The numerical solution of Stein (aka discrete Lyapunov) equations is the primary step in Newton's method for the solution of discrete-time algebraic Riccati equations (DARE). Here we present a low-rank Smith method as well as a low-rank alternating-direction-implicit-iteration to compute low-rank approximations to solutions of Stein equations arising in this context. Numerical results are given to verify the efficiency and accuracy of the proposed algorithms.

No description available.

Guaranteed Minimum Withdrawal Benefits(GMWB) have become popular riders on variable annuities. The pricing of a GMWB contract was originally formulated as a singular stochastic control problem which results in a Hamilton Jacobi Bellman (HJB) Variational Inequality (VI). A penalty method method can then be used to solve the HJB VI. We present a rigorous proof of convergence of the penalty method to the viscosity solution of the HJB VI assuming the underlying asset follows a Geometric Brownian Motion. A direct control method is an alternative formulation for the HJB VI. We also extend the HJB VI to the case of where the underlying asset follows a Poisson jump diffusion. The HJB VI is normally solved numerically by an implicit method, which gives rise to highly nonlinear discretized algebraic equations. The classic policy iteration approach works well for the Geometric Brownian Motion case. However it is not efficient in some circumstances such as when the underlying asset follows a Poisson jump diffusion process. We develop a combined fixed point policy iteration scheme which significantly increases the efficiency of solving the discretized equations. Sufficient conditions to ensure the convergence of the combined fixed point policy iteration scheme are derived both for the penalty method and direct control method. The GMWB formulated as a singular control problem has a special structure which results in a block matrix fixed point policy iteration converging about one order of magnitude faster than a full matrix fixed point policy iteration. Sufficient conditions for convergence of the block matrix fixed point policy iteration are derived. Estimates for bounds on the penalty parameter (penalty method) and scaling parameter (direct control method) are obtained so that convergence of the iteration can be expected in the presence of round-off error.

No description available.

Guaranteed Minimum Withdrawal Benefits(GMWB) have become popular riders on variable annuities. The pricing of a GMWB contract was originally formulated as a singular stochastic control problem which results in a Hamilton Jacobi Bellman (HJB) Variational Inequality (VI). A penalty method method can then be used to solve the HJB VI. We present a rigorous proof of convergence of the penalty method to the viscosity solution of the HJB VI assuming the underlying asset follows a Geometric Brownian Motion. A direct control method is an alternative formulation for the HJB VI. We also extend the HJB VI to the case of where the underlying asset follows a Poisson jump diffusion. The HJB VI is normally solved numerically by an implicit method, which gives rise to highly nonlinear discretized algebraic equations. The classic policy iteration approach works well for the Geometric Brownian Motion case. However it is not efficient in some circumstances such as when the underlying asset follows a Poisson jump diffusion process. We develop a combined fixed point policy iteration scheme which significantly increases the efficiency of solving the discretized equations. Sufficient conditions to ensure the convergence of the combined fixed point policy iteration scheme are derived both for the penalty method and direct control method. The GMWB formulated as a singular control problem has a special structure which results in a block matrix fixed point policy iteration converging about one order of magnitude faster than a full matrix fixed point policy iteration. Sufficient conditions for convergence of the block matrix fixed point policy iteration are derived. Estimates for bounds on the penalty parameter (penalty method) and scaling parameter (direct control method) are obtained so that convergence of the iteration can be expected in the presence of round-off error.

No description available.

In vorliegendem Artikel wird eine Verallgemeinerung eines Approximationsteorems von Chidume /6/ für streng pseudokontraktive Operatoren in Lp beziehungsweise lp-Räumen (mit P =2) angegeben. Es wird ein Verfahren betrachtet, welches von MANN /14/ für reelle Funktionen eingeführt würde. / This article gives a generalization of an approximation theorem by Chidume / 6 / for strictly pseudocontractive operators in Lp or lp spaces (with P = 2). We consider a method, which MANN / 14 / had introduced for real functions.

The numerical solution of Stein (aka discrete Lyapunov) equations is the primary step in Newton's method for the solution of discrete-time algebraic Riccati equations (DARE). Here we present a low-rank Smith method as well as a low-rank alternating-direction-implicit-iteration to compute low-rank approximations to solutions of Stein equations arising in this context. Numerical results are given to verify the efficiency and accuracy of the proposed algorithms.

Master of Science / Department of Statistics / Kun Chen / The singular value decomposition (SVD) is a commonly used matrix factorization technique in statistics, and it is very e ective in revealing many low-dimensional structures in a noisy data matrix or a coe cient matrix of a statistical model. In particular, it is often desirable to obtain a sparse SVD, i.e., only a few singular values are nonzero and their corresponding left and right singular vectors are also sparse. However, in several existing methods for sparse SVD estimation, the exact orthogonality among the singular vectors are often sacri ced due to the di culty in incorporating the non-convex orthogonality constraint in sparse estimation. Imposing orthogonality in addition to sparsity, albeit di cult, can be critical in restricting and guiding the search of the sparsity pattern and facilitating model interpretation. Combining the ideas of penalized regression and Bregman iterative methods, we propose two methods that strive to achieve the dual goal of sparse and orthogonal SVD estimation, in the general framework of high dimensional multivariate regression. We set up simulation studies to demonstrate the e cacy of the proposed methods.

The principal aim of this thesis is to provide a comprehensive account of the meaning of keep V-ing constructions, see (1a) and (1b). (1) a. Mary kept winning (again and again). (1) b. John kept running (for another ten minutes). On the basis of a systematic study of combinations of keep with predicates of different aktionsarts, it is shown that keep can give rise to two different readings which share the overall meaning of ‘continued activity’. It is argued that the two readings of keep V-ing arise from different aspectual properties of the predicate in the complement clause. Under the first reading, labelled the continuative-iterative reading, (1a), the event in the complement clause is telic, and the interpretation is an iterative reading. Under the second reading, labelled the continuative reading, (1b), the event in the complement clause is atelic, and the interpretation is a reading of nonstop continuation. It is argued that keep combines with activity predicates in the relevant construction type, that is, with dynamic, durative and atelic events, and that keep has the ability to induce aspect shift when combining with predicates that are not inherent activities. Thus, in (1a), a punctual and telic winning event is iterated, creating a series which in itself is durative and atelic. In (1b), the running event is already durative and atelic. By comparing keep V-ing with the progressive construction be V-ing, (2), and with two other continuative constructions, continue V-ing, (3), and V on, (4), it is shown that keep readily shifts a telic predicate into an atelic reading by taking scope over the entire event, (1a), but cannot take scope over an internal part of a telic event. Both be V-ing, (2), and continue V-ing, (3), are able to take scope over an internal part of a telic event. (2) John was building the house. (3) John continued building the house. (4) John ran on. In addition, unlike continue V-ing and V on, keep V-ing does not necessarily denote continuation of an event which has already been initiated.