Content

1 Wiener Integration with Respect to Fractional Brownian
Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Elements of Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Fractional Brownian Motion: Definition and Elementary
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Mandelbrot–van Ness Representation of fBm . . . . . . . . . . . . . . . . 9
1.4 Fractional Brownian Motion with H ∈ ( 1
2 , 1) on the White
Noise Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  10
1.5 Fractional Noise on White Noise Space . . . . . . . . . . . . . . . . . . . . . 12
1.6 Wiener Integration with Respect to fBm . . . . . . . . . . . . . . . . . . . 16
1.7 The Space of Gaussian Variables Generated by fBm. . . . . . . . . . 24
1.8 Representation of fBm via the Wiener Process on a Finite
Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.9 The Inequalities for the Moments of the Wiener Integrals
with Respect to fBm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.10 Maximal Inequalities for the Moments of Wiener Integrals
with Respect to fBm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.11 The Conditions of Continuity of Wiener Integrals with
Respect to fBm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.12 The Estimates of Moments of the Solution of Simple
Stochastic Differential Equations Involving fBm . . . . . . . . . . . . . 55
1.13 Stochastic Fubini Theorem for the Wiener Integrals
w.r.t fBm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
1.14 Martingale Transforms and Girsanov Theorem for Longmemory
Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
1.15 Nonsemimartingale Properties of fBm; How to Approximate
Them by Semimartingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
1.15.1 Approximation of fBm by Continuous Processes of
Bounded Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
1.15.2 Convergence BH,β → BH in Besov Space Wλ[a, b]. . . . . 73
1.15.3 Weak Convergence to fBm in the Schemes of Series . . . . 78
1.16 H¨older Properties of the Trajectories of fBm and of Wiener
Integrals w.r.t. fBm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
1.17 Estimates for Fractional Derivatives of fBm and of Wiener
Integrals w.r.t. Wiener Process via the Garsia–Rodemich–
Rumsey Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
1.18 Power Variations of fBm and of Wiener Integrals w.r.t. fBm . . 90
1.19 L′evy Theorem for fBm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
1.20 Multi-parameter Fractional Brownian Motion . . . . . . . . . . . . . . . 117
1.20.1 The Main Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
1.20.2 H¨older Properties of Two-parameter fBm . . . . . . . . . . . . . 117
1.20.3 Fractional Integrals and Fractional Derivatives of
Two-parameter Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 118
2 Stochastic Integration with Respect to fBm and Related
Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
2.1 Pathwise Stochastic Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
2.1.1 Pathwise Stochastic Integration in the Fractional
Sobolev-type Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
2.1.2 Pathwise Stochastic Integration in Fractional
Besov-type Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
2.2 Pathwise Stochastic Integration w.r.t. Multi-parameter fBm. . . 131
2.2.1 Some Additional Properties of Two-parameter
Fractional Integrals and Derivatives. . . . . . . . . . . . . . . . . . 131
2.2.2 Generalized Two-parameter Lebesgue–Stieltjes
Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  132
2.2.3 Generalized Integrals of Two-parameter fBm in the
Case of the Integrand Depending on fBm. . . . . . . . . . . . . 136
2.2.4 Pathwise Integration in Two-parameter Besov Spaces . . 136
2.2.5 The Existence of the Integrals of the Second Kind of a
Two-parameter fBm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
2.3 Wick Integration with Respect to fBm with H ∈ [1/2, 1) as
S?-integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
2.3.1 Wick Products and S?-integration . . . . . . . . . . . . . . . . . . . 141
2.3.2 Comparison of Wick and Pathwise Integrals for
“Markov” Integrands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
