The Experts below are selected from a list of 53721 Experts worldwide ranked by ideXlab platform
Ryan Kurniawan - One of the best experts on this subject based on the ideXlab platform.
-
Weak Convergence Rates for Euler-Type Approximations of Semilinear Stochastic Evolution Equations with Nonlinear Diffusion Coefficients
2020Co-Authors: Arnulf Jentzen, Ryan KurniawanAbstract:Strong Convergence rates for time-discrete numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the literature. Weak Convergence rates for time-discrete numerical approximations of such SEEs have, loosely speaking, been investigated since 2003 and are far away from being well understood: roughly speaking, no essentially sharp Weak Convergence rates are known for time-discrete numerical approximations of parabolic SEEs with nonlinear diffusion coefficient functions. In the recent article (Conus et al. in Ann Appl Probab 29(2):653–716, 2019 ) this Weak Convergence problem has been solved in the case of spatial spectral Galerkin approximations for semilinear SEEs with nonlinear diffusion coefficient functions. In this article we overcome this Weak Convergence problem in the case of a class of time-discrete Euler-type approximation methods (including exponential and linear-implicit Euler approximations as special cases) and, in particular, we establish essentially sharp Weak Convergence rates for linear-implicit Euler approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Key ingredients of our approach are applications of a mild Itô-type formula and the use of suitable semilinear integrated counterparts of the time-discrete numerical approximation processes.
-
Weak Convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients
2015Co-Authors: Arnulf Jentzen, Ryan KurniawanAbstract:Strong Convergence rates for time-discrete numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the literature. Weak Convergence rates for time-discrete numerical approximations of such SEEs have been investigated since about 12 years and are far away from being well understood: roughly speaking, no essentially sharp Weak Convergence rates are known for time-discrete numerical approximations of parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3 in [A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp. 80 (2011), no. 273, 89-117] for details. In the recent article [D. Conus, A. Jentzen & R. Kurniawan, Weak Convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients, arXiv:1408.1108] the Weak Convergence problem emerged from Debussche's article has been solved in the case of spatial spectral Galerkin approximations for semilinear SEEs with nonlinear diffusion coefficient functions. In this article we overcome the problem emerged from Debussche's article in the case of a class of time-discrete Euler-type approximation methods (including exponential and linear-implicit Euler approximations as special cases) and, in particular, we establish essentially sharp Weak Convergence rates for linear-implicit Euler approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Key ingredients of our approach are applications of a mild It\^o type formula and the use of suitable semilinear integrated counterparts of the time-discrete numerical approximation processes.
-
Weak Convergence rates of spectral galerkin approximations for spdes with nonlinear diffusion coefficients
2014Co-Authors: Daniel Conus, Arnulf Jentzen, Ryan KurniawanAbstract:Strong Convergence rates for (temporal, spatial, and noise) numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the scientific literature. Weak Convergence rates for numerical approximations of such SEEs have been investigated since about 11 years and are far away from being well understood: roughly speaking, no essentially sharp Weak Convergence rates are known for parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3 in [A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp. 80 (2011), no. 273, 89-117] for details. In this article we solve the Weak Convergence problem emerged from Debussche's article in the case of spectral Galerkin approximations and establish essentially sharp Weak Convergence rates for spatial spectral Galerkin approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Our solution to the Weak Convergence problem does not use Malliavin calculus. Rather, key ingredients in our solution to the Weak Convergence problem emerged from Debussche's article are the use of appropriately modified versions of the spatial Galerkin approximation processes and applications of a mild Ito type formula for solutions and numerical approximations of semilinear SEEs. This article solves the Weak Convergence problem emerged from Debussche's article merely in the case of spatial spectral Galerkin approximations instead of other more complicated numerical approximations. Our method of proof extends, however, to a number of other kind of spatial, temporal, and noise numerical approximations for semilinear SEEs.
