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Jeremie Unterberger - One of the best experts on this subject based on the ideXlab platform.
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Stochastic calculus for fractional brownian motion with hurst exponent h a rough path method by analytic extension
Annals of Probability, 2009Co-Authors: Jeremie UnterbergerAbstract:The d-dimensional fractional Brownian motion (FBM for short) B t = ((B (1) t ,.., B (d) t ), t ∈ R) with Hurst exponent α, α ∈ (0, 1), is a d-dimensional centered, self-similar Gaussian process with covariance E[B (i) s B (i) t ] = 1 2 δi,j (lsl 2α + ltl 2α ― |t ― s | 2α ). The long-standing problem of defining a Stochastic Integration with respect to FBM (and the related problem of solving Stochastic differential equations driven by FBM) has been addressed successfully by several different methods, although in each case with a restriction on the range of either d or α. The case α = ½ corresponds to the usual Stochastic Integration with respect to Brownian motion, while most computations become singular when α gets under various threshhold values, due to the growing irregularity of the trajectories as α → 0. We provide here a new method valid for any d and for α > 1/4 by constructing an approximation Γ(e) t , e → 0, of FBM which allows to define iterated integrals, and then applying the geometric rough path theory. The approximation relies on the definition of an analytic process Γ z on the cut plane z ∈ C \ R of which FBM appears to be a boundary value, and allows to understand very precisely the well-known (see [5]) but as yet a little mysterious divergence of Levy's area for α → 1/4
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Stochastic calculus for fractional brownian motion with hurst exponent h a rough path method by analytic extension
Annals of Probability, 2009Co-Authors: Jeremie UnterbergerAbstract:The d-dimensional fractional Brownian motion (FBM for short) Bt=((Bt(1), …, Bt(d)), t∈ℝ) with Hurst exponent α, α∈(0, 1), is a d-dimensional centered, self-similar Gaussian process with covariance ${\mathbb{E}}[B_{s}^{(i)}B_{t}^{(j)}]=\frac{1}{2}\delta_{i,j}(|s|^{2\alpha}+|t|^{2\alpha}-|t-s|^{2\alpha})$. The long-standing problem of defining a Stochastic Integration with respect to FBM (and the related problem of solving Stochastic differential equations driven by FBM) has been addressed successfully by several different methods, although in each case with a restriction on the range of either d or α. The case α=½ corresponds to the usual Stochastic Integration with respect to Brownian motion, while most computations become singular when α gets under various threshhold values, due to the growing irregularity of the trajectories as α→0. We provide here a new method valid for any d and for α>¼ by constructing an approximation Γ(ɛ)t, ɛ→0, of FBM which allows to define iterated integrals, and then applying the geometric rough path theory. The approximation relies on the definition of an analytic process Γz on the cut plane z∈ℂ∖ℝ of which FBM appears to be a boundary value, and allows to understand very precisely the well-known (see [5]) but as yet a little mysterious divergence of Levy’s area for α→¼.
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Stochastic calculus for fractional brownian motion with hurst exponent h 1 4 a rough path method by analytic extension
arXiv: Probability, 2007Co-Authors: Jeremie UnterbergerAbstract:The $d$-dimensional fractional Brownian motion (FBM for short) $B_t=((B_t^{(1)},...,B_t^{(d)}),t\in\mathbb{R})$ with Hurst exponent $\alpha$, $\alpha\in(0,1)$, is a $d$-dimensional centered, self-similar Gaussian process with covariance ${\mathbb{E}}[B_s^{(i)}B _t^{(j)}]={1/2}\delta_{i,j}(|s|^{2\alpha}+|t|^{2\alpha}-|t-s|^{2 \alpha}).$ The long-standing problem of defining a Stochastic Integration with respect to FBM (and the related problem of solving Stochastic differential equations driven by FBM) has been addressed successfully by several different methods, although in each case with a restriction on the range of either $d$ or $\alpha$. The case $\alpha={1/2}$ corresponds to the usual Stochastic Integration with respect to Brownian motion, while most computations become singular when $\alpha$ gets under various threshhold values, due to the growing irregularity of the trajectories as $\alpha\to0$. We provide here a new method valid for any $d$ and for $\alpha>{1/4}$ by constructing an approximation $\Gamma(\varepsilon)_t$, $\varepsilon\to0$, of FBM which allows to define iterated integrals, and then applying the geometric rough path theory. The approximation relies on the definition of an analytic process $\Gamma_z$ on the cut plane $z\in\mathbb{C}\setminus\mathbb{R}$ of which FBM appears to be a boundary value, and allows to understand very precisely the well-known (see \citeCQ02) but as yet a little mysterious divergence of L\'evy's area for $\alpha\to{1/4}$.
