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David Nualart - One of the best experts on this subject based on the ideXlab platform.
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rate of convergence and asymptotic error distribution of Euler approximation Schemes for fractional diffusions
Annals of Applied Probability, 2016Co-Authors: Yanghui Liu, David NualartAbstract:For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter H > 1 , it is known that the existing (naive) Euler Scheme has the rate of convergence n 1−2H. Since the limit H ! 1 of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler Scheme is the extension of the classical Euler Scheme for Ito SDEs for H = 1 , the convergence rate of the naive Euler Scheme deteriorates for H ! 1 . In this paper we introduce a new (modified Euler) approximation Scheme which is closer to the classical Euler Scheme for Stratonovich SDEs for H = 1 , and it has the rate of convergence −1 n , where n = n 2H−1/2 when H 3 4 . Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if {Xt,0 � tT} is the solution of a SDE driven by a fBm and if {X n t ,0 � tT} is its approximation obtained by the new modified Euler Scheme, then we prove that n(X n X) converges stably to the solution of a linear SDE driven by a matrix- valued Brownian motion, when H 2 ( 1 , 3 ). In the case H > 3 , we show the L p convergence of n(X n Xt), and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this Scheme. We also apply our approach to the naive Euler Scheme.
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rate of convergence and asymptotic error distribution of Euler approximation Schemes for fractional diffusions
Annals of Applied Probability, 2016Co-Authors: Yanghui Liu, David NualartAbstract:For a stochastic differential equation driven by a fractional Brownian motion with Hurst parameter H > 12 it is known that the existing (naive) Euler Scheme has the rate of convergence n1−2H , which means no convergence to zero of the error when H is formally set to 12 (the standard Brownian motion case). In this paper we introduce a new (modified Euler) approximation Scheme which is closer to the classical Euler Scheme for diffusion processes and it has the rate of convergence γ−1 n , where γn = n 2H− 12 when H 34 . In particular, the rate of convergence becomes n − 12 when H is formally set to 12 . Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if {Xt, 0 ≤ t ≤ T} is the solution of a stochastic differential equation driven by a fractional Brownian motion and if {X t , 0 ≤ t ≤ T} is its approximation obtained by the new modified Euler Scheme, then we prove that γn(X n − X) converges stably to the solution of a linear stochastic differential equation driven by a matrix-valued Brownian motion, when H ∈ ( 12 , 3 4 ]. In the case H > 34 , we show the L p convergence of n(X t − Xt) and the limiting process is identified as the solution of a linear stochastic differential equation driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this Scheme. We also apply our approach to the naive Euler Scheme. The main tools are fractional calculus, Malliavin calculus, and the fourth moment theorem.
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rate of convergence and asymptotic error distribution of Euler approximation Schemes for fractional diffusions
arXiv: Probability, 2014Co-Authors: Yanghui Liu, David NualartAbstract:For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter $H>\frac{1}{2}$, it is known that the existing (naive) Euler Scheme has the rate of convergence $n^{1-2H}$. Since the limit $H\rightarrow\frac{1}{2}$ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler Scheme is the extension of the classical Euler Scheme for Ito SDEs for $H=\frac{1}{2}$, the convergence rate of the naive Euler Scheme deteriorates for $H\rightarrow\frac{1}{2}$. In this paper we introduce a new (modified Euler) approximation Scheme which is closer to the classical Euler Scheme for Stratonovich SDEs for $H=\frac{1}{2}$, and it has the rate of convergence $\gamma_n^{-1}$, where $\gamma_n=n^{2H-{1}/2}$ when $H \frac{3}{4}$. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if $\{X_t,0\le t\le T\}$ is the solution of a SDE driven by a fBm and if $\{X_t^n,0\le t\le T\}$ is its approximation obtained by the new modified Euler Scheme, then we prove that $\gamma_n(X^n-X)$ converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when $H\in(\frac{1}{2},\frac{3}{4}]$. In the case $H>\frac{3}{4}$, we show the $L^p$ convergence of $n(X^n_t-X_t)$, and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this Scheme. We also apply our approach to the naive Euler Scheme.
Peter E Kloeden - One of the best experts on this subject based on the ideXlab platform.
