关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Stochastic equations in infinite dimensions

Stochastic optimization problems arise in decision-making problems under uncertainty, and find various applications in economics and finance. On the other hand, problems in finance have recently led to new developments in the theory of stochastic control. This volume provides a systematic treatment of stochastic optimization problems applied to finance by presenting the different existing methods: dynamic programming, viscosity solutions, backward stochastic differential equations, and martingale duality methods. The theory is discussed in the context of recent developments in this field, with complete and detailed proofs, and is illustrated by means of concrete examples from the world of finance: portfolio allocation, option hedging, real options, optimal investment, etc. This book is directed towards graduate students and researchers in mathematical finance, and will also benefit applied mathematicians interested in financial applications and practitioners wishing to know more about the use of stochastic optimization methods in finance.

Continuous-time stochastic control and optimization with financial applications / Huyen Pham

The author’s presentation of multilevel Monte Carlo path simulation at the MCQMC 2006 conference stimulated a lot of research into multilevel Monte Carlo methods. This paper reviews the progress since then, emphasising the simplicity, flexibility and generality of the multilevel Monte Carlo approach. It also offers a few original ideas and suggests areas for future research.

/pdf/multilevel-monte-carlo-methods-99zvgpxc2v.pdf

Multilevel Monte Carlo methods

We consider the numerical solution of elliptic partial differential equations with random coefficients Such problems arise, for example, in uncertainty quantification for groundwater flow We describe a novel variance reduction technique for the standard Monte Carlo method, called the multilevel Monte Carlo method, and demonstrate numerically its superiority The asymptotic cost of solving the stochastic problem with the multilevel method is always significantly lower than that of the standard method and grows only proportionally to the cost of solving the deterministic problem in certain circumstances Numerical calculations demonstrating the effectiveness of the method for one- and two-dimensional model problems arising in groundwater flow are presented

/pdf/multilevel-monte-carlo-methods-and-applications-to-elliptic-1jwicct1l9.pdf

Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients

In Monte Carlo methods quadrupling the sample size halves the error. In simulations of stochastic partial differential equations (SPDEs), the total work is the sample size times the solution cost of an instance of the partial differential equation. A Multi-level Monte Carlo method is introduced which allows, in certain cases, to reduce the overall work to that of the discretization of one instance of the deterministic PDE. The model problem is an elliptic equation with stochastic coefficients. Multi-level Monte Carlo errors and work estimates are given both for the mean of the solutions and for higher moments. The overall complexity of computing mean fields as well as k-point correlations of the random solution is proved to be of log-linear complexity in the number of unknowns of a single Multi-level solve of the deterministic elliptic problem. Numerical examples complete the theoretical analysis.

/pdf/multi-level-monte-carlo-finite-element-method-for-elliptic-5cjy4s9lmx.pdf

Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients

We analyze the convergence and complexity of multilevel Monte Carlo discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show under low regularity assumptions on the solution that the judicious combination of low order Galerkin discretizations in space and an Euler-Maruyama discretization in time yields mean square convergence of order one in space and of order 1/2 in time to the expected value of the mild solution. The complexity of the multilevel estimator is shown to scale log-linearly with respect to the corresponding work to generate a single path of the solution on the finest mesh, resp. of the corresponding deterministic parabolic problem on the finest mesh. © 2012 Springer Science + Business Media B.V.

Multilevel Monte Carlo method for parabolic stochastic partial differential equations

In this work, the approximation of Hilbert-space-valued random variables is combined with the approximation of the expectation by a multilevel Monte Carlo MLMC method. The number of samples on the different levels of the multilevel approximation are chosen such that the errors are balanced. The overall work then decreases in the optimal case to O h −2 if h is the error of the approximation. The MLMC method is applied to functions of solutions of parabolic and hyperbolic stochastic partial differential equations as needed, for example, for option pricing. Simulations complete the paper.

Multilevel Monte Carlo method with applications to stochastic partial differential equations

These notes describe numerical issues that may arise when implementing a simulation method for a stochastic partial differential equation (SPDE). It is shown that an additional approximation of the noise does not necessarily affect the order of convergence of a discretization method for a SPDE driven by Levy noise. Furthermore, finite element methods are explicitly given and simulations are done. In statistical tests, it is shown that the simulations obey the theoretical orders of convergence. © 2012 Copyright Taylor and Francis Group, LLC.

/pdf/simulation-of-stochastic-partial-differential-equations-3yjpz6re66.pdf

Simulation of stochastic partial differential equations using finite element methods

Objective—To evaluate long-term success of endoscopic injection of collagen into the urethral submucosa in female dogs with urinary incontinence caused by urethral sphincter incompetence. Design—Retrospective study. Animals—40 incontinent female dogs. Procedure—Medical records were reviewed for outcome and other results for dogs in which a cystoscope was passed into the urethra for deposition of 3 collagen deposits into the submucosa. Results—27 (68%) dogs were continent for 1 to 64 months (mean, 17 months) after the collagen injection. In another 10 dogs, incontinence improved and in 6 of these dogs, full continence was regained with administration of additional medication. In 3 dogs, incontinence was unchanged. As long as 12 months after injection, there was a deterioration in the initial result in 16 dogs, after which their condition stabilized. Mild and transient adverse effects developed in 6 (15%) dogs. Conclusions and Clinical Relevance—Long-term success of endoscopic injection of collagen was sati...

Evaluation of long-term effects of endoscopic injection of collagen into the urethral submucosa for treatment of urethral sphincter incompetence in female dogs: 40 cases (1993-2000).

Andrea Barth

Papers

Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients

Multilevel Monte Carlo method for parabolic stochastic partial differential equations

Multilevel Monte Carlo method with applications to stochastic partial differential equations

Simulation of stochastic partial differential equations using finite element methods

Evaluation of long-term effects of endoscopic injection of collagen into the urethral submucosa for treatment of urethral sphincter incompetence in female dogs: 40 cases (1993-2000).