-
Weak Convergence rates of spectral galerkin approximations for spdes with nonlinear diffusion coefficients
2014Co-Authors: Daniel Conus, Arnulf Jentzen, Ryan KurniawanAbstract:Strong Convergence rates for (temporal, spatial, and noise) numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the scientific literature. Weak Convergence rates for numerical approximations of such SEEs have been investigated for about two decades and are far away from being well understood: roughly speaking, no essentially sharp Weak Convergence rates are known for parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3 in [Math. Comp. 80 (2011) 89–117] for details. In this article, we solve the Weak Convergence problem emerged from Debussche’s article in the case of spectral Galerkin approximations and establish essentially sharp Weak Convergence rates for spatial spectral Galerkin approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Our solution to the Weak Convergence problem does not use Malliavin calculus. Rather, key ingredients in our solution to the Weak Convergence problem emerged from Debussche’s article are the use of appropriately modified versions of the spatial Galerkin approximation processes and applications of a mild Ito-type formula for solutions and numerical approximations of semilinear SEEs. This article solves the Weak Convergence problem emerged from Debussche’s article merely in the case of spatial spectral Galerkin approximations instead of other more complicated numerical approximations. Our method of proof extends, however, to a number of other kinds of spatial and temporal numerical approximations for semilinear SEEs.
Wataru Takahashi - One of the best experts on this subject based on the ideXlab platform.
-
strong and Weak Convergence theorems for equilibrium problems and relatively nonexpansive mappings in banach spaces
2009Co-Authors: Wataru Takahashi, Kei ZembayashiAbstract:Abstract In this paper, we introduce two iterative sequences for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a Banach space. Then we study the strong and Weak Convergence of the sequences.
-
Weak Convergence of an iterative sequence for accretive operators in banach spaces
2006Co-Authors: Koji Aoyama, Hideaki Iiduka, Wataru TakahashiAbstract:Let be a nonempty closed convex subset of a smooth Banach space and let be an accretive operator of into . We first introduce the problem of finding a point such that where is the duality mapping of . Next we study a Weak Convergence theorem for accretive operators in Banach spaces. This theorem extends the result by Gol'shteĭn and Tret'yakov in the Euclidean space to a Banach space. And using our theorem, we consider the problem of finding a fixed point of a strictly pseudocontractive mapping in a Banach space and so on.
-
Weak Convergence of an iterative sequence for accretive operators in banach spaces
2006Co-Authors: Koji Aoyama, Hideaki Iiduka, Wataru TakahashiAbstract:Let be a nonempty closed convex subset of a smooth Banach space and let be an accretive operator of into . We first introduce the problem of finding a point such that where is the duality mapping of . Next we study a Weak Convergence theorem for accretive operators in Banach spaces. This theorem extends the result by Gol'shteĭn and Tret'yakov in the Euclidean space to a Banach space. And using our theorem, we consider the problem of finding a fixed point of a strictly pseudocontractive mapping in a Banach space and so on.
Yan Dolinsky - One of the best experts on this subject based on the ideXlab platform.
-
extended Weak Convergence and utility maximisation with proportional transaction costs
2020Co-Authors: Erhan Bayraktar, Leonid Dolinskyi, Yan DolinskyAbstract:In this paper, we study utility maximisation with proportional transaction costs. Assuming extended Weak Convergence of the underlying processes, we prove the Convergence of the time-0 values of the corresponding utility maximisation problems. Moreover, we establish a limit theorem for the optimal trading strategies. The proofs are based on the extended Weak Convergence theory developed in Aldous (Weak Convergence of Stochastic Processes for Processes Viewed in the Strasbourg Manner, 1981) and on the Meyer–Zheng topology introduced in Meyer and Zheng (Ann. Inst. Henri Poincare Probab. Stat. 20:353–372, 1984).
-
Continuity of utility maximization under Weak Convergence
2020Co-Authors: Erhan Bayraktar, Yan DolinskyAbstract:In this paper we find tight sufficient conditions for the continuity of the value of the utility maximization problem from terminal wealth with respect to the Convergence in distribution of the underlying processes. We also establish a Weak Convergence result for the terminal wealths of the optimal portfolios. Finally, we apply our results to the computation of the minimal expected shortfall (shortfall risk) in the Heston model by building an appropriate lattice approximation.