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strong convergence of an explicit numerical method for sdes with nonglobally lipschitz continuous coefficients
Annals of Applied Probability, 2012Co-Authors: Martin Hutzenthaler, Arnulf Jentzen, Peter E KloedenAbstract:On the one hand, the explicit Euler Scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other hand, the implicit Euler Scheme is known to converge strongly to the exact solution of such an SDE. Implementations of the implicit Euler Scheme, however, require additional computational effort. In this article we therefore propose an explicit and easily implementable numerical method for such an SDE and show that this method converges strongly with the standard order one-half to the exact solution of the SDE. Simulations reveal that this explicit strongly convergent numerical Scheme is considerably faster than the implicit Euler Scheme.
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overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space time noise
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2009Co-Authors: Arnulf Jentzen, Peter E KloedenAbstract:We consider the numerical approximation of parabolic stochastic partial differential equations driven by additive space–time white noise. We introduce a new numerical Scheme for the time discretization of the finite-dimensional Galerkin stochastic differential equations, which we call the exponential Euler Scheme, and show that it converges (in the strong sense) faster than the classical numerical Schemes, such as the linear-implicit Euler Scheme or the Crank–Nicholson Scheme, for this equation with the general noise. In particular, we prove that our Scheme applied to a semilinear stochastic heat equation converges with an overall computational order 1/3 which exceeds the barrier order 1/6 for numerical Schemes using only basic increments of the noise process reported previously. By contrast, our Scheme takes advantage of the smoothening effect of the Laplace operator and of a linear functional of the noise and, therefore overcomes this order barrier.
Yanghui Liu - One of the best experts on this subject based on the ideXlab platform.
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rate of convergence and asymptotic error distribution of Euler approximation Schemes for fractional diffusions
Annals of Applied Probability, 2016Co-Authors: Yanghui Liu, David NualartAbstract:For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter H > 1 , it is known that the existing (naive) Euler Scheme has the rate of convergence n 1−2H. Since the limit H ! 1 of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler Scheme is the extension of the classical Euler Scheme for Ito SDEs for H = 1 , the convergence rate of the naive Euler Scheme deteriorates for H ! 1 . In this paper we introduce a new (modified Euler) approximation Scheme which is closer to the classical Euler Scheme for Stratonovich SDEs for H = 1 , and it has the rate of convergence −1 n , where n = n 2H−1/2 when H 3 4 . Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if {Xt,0 � tT} is the solution of a SDE driven by a fBm and if {X n t ,0 � tT} is its approximation obtained by the new modified Euler Scheme, then we prove that n(X n X) converges stably to the solution of a linear SDE driven by a matrix- valued Brownian motion, when H 2 ( 1 , 3 ). In the case H > 3 , we show the L p convergence of n(X n Xt), and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this Scheme. We also apply our approach to the naive Euler Scheme.
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rate of convergence and asymptotic error distribution of Euler approximation Schemes for fractional diffusions
Annals of Applied Probability, 2016Co-Authors: Yanghui Liu, David NualartAbstract:For a stochastic differential equation driven by a fractional Brownian motion with Hurst parameter H > 12 it is known that the existing (naive) Euler Scheme has the rate of convergence n1−2H , which means no convergence to zero of the error when H is formally set to 12 (the standard Brownian motion case). In this paper we introduce a new (modified Euler) approximation Scheme which is closer to the classical Euler Scheme for diffusion processes and it has the rate of convergence γ−1 n , where γn = n 2H− 12 when H 34 . In particular, the rate of convergence becomes n − 12 when H is formally set to 12 . Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if {Xt, 0 ≤ t ≤ T} is the solution of a stochastic differential equation driven by a fractional Brownian motion and if {X t , 0 ≤ t ≤ T} is its approximation obtained by the new modified Euler Scheme, then we prove that γn(X n − X) converges stably to the solution of a linear stochastic differential equation driven by a matrix-valued Brownian motion, when H ∈ ( 12 , 3 4 ]. In the case H > 34 , we show the L p convergence of n(X t − Xt) and the limiting process is identified as the solution of a linear stochastic differential equation driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this Scheme. We also apply our approach to the naive Euler Scheme. The main tools are fractional calculus, Malliavin calculus, and the fourth moment theorem.