Bernt Oksendal - One of the best experts on this subject based on the ideXlab platform.
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malliavin calculus for levy processes with applications to finance
2008Co-Authors: Giulia Di Nunno, Bernt Oksendal, Frank ProskeAbstract:The Continuous Case: Brownian Motion.- The Wiener-Ito Chaos Expansion.- The Skorohod Integral.- Malliavin Derivative via Chaos Expansion.- Integral Representations and the Clark-Ocone formula.- White Noise, the Wick Product, and Stochastic Integration.- The Hida-Malliavin Derivative on the Space ? = S?(?).- The Donsker Delta Function and Applications.- The Forward Integral and Applications.- The Discontinuous Case: Pure Jump Levy Processes.- A Short Introduction to Levy Processes.- The Wiener-Ito Chaos Expansion.- Skorohod Integrals.- The Malliavin Derivative.- Levy White Noise and Stochastic Distributions.- The Donsker Delta Function of a Levy Process and Applications.- The Forward Integral.- Applications to Stochastic Control: Partial and Inside Information.- Regularity of Solutions of SDEs Driven by Levy Processes.- Absolute Continuity of Probability Laws.
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an introduction to white noise theory and malliavin calculus for fractional brownian motion
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2004Co-Authors: Francesca Biagini, Bernt Oksendal, Agnes Sulem, Naomi WallnerAbstract:Fractional Brownian motion (FBM) with Hurst parameter index between 0 and 1 is a Stochastic process originally introduced by Kolmogorov in a study of turbulence. Many other applications have subsequently been suggested. In order to obtain good mathematical models based on FBM, it is necessary to have a Stochastic calculus for such processes. The purpose of this paper is to give an introduction to this newly developed theory of Stochastic Integration for FBM based on white-noise theory and (Malliavin–type) differentiation.
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fractional white noise calculus and applications to finance
Infinite Dimensional Analysis Quantum Probability and Related Topics, 2003Co-Authors: Yaozhong Hu, Bernt OksendalAbstract:The purpose of this paper is to develop a fractional white noise calculus and to apply this to markets modeled by (Wick–) Ito type of Stochastic differential equations driven by fractional Brownian motion BH(t); 1/2 < H < 1. We show that if we use an Ito type of Stochastic Integration with respect to BH(t) (as developed in Ref. 8), then the corresponding Ito fractional Black–Scholes market has no arbitrage, contrary to the situation when the pathwise Integration is used. Moreover, we prove that our Ito fractional Black–Scholes market is complete and we compute explicitly the price and replicating portfolio of a European option in this market. The results are compared to the classical results based on standard Brownian motion B(t).
Yonggang Zhang - One of the best experts on this subject based on the ideXlab platform.
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robust student s t based Stochastic cubature filter for nonlinear systems with heavy tailed process and measurement noises
IEEE Access, 2017Co-Authors: Yulong Huang, Yonggang ZhangAbstract:In this paper, a new robust Student’s t-based Stochastic cubature filter (RSTSCF) is proposed for a nonlinear state–space model with heavy-tailed process and measurement noises. The heart of the RSTSCF is a Stochastic Student’s t-spherical radial cubature rule (SSTSRCR), which is derived based on the third-degree unbiased spherical rule and the proposed third-degree unbiased radial rule. The existing Stochastic Integration rule is a special case of the proposed SSTSRCR when the degrees of freedom parameter tends to infinity. The proposed filter is applied to a maneuvering bearings-only tracking example, in which an agile target is tracked and the bearing is observed in clutter. Simulation results show that the proposed RSTSCF can achieve higher estimation accuracy than the existing Gaussian approximate filter, Gaussian sum filter, Huber-based nonlinear Kalman filter, maximum correntropy criterion-based Kalman filter, and robust Student’s t-based nonlinear filters, and is computationally much more efficient than the existing particle filter.
