Fractional Brownian motion Jorge A. Len Departamento de Control - PowerPoint PPT Presentation

Fractional Brownian motion Jorge A. León Departamento de Control Automático Cinvestav del IPN Spring School “Stochastic Control in Finance”, Roscoff 2010 Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 1 / 62

Contents Introduction 1 FBM and Some Properties 2 Integral Representation 3 Wiener Integrals 4 Malliavin Calculus 5 Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 2 / 62

Contents Introduction 1 FBM and Some Properties 2 Integral Representation 3 Wiener Integrals 4 Malliavin Calculus 5 Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 3 / 62

Stochastic integration We consider � T 0 · dB s . Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 4 / 62

Stochastic integration We consider � T 0 · dB s . Here B is a fractional Brownian motion. Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 5 / 62

Contents Introduction 1 FBM and Some Properties 2 Integral Representation 3 Wiener Integrals 4 Malliavin Calculus 5 Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 6 / 62

Fractional Brownian motion Definition A Gaussian stochastic process B = { B t ; t ≥ 0 } is called a fractional Brownian motion (fBm) of Hurst parameter H ∈ ( 0 , 1 ) if it has zero mean and covariance fuction R H ( t , s ) = E ( B t B s ) = 1 t 2 H + s 2 H − | t − s | 2 H � � . 2 Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 7 / 62

Properties of fBm Definition A Gaussian stochastic process B = { B t ; t ≥ 0 } is called a fractional Brownian motion (fBm) of Hurst parameter H ∈ ( 0 , 1 ) if it has zero mean and covariance fuction R H ( t , s ) = E ( B t B s ) = 1 t 2 H + s 2 H − | t − s | 2 H � � . 2 B is a Brownian motion for H = 1 / 2. Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 8 / 62

Properties of fBm Definition A Gaussian stochastic process B = { B t ; t ≥ 0 } is called a fractional Brownian motion (fBm) of Hurst parameter H ∈ ( 0 , 1 ) if it has zero mean and covariance fuction R H ( t , s ) = E ( B t B s ) = 1 t 2 H + s 2 H − | t − s | 2 H � � . 2 B is a Brownian motion for H = 1 / 2. B has stationary increments. Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 9 / 62

Properties of fBm Definition A Gaussian stochastic process B = { B t ; t ≥ 0 } is called a fractional Brownian motion (fBm) of Hurst parameter H ∈ ( 0 , 1 ) if it has zero mean and covariance fuction R H ( t , s ) = E ( B t B s ) = 1 t 2 H + s 2 H − | t − s | 2 H � � . 2 B is a Brownian motion for H = 1 / 2. B has stationary increments : � | B t − B s | 2 � = | t − s | 2 H . E Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 10 / 62

Properties of fBm Definition A Gaussian stochastic process B = { B t ; t ≥ 0 } is called a fractional Brownian motion (fBm) of Hurst parameter H ∈ ( 0 , 1 ) if it has zero mean and covariance fuction R H ( t , s ) = E ( B t B s ) = 1 t 2 H + s 2 H − | t − s | 2 H � � . 2 B is a Brownian motion for H = 1 / 2. B has stationary increments : � | B t − B s | 2 � = | t − s | 2 H . E For any ε ∈ ( 0 , H ) and T > 0, there exists G ε, T such that | B t − B s | ≤ G ε, T | t − s | H − ε , t , s ∈ [ 0 , T ] . Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 11 / 62

Properties of fBm Definition A Gaussian stochastic process B = { B t ; t ≥ 0 } is called a fractional Brownian motion (fBm) of Hurst parameter H ∈ ( 0 , 1 ) if it has zero mean and covariance fuction R H ( t , s ) = E ( B t B s ) = 1 t 2 H + s 2 H − | t − s | 2 H � � . 2 B is a Brownian motion for H = 1 / 2. B has stationary increments. B is Hölder continuous for any exponent less than H . B is self-similar (with index H ). That is, for any a > 0, { a − H B at ; t ≥ 0 } and { B t ; t ≥ 0 } have the same distribution. Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 12 / 62

Properties of fBm Definition A Gaussian stochastic process B = { B t ; t ≥ 0 } is called a fractional Brownian motion (fBm) of Hurst parameter H ∈ ( 0 , 1 ) if it has zero mean and covariance fuction R H ( t , s ) = E ( B t B s ) = 1 t 2 H + s 2 H − | t − s | 2 H � � . 2 B has stationary increments. B is Hölder continuous for any exponent less than H . B is self-similar (with index H ). That is, for any a > 0, { a − H B at ; t ≥ 0 } and { B t ; t ≥ 0 } have the same distribution. The covariance of its increments on intervals decays asymptotically as a negative power of the distance between the intervals. Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 13 / 62

Properties of fBm B is a Brownian motion for H = 1 / 2. B has stationary increments. B is Hölder continuous for any exponent less than H . B is self-similar (with index H ). That is, for any a > 0, { a − H B at ; t ≥ 0 } and { B t ; t ≥ 0 } have the same distribution. The covariance of its increments on intervals decays asymptotically as a negative power of the distance between the intervals : Let t − s = nh and ρ H ( n ) = E [( B t + h − B t )( B s + h − B s )] h 2 H H ( 2 H − 1 ) n 2 H − 2 → 0 . ≈ Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 14 / 62