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rate of convergence and asymptotic error distribution of Euler approximation Schemes for fractional diffusions
arXiv: Probability, 2014Co-Authors: Yanghui Liu, David NualartAbstract:For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter $H>\frac{1}{2}$, it is known that the existing (naive) Euler Scheme has the rate of convergence $n^{1-2H}$. Since the limit $H\rightarrow\frac{1}{2}$ of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler Scheme is the extension of the classical Euler Scheme for Ito SDEs for $H=\frac{1}{2}$, the convergence rate of the naive Euler Scheme deteriorates for $H\rightarrow\frac{1}{2}$. In this paper we introduce a new (modified Euler) approximation Scheme which is closer to the classical Euler Scheme for Stratonovich SDEs for $H=\frac{1}{2}$, and it has the rate of convergence $\gamma_n^{-1}$, where $\gamma_n=n^{2H-{1}/2}$ when $H \frac{3}{4}$. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if $\{X_t,0\le t\le T\}$ is the solution of a SDE driven by a fBm and if $\{X_t^n,0\le t\le T\}$ is its approximation obtained by the new modified Euler Scheme, then we prove that $\gamma_n(X^n-X)$ converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when $H\in(\frac{1}{2},\frac{3}{4}]$. In the case $H>\frac{3}{4}$, we show the $L^p$ convergence of $n(X^n_t-X_t)$, and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this Scheme. We also apply our approach to the naive Euler Scheme.
Gilles Pages - One of the best experts on this subject based on the ideXlab platform.
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Recursive computation of invariant distributions of Feller processes
Stochastic Processes and their Applications, 2020Co-Authors: Gilles Pages, Clément ReyAbstract:Abstract This paper provides a general and abstract approach to compute invariant distributions for Feller processes. More precisely, we show that the recursive algorithm presented in Lamberton and Pages (2002) and based on simulation algorithms of stochastic Schemes with decreasing steps can be used to build invariant measures for general Feller processes. We also propose various applications: Approximation of Markov Brownian diffusion stationary regimes with a Milstein or an Euler Scheme and approximation of a Markov switching Brownian diffusion stationary regimes using an Euler Scheme.
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An optimal markovian quantization algorithm for multidimensional stochastic control problems
2020Co-Authors: Gilles Pages, Huyên Pham, Université Paris, Jacques PrintemsAbstract:Abstract We propose a probabilistic numerical method based on optimal quantization to solve some multidimensional stochastic control problems that arise, for example, in Mathematical Finance for portfolio optimization. We then consider some controlled diffusions with most control free components. The Euler Scheme of the uncontrolled diffusion part is approximated by a discrete time process obtained by a nearest neighbor projection on some grids optimally fitted to its dynamics. The resulting process is also designed to preserve the Markov property with respect to the filtration of the Euler Scheme. This Markovian quantization approach leads to an approximate control problem that can be solved numerically by the dynamic programming formula. This approach seems promising in higher dimension. A priori L p -error bounds are stated and we show that the spatial discretization error term is minimal at some specific grids. A simple recursive algorithm is devised to compute these optimal grids by induction based on a Monte Carlo simulation. Some numerical illustrations are processed for solving a mean-variance hedging problem
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discretization Scheme s of a brownian diffusion
2018Co-Authors: Gilles PagesAbstract:This chapter is devoted to the discretization Schemes of the solutions of a stochastic differential equation driven by a Brownian motion (diffusion): the (discrete time and continuous) Euler Scheme and the Milstein Scheme. The existence of moments, the strong (or pathwise) convergence rate in \(L_{p}\) (and a:s:) of both Schemes are established under Lipschitz assumptions of the diffusion coeffcients (Euler Scheme) or of their partial derivatives (Milstein Scheme). Several other important properties of these Schemes are established (e.g., conditions for the simulability of the Milstein Scheme in higher dimension, Lipschitz property of the flow, etc). The main weak error results for the Euler Scheme, either under smoothness (Talay–Tubaro) or ellipticity (Bally–Talay) assumptions, are stated, with a detailed proof in the first setting. Applications to the Richardson-Romberg extrapolation to reduce the bias in Monte Carlo simulations is presented and illustrated in an example.
