A Strong Uniform Approximation of FBM by Means of Transport - PowerPoint PPT Presentation

A Strong Uniform Approximation of FBM by Means of Transport Processes Jorge A. León Departamento de Control Automático Cinvestav del IPN Spring School “Stochastic Control in Finance”, Roscoff 2010 Jointly with Johanna Garzón Merchán and Luis G. Gorostiza Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 1 / 82

Contents Some Approximations of FBM 1 Transport Processes 2 Approximation of FBM by Means of Transport Processes 3 Approximatios of Fractional Stochastic Differential Equations 4 Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 2 / 82

Contents Some Approximations of FBM 1 Transport Processes 2 Approximation of FBM by Means of Transport Processes 3 Approximatios of Fractional Stochastic Differential Equations 4 Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 3 / 82

Taqqu 1975 Let { Y i } be a sequence of stationary Gaussian random variables such that n n � � E ( Y i Y j ) ∼ An 2 H L ( n ) i = 1 j = 1 as n → ∞ . Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 4 / 82

Taqqu 1975 Let { Y i } be a sequence of stationary Gaussian random variables such that n n � � E ( Y i Y j ) ∼ An 2 H L ( n ) i = 1 j = 1 as n → ∞ . Here 0 < H < 1, A > 0 and L is a slowly varying function (i.e., for all L ( ax ) a > 0, lim x →∞ L ( x ) = 1). Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 5 / 82

Taqqu 1975 Let { Y i } be a sequence of stationary Gaussian random variables such that n n � � E ( Y i Y j ) ∼ An 2 H L ( n ) i = 1 j = 1 as n → ∞ . Here 0 < H < 1, A > 0 and L is a slowly varying function Then ⌊ nt ⌋ X n ( t ) = 1 � Y i d n i = 1 √ AB H t , where d 2 n ∼ n 2 H L ( n ) . converges weakly to Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 6 / 82

Tommi Sottinen 2001 Let { ξ i : i > 0 } be a sequence of i.i.d. random variables with E [ ξ i ] = 0 and Var [ ξ i ] = 1, and � 1 � � ⌊ nt ⌋ � ⌊ nt ⌋ i � � B ( n ) n √ n ξ i . = n K H n , s ds t i − 1 i = 1 n Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 7 / 82

Tommi Sottinen 2001 Let { ξ i : i > 0 } be a sequence of i.i.d. random variables with E [ ξ i ] = 0 and Var [ ξ i ] = 1, and � 1 � � ⌊ nt ⌋ � ⌊ nt ⌋ i � � B ( n ) n = n , s √ n ξ i . n K H ds t i − 1 i = 1 n Here, H < 1 / 2 and � ( t s ) H − 1 2 ( t − s ) H − 1 K H ( t , s ) = d H 2 � t � − ( H − 1 1 s u H − 3 2 ( u − s ) H − 1 2 − H 2 ) s 2 du Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 8 / 82

Tommi Sottinen 2001 Let { ξ i : i > 0 } be a sequence of i.i.d. random variables with E [ ξ i ] = 0 and Var [ ξ i ] = 1, and � 1 � � ⌊ nt ⌋ � ⌊ nt ⌋ � i � B ( n ) n √ n ξ i . = n K H n , s ds t i − 1 i = 1 n Here, H < 1 / 2 and � ( t s ) H − 1 2 ( t − s ) H − 1 K H ( t , s ) = d H 2 � t − ( H − 1 � 1 s u H − 3 2 ( u − s ) H − 1 2 − H 2 ) s 2 du Then B ( n ) goes weakly to B H . Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 9 / 82

Stroock 1982 The family � t � � Z ǫ ( t ) = 1 0 ( − 1 ) N ( s ǫ 2 ) ds , t ∈ [ 0 , T ] Z ǫ := ǫ converges weakly to the Brownian motion, as ǫ → 0. Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 10 / 82

Stroock 1982 The family � t � � Z ǫ ( t ) = 1 0 ( − 1 ) N ( s ǫ 2 ) ds , t ∈ [ 0 , T ] Z ǫ := ǫ converges weakly to the Brownian motion, as ǫ → 0. Here { N ( t ) , t ≥ 0 } is a Poisson process Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 11 / 82

Delgado y Jolis (2000) The family � t � � X ǫ ( t ) = 1 0 K H ( t , s )( − 1 ) N ( s ǫ 2 ) ds , t ∈ [ 0 , T ] X ǫ := ǫ converges to B H , as ǫ → 0 . Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 12 / 82

Delgado y Jolis (2000) The family � t � � X ǫ ( t ) = 1 0 K H ( t , s )( − 1 ) N ( s ǫ 2 ) ds , t ∈ [ 0 , T ] X ǫ := ǫ converges to B H , as ǫ → 0 . Here K H ( t , s ) is given by : �� t � H > 1 1 s ( u − s ) H − 3 2 u H − 1 2 − H c H s 2 du 1 ( 0 , t ) ( s ) , 2 , Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 13 / 82

Delgado y Jolis (2000) � t � � X ǫ ( t ) = 1 0 K H ( t , s )( − 1 ) N ( s ǫ 2 ) ds , t ∈ [ 0 , T ] X ǫ := ǫ converges to B H , as ǫ → 0 . Here K H ( t , s ) is given by : �� t � H > 1 1 s ( u − s ) H − 3 2 u H − 1 2 − H 1 ( 0 , t ) ( s ) , 2 , c H s 2 du and � t � 2 − ( H − 1 � ( t s ) H − 1 2 ( t − s ) H − 1 1 s u H − 3 2 ( u − s ) H − 1 2 − H 2 ) s , d H 2 du for H < 1 2 . Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 14 / 82

