′ ( F ( s )) dF ( s ) + 1 g ′′ ( F ( s )) dB ( s ) , g ( F ( t )) = g ( F ( 0 )) + 2 0 0 where B is a Brownian motion independent of F . Jason Swanson A change of variable formula with Itˆ o correction term*
Introduction and setup Heuristics and preliminaries Results Alternating quadratic variation Proof methods Classical Itˆ o formula for Brownian motion: � t � t g ′ ( B ( s )) dB ( s ) + 1 g ′′ ( B ( s )) ds g ( B ( t )) = g ( B ( 0 )) + 2 0 0 Correction term due to E | B ( t + ∆ t ) − B ( t ) | 2 = ∆ t . For a process F with E | F ( t + ∆ t ) − F ( t ) | 4 ≈ ∆ t , we construct an integral such that � t � t g ′ ( F ( s )) dF ( s ) + 1 g ′′ ( F ( s )) dB ( s ) , g ( F ( t )) = g ( F ( 0 )) + 2 0 0 where B is a Brownian motion independent of F . Jason Swanson A change of variable formula with Itˆ o correction term*
Introduction and setup Heuristics and preliminaries Results Alternating quadratic variation Proof methods Definition of F ∂ t u = 1 x u + ˙ 2 ∂ 2 W ( x , t ) , x ∈ R ; u ( x , 0 ) ≡ 0 F ( t ) := u ( x , t ) F is a centered Gaussian process with covariance 1 ( | t + s | 1 / 2 − | t − s | 1 / 2 ) ρ ( s , t ) = E [ F ( s ) F ( t )] = √ 2 π F is a bifractional Brownian motion, qualitatively similar to fractional Brownian motion (fBm) B H with H = 1 / 4. Jason Swanson A change of variable formula with Itˆ o correction term*
Introduction and setup Heuristics and preliminaries Results Alternating quadratic variation Proof methods Quartic variation of F Let Π = { 0 = t 0 < t 1 < t 2 < · · · } with t j ↑ ∞ and | Π | := sup ( t j − t j − 1 ) < ∞ . j ∈ N � | F ( t j ) − F ( t j − 1 ) | 4 . Define V Π ( t ) = 0 < t j ≤ t Theorem (S 2007) � � � 2 � � � � V Π ( t ) − 6 � � | Π |→ 0 E lim sup π t = 0 , ∀ T > 0 . � 0 ≤ t ≤ T F is not a semimartingale; cannot construct classical stochastic integral. We construct an integral using Riemann sums. Jason Swanson A change of variable formula with Itˆ o correction term*
Introduction and setup Heuristics and preliminaries Results Alternating quadratic variation Proof methods ∆ t = n − 1 t j = j ∆ t ∆ F j = F ( t j ) − F ( t j − 1 ) Left-endpoint and right-endpoint Riemann sums diverge. � ⌊ nt ⌋ ⌊ nt ⌋ � � 1 n ∆ t 1 / 2 ≈ − t E F ( t j − 1 )∆ F j ≈ − √ 2 π 2 π j = 1 j = 1 � ⌊ nt ⌋ ⌊ nt ⌋ � � 1 n ∆ t 1 / 2 ≈ t F ( t j )∆ F j ≈ √ E 2 π 2 π j = 1 j = 1 Need a symmetric Riemann sum to generate cancellations Jason Swanson A change of variable formula with Itˆ o correction term*
Introduction and setup Heuristics and preliminaries Results Alternating quadratic variation Proof methods ∆ t = n − 1 t j = j ∆ t ∆ F j = F ( t j ) − F ( t j − 1 ) Let θ ( t ) = g ( F ( t ) , t ) and t j = ( t j − 1 + t j ) / 2. We will consider � t ⌊ nt ⌋ � ? I n ( g , t ) = θ ( t j )∆ F j − → θ ( s ) dF ( s ) 0 j = 1 � t ⌊ nt ⌋ � θ ( t j − 1 ) + θ ( t j ) ? I T θ ( s ) d T F ( s ) n ( g , t ) = ∆ F j − → 2 0 j = 1 Jason Swanson A change of variable formula with Itˆ o correction term*
Introduction and setup Heuristics and preliminaries Results Alternating quadratic variation Proof methods Quadratic variation of F is infinite. Define ⌊ nt ⌋ ⌊ nt ⌋ � � ( − 1 ) j ∆ F 2 (∆ F 2 j − ∆ F 2 Q n ( t ) = j ≈ j − 1 ) . j = 1 j = 1 j even Terms have (approximately) mean 0, variance ∆ t . Jason Swanson A change of variable formula with Itˆ o correction term*
Introduction and setup Heuristics and preliminaries Results Alternating quadratic variation Proof methods Theorem (S 2007) If Q n ( t ) = � ⌊ nt ⌋ j = 1 ( − 1 ) j ∆ F 2 j , then ( F , Q n ) → ( F , κ B ) in law in D R 2 [ 0 , ∞ ) , the Skorohod space of cadlag functions from [ 0 , ∞ ) to R d , where B is a standard Brownian motion independent of F, and � 4 � 1 / 2 ∞ � π + 2 ( − 1 ) j ( 2 j 1 / 2 − | j − 1 | 1 / 2 + | j + 1 | 1 / 2 ) 2 κ = π � �� � j = 1 derived from the covariance of F ≈ 1 . 029 . We define [ [ F ] ] t := κ B ( t ) to be the alternating quadratic variation of F . Jason Swanson A change of variable formula with Itˆ o correction term*