-
extended Weak Convergence and utility maximization with proportional transaction costs
2019Co-Authors: Erhan Bayraktar, Yan Dolinsky, Leonid DolinskyiAbstract:In this paper we study utility maximization with proportional transaction costs. Assuming extended Weak Convergence of the underlying processes we prove the Convergence of the corresponding utility maximization problems. Moreover, we establish a limit theorem for the optimal trading strategies. The proofs are based on the extended Weak Convergence theory developed by Aldous and the Meyer-Zheng topology.
Erhan Bayraktar - One of the best experts on this subject based on the ideXlab platform.
-
extended Weak Convergence and utility maximisation with proportional transaction costs
2020Co-Authors: Erhan Bayraktar, Leonid Dolinskyi, Yan DolinskyAbstract:In this paper, we study utility maximisation with proportional transaction costs. Assuming extended Weak Convergence of the underlying processes, we prove the Convergence of the time-0 values of the corresponding utility maximisation problems. Moreover, we establish a limit theorem for the optimal trading strategies. The proofs are based on the extended Weak Convergence theory developed in Aldous (Weak Convergence of Stochastic Processes for Processes Viewed in the Strasbourg Manner, 1981) and on the Meyer–Zheng topology introduced in Meyer and Zheng (Ann. Inst. Henri Poincare Probab. Stat. 20:353–372, 1984).
-
Continuity of utility maximization under Weak Convergence
2020Co-Authors: Erhan Bayraktar, Yan DolinskyAbstract:In this paper we find tight sufficient conditions for the continuity of the value of the utility maximization problem from terminal wealth with respect to the Convergence in distribution of the underlying processes. We also establish a Weak Convergence result for the terminal wealths of the optimal portfolios. Finally, we apply our results to the computation of the minimal expected shortfall (shortfall risk) in the Heston model by building an appropriate lattice approximation.
-
extended Weak Convergence and utility maximization with proportional transaction costs
2019Co-Authors: Erhan Bayraktar, Yan Dolinsky, Leonid DolinskyiAbstract:In this paper we study utility maximization with proportional transaction costs. Assuming extended Weak Convergence of the underlying processes we prove the Convergence of the corresponding utility maximization problems. Moreover, we establish a limit theorem for the optimal trading strategies. The proofs are based on the extended Weak Convergence theory developed by Aldous and the Meyer-Zheng topology.
Leonid Dolinskyi - One of the best experts on this subject based on the ideXlab platform.
-
extended Weak Convergence and utility maximisation with proportional transaction costs
2020Co-Authors: Erhan Bayraktar, Leonid Dolinskyi, Yan DolinskyAbstract:In this paper, we study utility maximisation with proportional transaction costs. Assuming extended Weak Convergence of the underlying processes, we prove the Convergence of the time-0 values of the corresponding utility maximisation problems. Moreover, we establish a limit theorem for the optimal trading strategies. The proofs are based on the extended Weak Convergence theory developed in Aldous (Weak Convergence of Stochastic Processes for Processes Viewed in the Strasbourg Manner, 1981) and on the Meyer–Zheng topology introduced in Meyer and Zheng (Ann. Inst. Henri Poincare Probab. Stat. 20:353–372, 1984).
-
extended Weak Convergence and utility maximization with proportional transaction costs
2019Co-Authors: Erhan Bayraktar, Yan Dolinsky, Leonid DolinskyiAbstract:In this paper we study utility maximization with proportional transaction costs. Assuming extended Weak Convergence of the underlying processes we prove the Convergence of the corresponding utility maximization problems. Moreover, we establish a limit theorem for the optimal trading strategies. The proofs are based on the extended Weak Convergence theory developed by Aldous and the Meyer-Zheng topology.