2.3.3 Comparison of Wick and Stratonovich Integrals for
“General” Integrands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
2.3.4 Reduction of Wick Integration w.r.t. Fractional Noise
to the Integration w.r.t. White Noise . . . . . . . . . . . . . . . . . 157
2.4 Skorohod, Forward, Backward and Symmetric Integration
w.r.t. fBm. Two Approaches to Skorohod Integration . . . . . . . . 158
2.5 Isometric Approach to Stochastic Integration with Respect
to fBm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 162
2.5.1 The Basic Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .162
2.5.2 First- and Higher-order Integrals with Respect to X . . . 164
2.5.3 Generalized Integrals with Respect to fBm. . . . . . . . . . . . 169
2.6 Stochastic Fubini Theorem for Stochastic Integrals w.r.t.
Fractional Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
2.7 The It?o Formula for Fractional Brownian Motion . . . . . . . . . . . . 182
2.7.1 The Simplest Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
2.7.2 It?o Formula for Linear Combination of Fractional
Brownian Motions with Hi ∈ [1/2, 1) in Terms of
Pathwise Integrals and It?o Integral . . . . . . . . . . . . . . . . . . 183
2.7.3 The It?o Formula in Terms of Wick Integrals . . . . . . . . . . 184
2.7.4 The It?o Formula for H ∈ (0, 1/2) . . . . . . . . . . . . . . . . . . . . 185
2.7.5 It?o Formula for Fractional Brownian Fields . . . . . . . . . . . 186
2.7.6 The It?o Formula for H ∈ (0, 1) in Terms of Isometric
Integrals, and Its Applications . . . . . . . . . . . . . . . . . . . . . . 189
2.8 The Girsanov Theorem for fBm and Its Applications . . . . . . . . . 191
2.8.1 The Girsanov Theorem for fBm . . . . . . . . . . . . . . . . . . . . . 191
2.8.2 When the Conditions of the Girsanov Theorem Are
Fulfilled? Differentiability of the Fractional Integrals . . . 193
3 Stochastic Differential Equations Involving Fractional
Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
3.1 Stochastic Differential Equations Driven by Fractional
Brownian Motion with Pathwise Integrals . . . . . . . . . . . . . . . . . . 197
3.1.1 Existence and Uniqueness of Solutions: the Results of
Nualart and Rˇa?scanu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
3.1.2 Norm and Moment Estimates of Solution . . . . . . . . . . . . . 202
3.1.3 Some Other Results on Existence and Uniqueness of
Solution of SDE Involving Processes Related to fBm
with (H ∈ (1/2,1)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
3.1.4 Some Properties of the Stochastic Differential
Equations with Stationary Coefficients . . . . . . . . . . . . . . . 206
3.1.5 Semilinear Stochastic Differential Equations Involving
Forward Integral w.r.t. fBm. . . . . . . . . . . . . . . . . . . . . . . . . 220
3.1.6 Existence and Uniqueness of Solutions of SDE with
Two-Parameter Fractional Brownian Field . . . . . . . . . . . . 223
3.2 The Mixed SDE Involving Both the Wiener Process
and fBm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
3.2.1 The Existence and Uniqueness of the Solution of the
Mixed Semilinear SDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
3.2.2 The Existence and Uniqueness of the Solution of the
Mixed SDE for fBm with H ∈ (3/4,1) . . . . . . . . . . . . . . . 227
3.2.3 The Girsanov Theorem and the Measure
Transformation for the Mixed Semilinear SDE . . . . . . . . 238
3.3 Stochastic Differential Equations with Fractional
White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  240
3.3.1 The Lipschitz and the Growth Conditions on the
Negative Norms of Coefficients . . . . . . . . . . . . . . . . . . . . . . 240
3.3.2 Quasilinear SDE with Fractional Noise . . . . . . . . . . . . . . . 241
3.4 The Rate of Convergence of Euler Approximations of
Solutions of SDE Involving fBm . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
3.4.1 Approximation of Pathwise Equations . . . . . . . . . . . . . . . . 244
3.4.2 Approximation of Quasilinear Skorohod-type
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
3.5 SDE with the Additive Wiener Integral w.r.t. Fractional
Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
3.5.1 Existence of a Weak Solution for Regular Coefficients . . 263