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the diffusion bridge method application to path dependent options ii
2018Co-Authors: Gilles PagesAbstract:This chapter provides a (partial) answer to the following question: can we simulate the continuous – or genuine – Euler Scheme? To this end, we first investigate the Brownian bridge and its avatar for diffusion processes which. It allows to simulate in an exat way some functionals of the genuine Euler Scheme involving its maximum or its minimum over a given time interval and provide sharper approximations of functionals involving time integrals. Several first order weak error are stated with precise references. Applications to several families of path-dependent European options (Asian, lookback, barrier) are given, including some variance reduction methods for barrier options.
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recursive computation of the invariant distributions of feller processes original applications
2017Co-Authors: Gilles PagesAbstract:In this paper, we show that the abstract framework developed in \cite{Pages_Rey_2017} and inspired by \cite{Lamberton_Pages_2002} can be used to build invariant distributions for Brownian diffusion processes using the Milstein Scheme and for diffusion processes with censored jump using the Euler Scheme. Both studies rely on a weakly mean reverting setting for both cases. For the Milstein Scheme we prove the convergence for test functions with polynomial (Wasserstein convergence) and exponential growth. For the Euler Scheme of diffusion processes with censored jump we prove the convergence for test functions with polynomial growth.
Arnulf Jentzen - One of the best experts on this subject based on the ideXlab platform.
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on stochastic differential equations with arbitrary slow convergence rates for strong approximation
arXiv: Numerical Analysis, 2015Co-Authors: Arnulf Jentzen, Thomas Mullergronbach, Larisa YaroslavtsevaAbstract:In the recent article [Hairer, M., Hutzenthaler, M., Jentzen, A., Loss of regularity for Kolmogorov equations, Ann. Probab. 43 (2015), no. 2, 468--527] it has been shown that there exist stochastic differential equations (SDEs) with infinitely often differentiable and globally bounded coefficients such that the Euler Scheme converges to the solution in the strong sense but with no polynomial rate. Hairer et al.'s result naturally leads to the question whether this slow convergence phenomenon can be overcome by using a more sophisticated approximation method than the simple Euler Scheme. In this article we answer this question to the negative. We prove that there exist SDEs with infinitely often differentiable and globally bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion converges in absolute mean to the solution with a polynomial rate. Even worse, we prove that for every arbitrarily slow convergence speed there exist SDEs with infinitely often differentiable and globally bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion can converge in absolute mean to the solution faster than the given speed of convergence.
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strong convergence of an explicit numerical method for sdes with nonglobally lipschitz continuous coefficients
Annals of Applied Probability, 2012Co-Authors: Martin Hutzenthaler, Arnulf Jentzen, Peter E KloedenAbstract:On the one hand, the explicit Euler Scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other hand, the implicit Euler Scheme is known to converge strongly to the exact solution of such an SDE. Implementations of the implicit Euler Scheme, however, require additional computational effort. In this article we therefore propose an explicit and easily implementable numerical method for such an SDE and show that this method converges strongly with the standard order one-half to the exact solution of the SDE. Simulations reveal that this explicit strongly convergent numerical Scheme is considerably faster than the implicit Euler Scheme.
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overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space time noise
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2009Co-Authors: Arnulf Jentzen, Peter E KloedenAbstract:We consider the numerical approximation of parabolic stochastic partial differential equations driven by additive space–time white noise. We introduce a new numerical Scheme for the time discretization of the finite-dimensional Galerkin stochastic differential equations, which we call the exponential Euler Scheme, and show that it converges (in the strong sense) faster than the classical numerical Schemes, such as the linear-implicit Euler Scheme or the Crank–Nicholson Scheme, for this equation with the general noise. In particular, we prove that our Scheme applied to a semilinear stochastic heat equation converges with an overall computational order 1/3 which exceeds the barrier order 1/6 for numerical Schemes using only basic increments of the noise process reported previously. By contrast, our Scheme takes advantage of the smoothening effect of the Laplace operator and of a linear functional of the noise and, therefore overcomes this order barrier.