Szabados (2001) i.i.d r.v. random walks ∆ t ∆ x Modification S 0 = � n ˜ X 0 ( 1 ) , X 0 ( 2 ) , · · · k = 1 X 0 ( k ) 1 1 S 0 ( n ) ˜ � n 2 − 2 2 − 1 X 1 ( 1 ) , X 1 ( 2 ) , · · · S 1 = k = 1 X 1 ( k ) S 1 ( n ) . . . . . . . . . . . . . . . S m = � n ˜ 2 − 2 m 2 − m X m ( 1 ) , X m ( 2 ) , · · · k = 1 X m ( k ) S m ( n ) P ( { X n ( k ) = 1 } ) = P ( { X n ( k ) = − 1 } ) = 1 2 . and S m ( t ) = S m ( j 2 2 n ) + 2 2 n ( t − j 2 2 n ≤ t < j + 1 j 2 2 n ) X m ( j + 1 ) , 2 2 n , with j S m ( j � 2 2 n ) = X m ( i ) . i = 1 Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 15 / 82

Szabados (2001) i.i.d r.v. random walks ∆ t ∆ x Modification ˜ � n X 0 ( 1 ) , X 0 ( 2 ) , · · · S 0 = k = 1 X 0 ( k ) 1 1 S 0 ( n ) S 1 = � n ˜ 2 − 2 2 − 1 X 1 ( 1 ) , X 1 ( 2 ) , · · · k = 1 X 1 ( k ) S 1 ( n ) . . . . . . . . . . . . . . . ˜ S m = � n 2 − 2 m 2 − m X m ( 1 ) , X m ( 2 ) , · · · k = 1 X m ( k ) S m ( n ) Set B m ( t ) = 2 − m ˜ S m ( t 2 2 m ) Theorem B m → W as m → ∞ a.s. uniformly on compact sets. Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 16 / 82

Szabados (2001) B m ( t ) = 2 − m ˜ S m ( t 2 2 m ) Set k − 1 � B H m ( t k ) = h ( t r , t k )[ B m ( t r + ∆ t ) − B m ( t r )] r = −∞ and B H m ( 0 ) = 0 . Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 17 / 82

Szabados (2001) Set k − 1 � B H m ( t k ) = h ( t r , t k )[ B m ( t r + ∆ t ) − B m ( t r )] , r = −∞ and B H m ( 0 ) = 0 , with ∆ t = 2 − 2 m , t x = x ∆ t , x ∈ R , and � ( t − s ) H − 1 / 2 − ( − s ) H − 1 / 2 � h ( s , t ) = C H , s ≤ t . + Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 18 / 82

Szabados (2001) Set k − 1 � B H m ( t k ) = h ( t r , t k )[ B m ( t r + ∆ t ) − B m ( t r )] , r = −∞ and B H m ( 0 ) = 0 , with ∆ t = 2 − 2 m , t x = x ∆ t , x ∈ R , and � ( t − s ) H − 1 / 2 − ( − s ) H − 1 / 2 � h ( s , t ) = C H , s ≤ t . + Theorem For H ∈ ( 1 / 4 , 1 ) , B H m → B H as m → ∞ a.s. uniformly on compact sets. Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 19 / 82

Szabados (2001) Set k − 1 � B H m ( t k ) = h ( t r , t k )[ B m ( t r + ∆ t ) − B m ( t r )] , r = −∞ and B H m ( 0 ) = 0 , with ∆ t = 2 − 2 m , t x = x ∆ t , x ∈ R , and � ( t − s ) H − 1 / 2 − ( − s ) H − 1 / 2 � h ( s , t ) = C H , s ≤ t . + Theorem For H ∈ ( 1 / 4 , 1 ) , B H m → B H as m → ∞ a.s. uniformly on compact sets. The rate of convergence is O ( n − min { H − 1 / 4 , 1 / 4 } 2 log 2 log n ) . Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 20 / 82

Contents Some Approximations of FBM 1 Transport Processes 2 Approximation of FBM by Means of Transport Processes 3 Approximatios of Fractional Stochastic Differential Equations 4 Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 21 / 82

Transport processes � � X ( n ) ( t ) , t ≥ 0 is a transport process iff : X ( n ) ( 0 ) = 0, Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 22 / 82

Transport processes � � X ( n ) ( t ) , t ≥ 0 is a transport process iff : X ( n ) ( 0 ) = 0, X ( n ) is continuous, Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 23 / 82

Transport processes � � X ( n ) ( t ) , t ≥ 0 is a transport process iff : X ( n ) ( 0 ) = 0, X ( n ) is continuous, X ( n ) is a piecewise linear function with slopes ± n , Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 24 / 82

Transport processes � � X ( n ) ( t ) , t ≥ 0 is a transport process iff : X ( n ) ( 0 ) = 0, X ( n ) is continuous, X ( n ) is a piecewise linear function with slopes ± n , The slope at 0 + is random. It is n or − n with probability 1 / 2, Jorge A. León (Cinvestav–IPN) A strong uniform approximation Roscoff 2010 25 / 82