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David Nualart - One of the best experts on this subject based on the ideXlab platform.
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fractional brownian motion Stochastic Calculus and applications
2010Co-Authors: David NualartAbstract:Fractional Brownian motion (fBm) is a centered self-similar Gaussian process with stationary increments, which depends on a parameter H ∈ (0, 1) called the Hurst index. In this note we will survey some facts about the Stochastic Calculus with respect to fBm using a pathwise approach and the techniques of the Malliavin Calculus. Some applications in turbulence and finance will be discussed. Mathematics Subject Classification (2000). Primary 60H30; Secondary 60G18.
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Stochastic differential equations driven by fractional brownian motion and standard brownian motion
arXiv: Probability, 2008Co-Authors: Joao Guerra, David NualartAbstract:We prove an existence and uniqueness theorem for solutions of multidimensional, time dependent, Stochastic differential equations driven simultaneously by a multidimensional fractional Brownian motion with Hurst parameter H>1/2 and a multidimensional standard Brownian motion. The proof relies on some a priori estimates, which are obtained using the methods of fractional integration, and the classical Ito Stochastic Calculus. The existence result is based on the Yamada-Watanabe theorem.
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Stochastic integration with respect to the fractional Brownian motion
Stochastics and Stochastic Reports, 2003Co-Authors: Elisa Alòs, David NualartAbstract:We develop a Stochastic Calculus for the fractional Brownian motion with Hurst parameter H > 1/2 using the techniques of the Malliavin Calculus. We establish estimates in L p , maximal inequalities and a continuity criterion for the Stochastic integral. Finally, we derive an Ito's formula for integral processes.
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Stochastic Calculus with respect to gaussian processes
Annals of Probability, 2001Co-Authors: Elisa Alòs, Olivier Mazet, David NualartAbstract:In this paper we develop a Stochastic Calculus with respect to a Gaussian process of the form B t = ∫ t 0 K(t, s) dW s , where W is a Wiener process and K(t, s) is a square integrable kernel, using the techniques of the Stochastic Calculus of variations. We deduce change-of-variable formulas for the indefinite integrals and we study the approximation by Riemann sums. The particular case of the fractional Brownian motion is discussed.
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the malliavin Calculus and related topics
1995Co-Authors: David NualartAbstract:Analysis on the Wiener space.- Regularity of probability laws.- Anticipating Stochastic Calculus.- Transformations of the Wiener measure.- Fractional Brownian motion.- Malliavin Calculus in finance.- Malliavin Calculus in finance.
Francesco Russo - One of the best experts on this subject based on the ideXlab platform.
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functional ito versus banach space Stochastic Calculus and strict solutions of semilinear path dependent equations
Infinite Dimensional Analysis Quantum Probability and Related Topics, 2016Co-Authors: Andrea Cosso, Francesco RussoAbstract:Functional It\^o Calculus was introduced in order to expand a functional $F(t, X_{\cdot+t}, X_t)$ depending on time $t$, past and present values of the process $X$. Another possibility to expand $F(t, X_{\cdot+t}, X_t)$ consists in considering the path $X_{\cdot+t}=\{X_{x+t},\,x\in[-T,0]\}$ as an element of the Banach space of continuous functions on $C([-T,0])$ and to use Banach space Stochastic Calculus. The aim of this paper is threefold. 1) To reformulate functional It\^o Calculus, separating time and past, making use of the regularization procedures which matches more naturally the notion of horizontal derivative which is one of the tools of that Calculus. 2) To exploit this reformulation in order to discuss the (not obvious) relation between the functional and the Banach space approaches. 3) To study existence and uniqueness of smooth solutions to path-dependent partial differential equations which naturally arise in the study of functional It\^o Calculus. More precisely, we study a path-dependent equation of Kolmogorov type which is related to the window process of the solution to an It\^o Stochastic differential equation with path-dependent coefficients. We also study a semilinear version of that equation.
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functional it o versus banach space Stochastic Calculus and strict solutions of semilinear path dependent equations
arXiv: Probability, 2015Co-Authors: Andrea Cosso, Francesco RussoAbstract:Functional It\^o Calculus was introduced in order to expand a functional $F(t, X\_{\cdot+t}, X\_t)$ depending on time $t$, past and present values of the process $X$. Another possibility to expand $F(t, X\_{\cdot+t}, X\_t)$ consists in considering the path $X\_{\cdot+t}=\{X\_{x+t},\,x\in[-T,0]\}$ as an element of the Banach space of continuous functions on $C([-T,0])$ and to use Banach space Stochastic Calculus. The aim of this paper is threefold. 1) To reformulate functional It\^o Calculus, separating time and past, making use of the regularization procedures which matches more naturally the notion of horizontal derivative which is one of the tools of that Calculus. 2) To exploit this reformulation in order to discuss the (not obvious) relation between the functional and the Banach space approaches. 3) To study existence and uniqueness of smooth solutions to path-dependent partial differential equations which naturally arise in the study of functional It\^o Calculus. More precisely, we study a path-dependent equation of Kolmogorov type which is related to the window process of the solution to an It\^o Stochastic differential equation with path-dependent coefficients. We also study a semilinear version of that equation.