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robust student s t based Stochastic cubature filter for nonlinear systems with heavy tailed process and measurement noises
arXiv: Applications, 2017Co-Authors: Yulong Huang, Yonggang ZhangAbstract:In this paper, a new robust Student's t based Stochastic cubature filter (RSTSCF) is proposed for nonlinear state-space model with heavy-tailed process and measurement noises. The heart of the RSTSCF is a Stochastic Student's t spherical radial cubature rule (SSTSRCR), which is derived based on the third-degree unbiased spherical rule and the proposed third-degree unbiased radial rule. The existing Stochastic Integration rule is a special case of the proposed SSTSRCR when the degrees of freedom parameter tends to infinity. The proposed filter is applied to a manoeuvring bearings-only tracking example, where an agile target is tracked and the bearing is observed in clutter. Simulation results show that the proposed RSTSCF can achieve higher estimation accuracy than the existing Gaussian approximate filter, Gaussian sum filter, Huber-based nonlinear Kalman filter, maximum correntropy criterion based Kalman filter, and robust Student's t based nonlinear filters, and is computationally much more efficient than the existing particle filter.
H J Kappen - One of the best experts on this subject based on the ideXlab platform.
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path integrals and symmetry breaking for optimal control theory
Journal of Statistical Mechanics: Theory and Experiment, 2005Co-Authors: H J KappenAbstract:This paper considers linear-quadratic control of a non-linear dynamical system subject to arbitrary cost. I show that for this class of Stochastic control problems the non-linear Hamilton–Jacobi–Bellman equation can be transformed into a linear equation. The transformation is similar to the transformation used to relate the classical Hamilton–Jacobi equation to the Schrodinger equation. As a result of the linearity, the usual backward computation can be replaced by a forward diffusion process that can be computed by Stochastic Integration or by the evaluation of a path integral. It is shown how in the deterministic limit the Pontryagin minimum principle formalism is recovered. The significance of the path integral approach is that it forms the basis for a number of efficient computational methods, such as Monte Carlo sampling, the Laplace approximation and the variational approximation. We show the effectiveness of the first two methods in a number of examples. Examples are given that show the qualitative difference between Stochastic and deterministic control and the occurrence of symmetry breaking as a function of the noise.
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path integrals and symmetry breaking for optimal control theory
arXiv: General Physics, 2005Co-Authors: H J KappenAbstract:This paper considers linear-quadratic control of a non-linear dynamical system subject to arbitrary cost. I show that for this class of Stochastic control problems the non-linear Hamilton-Jacobi-Bellman equation can be transformed into a linear equation. The transformation is similar to the transformation used to relate the classical Hamilton-Jacobi equation to the Schr\"odinger equation. As a result of the linearity, the usual backward computation can be replaced by a forward diffusion process, that can be computed by Stochastic Integration or by the evaluation of a path integral. It is shown, how in the deterministic limit the PMP formalism is recovered. The significance of the path integral approach is that it forms the basis for a number of efficient computational methods, such as MC sampling, the Laplace approximation and the variational approximation. We show the effectiveness of the first two methods in number of examples. Examples are given that show the qualitative difference between Stochastic and deterministic control and the occurrence of symmetry breaking as a function of the noise.
Frederi Viens - One of the best experts on this subject based on the ideXlab platform.
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Statistical aspects of the fractional Stochastic calculus
Annals of Statistics, 2007Co-Authors: Ciprian Tudor, Frederi ViensAbstract:We apply the techniques of Stochastic Integration with respect to the fractional Brownian motion and the Gaussian theory of regularity and supremum estimation to study the maximum likelihood estimator (MLE) for the drift parameter of Stochastic processes satisfying Stochastic equations driven by fractional Brownian motion with any level of H\"{o}lder-regularity (any \emph{Hurst} parameter). We prove existence and strong consistency of the MLE for linear and nonlinear equations. We\ also prove that a basic discretized version of the MLE, is still a strongly consistent estimator.
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statistical aspects of the fractional Stochastic calculus
arXiv: Statistics Theory, 2006Co-Authors: Ciprian A. Tudor, Frederi ViensAbstract:We apply the techniques of Stochastic Integration with respect to fractional Brownian motion and the theory of regularity and supremum estimation for Stochastic processes to study the maximum likelihood estimator (MLE) for the drift parameter of Stochastic processes satisfying Stochastic equations driven by a fractional Brownian motion with any level of H\"{o}lder-regularity (any Hurst parameter). We prove existence and strong consistency of the MLE for linear and nonlinear equations. We also prove that a version of the MLE using only discrete observations is still a strongly consistent estimator.