3.5.2 Existence of a Weak Solution for SDE with
Discontinuous Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
3.5.3 Uniqueness in Law and Pathwise Uniqueness for
Regular Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
3.5.4 Existence of a Strong Solution for the Regular Case. . . . 272
3.5.5 Existence of a Strong Solution
for Discontinuous Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
3.5.6 Estimates of Moments of Solutions for Regular Case
and H ∈ (0, 1/2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
3.5.7 The Estimates of the Norms of the Solution in the
Orlicz Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
3.5.8 The Distribution of the Supremum of the Process X
on [0, T] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
3.5.9 Modulus of Continuity of Solution of Equation
Involving Fractional Brownian Motion . . . . . . . . . . . . . . . 287
4 Filtering in Systems with Fractional Brownian Noise . . . . . . 291
4.1 Optimal Filtering of a Mixed Brownian–Fractional-Brownian
Model with Fractional Brownian Observation Noise . . . . . . . . . . 291
4.2 Optimal Filtering in Conditionally Gaussian Linear Systems
with Mixed Signal and Fractional Brownian Observation
Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 295
4.3 Optimal Filtering in Systems with Polynomial Fractional
Brownian Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 298
5 Financial Applications of Fractional Brownian Motion . . . . . 301
5.1 Discussion of the Arbitrage Problem . . . . . . . . . . . . . . . . . . . . . . . 301
5.1.1 Long-range Dependence in Economics and Finance . . . . 301
5.1.2 Arbitrage in “Pure” Fractional Brownian Model.
The Original Rogers Approach . . . . . . . . . . . . . . . . . . . . . . 302
5.1.3 Arbitrage in the “Pure” Fractional Model.
Results of Shiryaev and Dasgupta . . . . . . . . . . . . . . . . . . . 304
5.1.4 Mixed Brownian–Fractional-Brownian Model:
Absence of Arbitrage and Related Topics . . . . . . . . . . . . . 305
5.1.5 Equilibrium of Financial Market. The Fractional
Burgers Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
5.2 The Different Forms of the Black–Scholes Equation . . . . . . . . . . 322
5.2.1 The Black–Scholes Equation for the Mixed
Brownian–Fractional-Brownian Model . . . . . . . . . . . . . . . . 322
5.2.2 Discussion of the Place of Wick Products and Wick–
It?o–Skorohod Integral in the Problems of Arbitrage
and Replication in the Fractional Black–Scholes
Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
6 Statistical Inference with Fractional Brownian Motion . . . . . 327
6.1 Testing Problems for the Density Process for fBm with
Different Drifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
6.1.1 Observations Based on the Whole Trajectory with σ
and H Known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
6.1.2 Discretely Observed Trajectory and σ Unknown . . . . . . . 331
6.2 Goodness-of-fit Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
6.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
6.2.2 The Whole Trajectory Is Observed and the Parameters
μ and σ Are Known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
6.2.3 Goodness-of-fit Tests with Discrete Observations . . . . . . 337
6.2.4 On Volatility Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
6.2.5 Goodness-of-fit Test with Unknown μ and σ . . . . . . . . . . 342
6.3 Parameter Estimates in the Models Involving fBm . . . . . . . . . . 343
6.3.1 Consistency of the Drift Parameter Estimates in the
Pure Fractional Brownian Diffusion Model . . . . . . . . . . . . 344
6.3.2 Consistency of the Drift Parameter Estimates in the
Mixed Brownian–fractional-Brownian Diffusion Model
with “Linearly” Dependent Wt and BH
t . . . . . . . . . . . . . . 349
6.3.3 The Properties of Maximum Likelihood Estimates
in Diffusion Brownian–Fractional-Brownian Models
with Independent Components . . . . . . . . . . . . . . . . . . . . . . 354
A Mandelbrot–van Ness Representation: Some Related
Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
B Approximation of Beta Integrals and Estimation of Kernels . . . ..365

Index . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . 391

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