Properties of fBm B is a Brownian motion for H = 1 / 2. B has stationary increments. B is Hölder continuous for any exponent less than H . B is self-similar (with index H ). That is, for any a > 0, { a − H B at ; t ≥ 0 } and { B t ; t ≥ 0 } have the same distribution. The covariance of its increments on intervals decays asymptotically as a negative power of the distance between the intervals : Let t − s = nh and ρ H ( n ) = E [( B t + h − B t )( B s + h − B s )] h 2 H H ( 2 H − 1 ) n 2 H − 2 → 0 . ≈ i) If H > 1 / 2, ρ H ( n ) > 0 and � ∞ n = 1 ρ ( n ) = ∞ . Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 15 / 62

Properties of fBm B is a Brownian motion for H = 1 / 2. B has stationary increments. B is Hölder continuous for any exponent less than H . B is self-similar (with index H ). That is, for any a > 0, { a − H B at ; t ≥ 0 } and { B t ; t ≥ 0 } have the same distribution. The covariance of its increments on intervals decays asymptotically as a negative power of the distance between the intervals : Let t − s = nh and ρ H ( n ) = E [( B t + h − B t )( B s + h − B s )] h 2 H H ( 2 H − 1 ) n 2 H − 2 → 0 . ≈ i) If H > 1 / 2, ρ H ( n ) > 0 and � ∞ n = 1 ρ ( n ) = ∞ . ii) If H < 1 / 2, ρ H ( n ) < 0 and � ∞ n = 1 ρ ( n ) < ∞ . Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 16 / 62

Properties of fBm B is a Brownian motion for H = 1 / 2. B has stationary increments. B is Hölder continuous for any exponent less than H . B is self-similar (with index H ). That is, for any a > 0, { a − H B at ; t ≥ 0 } and { B t ; t ≥ 0 } have the same distribution. The covariance of its increments on intervals decays asymptotically as a negative power of the distance between the intervals : Let t − s = nh and ρ H ( n ) = E [( B t + h − B t )( B s + h − B s )] h 2 H H ( 2 H − 1 ) n 2 H − 2 → 0 . ≈ i) If H > 1 / 2, ρ H ( n ) > 0 and � ∞ n = 1 ρ ( n ) = ∞ . ii) If H < 1 / 2, ρ H ( n ) < 0 and � ∞ n = 1 ρ ( n ) < ∞ . B has no bounded variation paths. Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 17 / 62

FBM is not a semimartingale Theorem B is not a semimartingale for H � = 1 / 2 . Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 18 / 62

FBM is not a semimartingale Theorem B is not a semimartingale for H � = 1 / 2 . Proof : (i) Case H > 1 / 2. Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 19 / 62

FBM is not a semimartingale Theorem B is not a semimartingale for H � = 1 / 2 . Proof : (i) Case H > 1 / 2. Let Π t = { 0 = t 0 < t 1 < . . . < t n = t } be a partition of [ 0 , t ] . Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 20 / 62

FBM is not a semimartingale Theorem B is not a semimartingale for H � = 1 / 2 . Proof : (i) Case H > 1 / 2. Let Π t = { 0 = t 0 < t 1 < . . . < t n = t } be a partition of [ 0 , t ] . Then, � n n � | B t i − B t i − 1 | 2 | t i − t i − 1 | 2 H � � E = i = 1 i = 1 n | Π | 2 H − 1 � ≤ | t i − t i − 1 | i = 1 t | Π | 2 H − 1 → 0 = Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 21 / 62

FBM is not a semimartingale Theorem B is not a semimartingale for H � = 1 / 2 . Proof : (i) Case H > 1 / 2. Let Π t = { 0 = t 0 < t 1 < . . . < t n = t } be a partition of [ 0 , t ] . Then, � n � n � | B t i − B t i − 1 | 2 � | t i − t i − 1 | 2 H E = i = 1 i = 1 n | Π | 2 H − 1 � ≤ | t i − t i − 1 | i = 1 t | Π | 2 H − 1 → 0 = If B were a semimartingale. Then, B = M + V . Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 22 / 62

FBM is not a semimartingale Theorem B is not a semimartingale for H � = 1 / 2 . Proof : (i) Case H > 1 / 2. Let Π t = { 0 = t 0 < t 1 < . . . < t n = t } be a partition of [ 0 , t ] . Then, � n � n � | B t i − B t i − 1 | 2 � | t i − t i − 1 | 2 H = E i = 1 i = 1 n | Π | 2 H − 1 � ≤ | t i − t i − 1 | i = 1 t | Π | 2 H − 1 → 0 = If B were a semimartingale. Then, B = M + V . Thus 0 = [ B ] = [ M ] . Jorge A. León (Cinvestav–IPN) FBM Roscoff 2010 23 / 62