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wiener integrals malliavin Calculus and covariance measure structure
Journal of Functional Analysis, 2007Co-Authors: Ida Kruk, Francesco Russo, Ciprian A TudorAbstract:We introduce the notion of covariance measure structure for square integrable Stochastic processes. We define Wiener integral, we develop a suitable formalism for Stochastic Calculus of variations and we make Gaussian assumptions only when necessary. Our main examples are finite quadratic variation processes with stationary increments and the bifractional Brownian motion.
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wiener integrals malliavin Calculus and covariance measure structure
arXiv: Probability, 2006Co-Authors: Ida Kruk, Francesco Russo, Ciprian A TudorAbstract:We introduce the notion of {\em covariance measure structure} for square integrable Stochastic processes. We define Wiener integral, we develop a suitable formalism for Stochastic Calculus of variations and we make Gaussian assumptions only when necessary. Our main examples are finite quadratric variation processes with stationary increments and the bifractional Brownian motion.
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Stochastic Calculus with respect to continuous finite quadratic variation processes
Stochastics An International Journal of Probability and Stochastic Processes, 2000Co-Authors: Francesco Russo, Pierre ValloisAbstract:The quadratic variation of a continuous process (when it exists) is defined through a regularization procedure. A large class of finite quadratic variation processes is provided, with a particular emphasis on Gaussian processes. For such processes a Calculus is developed with application to the study of some Stochastic differential equations.
Francesca Biagini - One of the best experts on this subject based on the ideXlab platform.
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Stochastic Calculus for fractional brownian motion and applications
2008Co-Authors: Francesca Biagini, Yaozhong Hu, Bernt Oksendal, Tusheng ZhangAbstract:Fractional Brownian motion.- Intrinsic properties of the fractional Brownian motion.- Stochastic Calculus.- Wiener and divergence-type integrals for fractional Brownian motion.- Fractional Wick Ito Skorohod (fWIS) integrals for fBm of Hurst index H >1/2.- WickIto Skorohod (WIS) integrals for fractional Brownian motion.- Pathwise integrals for fractional Brownian motion.- A useful summary.- Applications of Stochastic Calculus.- Fractional Brownian motion in finance.- Stochastic partial differential equations driven by fractional Brownian fields.- Stochastic optimal control and applications.- Local time for fractional Brownian motion.
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a general Stochastic Calculus approach to insider trading
Applied Mathematics and Optimization, 2005Co-Authors: Francesca Biagini, Bernt OksendalAbstract:The purpose of this paper is to present a general Stochastic Calculus approach to insider trading. We consider a market driven by a standard Brownian motion $B(t)$ on a filtered probability space $\displaystyle (\Omega,\F,\left\{\F\right\}_{t\geq 0},P)$ where the coefficients are adapted to a filtration ${\Bbb G}=\left\{\G_t\right\}_{0\leq t\leq T}$, with $\F_t\subset\G_t$ for all $t\in [0,T]$, $T>0$ being a fixed terminal time. By an {\it insider} in this market we mean a person who has access to a filtration (information) $\displaystyle{\Bbb H}=\left\{\H_t\right\}_{0\leq t\leq T}$ which is strictly bigger than the filtration $\displaystyle{\Bbb G}=\left\{\G_t\right\}_{0\leq t\leq T}$. In this context an insider strategy is represented by an $\H_t$-adapted process $\phi(t)$ and we interpret all anticipating integrals as the forward integral defined in [23] and [25]. We consider an optimal portfolio problem with general utility for an insider with access to a general information $\H_t \supset\G_t$ and show that if an optimal insider portfolio $\pi^*(t)$ of this problem exists, then $B(t)$ is an $\H_t$-semimartingale, i.e. the enlargement of filtration property holds. This is a converse of previously known results in this field. Moreover, if $\pi^*$ exists we obtain an explicit expression in terms of $\pi^*$ for the semimartingale decomposition of $B(t)$ with respect to $\H_t$. This is a generalization of results in [16], [20] and [2].
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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, 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.
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.
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skorohod integration and Stochastic Calculus beyond the fractional brownian scale
Journal of Functional Analysis, 2005Co-Authors: Oana Mocioalca, Frederi ViensAbstract:Abstract We extend the Skorohod integral, allowing integration with respect to Gaussian processes that can be more irregular than any fractional Brownian motion. This is done by restricting the class of test random variables used to define Skorohod integrability. A detailed analysis of the size of this class is given; it is proved to be non-empty even for Gaussian processes which are not continuous on any closed interval. Despite the extreme irregularity of these Stochastic integrators, the Skorohod integral is shown to be uniquely defined, and to be useful: an Ito formula is established; it is employed to derive a Tanaka formula for a corresponding local time; linear additive and multiplicative Stochastic differential equations are solved; an analysis of existence for the Stochastic heat equation is given.
B. Pasik-duncan - One of the best experts on this subject based on the ideXlab platform.
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Stochastic Calculus for fractional Brownian motion. I. Theory
Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187), 2000Co-Authors: Tyrone E. Duncan, Y.z. Hu, B. Pasik-duncanAbstract:Describes some of the results in Duncan et al. (2000) for a Stochastic Calculus for a fractional Brownian motion with the Hurst parameter in the interval (1/2, 1). Two Stochastic integrals are defined with explicit expressions for their first two moments. Multiple and iterated integrals of a fractional Brownian motion are defined and various properties of these integrals are given. A square integrable functional on a probability space of a fractional Brownian motion is expressed as an infinite series of multiple integrals.