Introduction and setup Heuristics and preliminaries Results Alternating quadratic variation Proof methods Key idea of proof: 1/ 4 s s s = � k n 0 1 k = = t t 1 0 t t � 1 2 n Δ = t 1/ n 1/4 1/4 Δ t = 1/ n Let t = 1 and suppose k = n 1 / 4 ∈ N . If s j = j ∆ t 1 / 4 , then � � ⌊ nt ⌋ ⌊ ns k ⌋ k � � � ( − 1 ) j ∆ F 2 ( − 1 ) j ∆ F 2 j ≈ j j = 1 i = 1 j = ⌊ ns k − 1 ⌋ � �� � These terms are asymptotically independent Jason Swanson A change of variable formula with Itˆ o correction term*
Main result Introduction and setup Extended main result Results Trapezoid sums Proof methods Comparison with regularization Theorem (main result, informal version) If g is “nice enough”, then I n ( g , t ) converges in law to a process � t 0 g ( F ( s ) , s ) dF ( s ) satisfying � t g ( F ( t ) , t ) = g ( F ( 0 ) , 0 ) + ∂ t g ( F ( s ) , s ) ds 0 � t � t ∂ x g ( F ( s ) , s ) dF ( s ) + 1 ∂ 2 + x g ( F ( s ) , s ) d [ [ F ] ] s . ( ∗ ) 2 0 0 Jason Swanson A change of variable formula with Itˆ o correction term*
Main result Introduction and setup Extended main result Results Trapezoid sums Proof methods Comparison with regularization Definition Let k and r be integers such that 0 ≤ r ≤ k . We write g ∈ C k , 1 ( R × [ 0 , ∞ )) if r g : R × [ 0 , ∞ ) → R is continuous. ∂ j x g exists and is cont. on R × [ 0 , ∞ ) for all 0 ≤ j ≤ k . ∂ t ∂ j x g exists and is cont. on R × ( 0 , ∞ ) for all 0 ≤ j ≤ r . | ∂ t ∂ j t → 0 sup lim x g ( x , t ) | < ∞ for all cpct K and all 0 ≤ j ≤ r . x ∈ K = C k , 1 = functions with k spatial derivs, 1 time deriv C k , 1 0 g ∈ C k , 1 ⇒ ∂ x g ∈ C k − 1 , 1 1 g ∈ C k , 1 ⇒ ∂ x g ∈ C k − 1 , 1 and ∂ 2 x g ∈ C k − 2 , 1 2 1 . . . etc. Jason Swanson A change of variable formula with Itˆ o correction term*
Main result Introduction and setup Extended main result Results Trapezoid sums Proof methods Comparison with regularization Definition Given g ∈ C 8 , 1 3 , choose G ∈ C 9 , 1 such that ∂ x G = g . Let B be a 4 standard Brownian motion independent of F . Define � � t g ( F , s ) dF = g ( F ( s ) , s ) dF ( s ) 0 � t := G ( F ( t ) , t ) − G ( F ( 0 ) , 0 ) − ∂ t G ( F ( s ) , s ) ds 0 � t − κ ∂ 2 x G ( F ( s ) , s ) dB ( s ) . 2 0 By definition, then, for every g ∈ C 9 , 1 4 , the Itˆ o formula ( ∗ ) holds. � The issue is therefore whether I n ( g , t ) → g ( F , s ) dF . Jason Swanson A change of variable formula with Itˆ o correction term*
Main result Introduction and setup Extended main result Results Trapezoid sums Proof methods Comparison with regularization Theorem (S 2007) ( F , Q n ) → ( F , κ B ) in law in D R 2 [ 0 , ∞ ) , where B is a standard Brownian motion independent of F. Theorem (Burdzy, S 2010) � If g ∈ C 8 , 1 3 , then ( F , Q n , I n ( g , · )) → ( F , κ B , g ( F , s ) dF ) in law in D R 3 [ 0 , ∞ ) , where B is a standard BM, independent of F. (Note: The B that appears in the second component of the limit � is the same B used in the definition of g ( F , s ) dF.) The method of proof (based in part on (Kurtz, Protter 1991)) actually gives something somewhat stronger. Jason Swanson A change of variable formula with Itˆ o correction term*
Main result Introduction and setup Extended main result Results Trapezoid sums Proof methods Comparison with regularization Theorem (Burdzy, S 2010) � If g ∈ C 8 , 1 3 , then ( F , Q n , I n ( g , · )) → ( F , κ B , g ( F , s ) dF ) in law in D R 3 [ 0 , ∞ ) , where B is a standard BM, independent of F. Define F t = σ { W ( A ) : A ⊂ R × [ 0 , t ] , m ( A ) < ∞} , where m is Lebesgue measure. Suppose: { W n ( · ) } ⊂ D R d [ 0 , ∞ ) W n ( t ) ∈ F t ∨ G n t , where G n t is independent of F t ( W n , F , Q n ) → ( W , F , κ B ) in law in D R d + 2 [ 0 , ∞ ) . � Then ( W n , F , Q n , I n ( g , · )) → ( W , F , κ B , g ( F , s ) ds ) in law in D R d + 3 [ 0 , ∞ ) . Jason Swanson A change of variable formula with Itˆ o correction term*
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