The composite Euler method for stiff stochastic differential equations

In this paper we present the composite Euler method for the strong solution of stochastic differential equations driven by d-dimensional Wiener processes. This method is a combination of the semi-implicit Euler method and the implicit Euler method. At each step either the semi-implicit Euler method or the implicit Euler method is used in order to obtain better stability properties. We give criteria for selecting the semi-implicit Euler method or the implicit Euler method. For the linear test equation, the convergence properties of the composite Euler method depend on the criteria for selecting the methods. Numerical results suggest that the convergence properties of the composite Euler method applied to nonlinear SDEs is the same as those applied to linear equations. The stability properties of the composite Euler method are shown to be far superior to those of the Euler methods, and numerical results show that the composite Euler method is a very promising method. (C) 2001 Elsevier Science B.V. All rights reserved.

Implicit Taylor methods for stiff stochastic differential equations

In this paper we discuss implicit Taylor methods for stiff Ito stochastic differential equations. Based on the relationship between Ito stochastic integrals and backward stochastic integrals, we introduce three implicit Taylor methods: the implicit Euler-Taylor method with strong order 0.5, the implicit Milstein-Taylor method with strong order 1.0 and the implicit Taylor method with strong order 1.5. The mean-square stability properties of the implicit Euler-Taylor and Milstein-Taylor methods are much better than those of the corresponding semi-implicit Euler and Milstein methods and these two implicit methods can be used to solve stochastic differential equations which are stiff in both the deterministic and the stochastic components. Numerical results are reported to show the convergence properties and the stability properties of these three implicit Taylor methods. The stability analysis and numerical results show that the implicit Euler-Taylor and Milstein-Taylor methods are very promising methods for stiff stochastic differential equations.

Three-point boundary value problems for second-order, ordinary, differential equations

We establish existence results for solutions to three-point boundary value problems for nonlinear, second-order, ordinary differential equations with nonlinear boundary conditions. (C) 2001 Elsevier Science Ltd. All rights reserved.

Differential inclusions and robust control theory

Uncontrolled systems (x) over dot is an element of Ax, where A is a non-empty compact set of matrices, and controlled systems (x) over dot is an element of Ax + Bu are considered. Higher-order systems 0 is an element of Px - Du, where and are sets of differential polynomials, are also studied. It is shown that, under natural conditions commonly occurring in robust control theory, with some mild additional restrictions, asymptotic stability of differential inclusions is guaranteed. The main results are variants of small-gain theorems and the principal technique used is the Krasnosel'skii-Pokrovskii principle of absence of bounded solutions.

Stability of linear difference and differential inclusions

Any given n X n matrix A is shown to be a restriction, to the A-invariant subspace, of a nonnegative N x N matrix B of spectral radius p(B) arbitrarily close to p(A). A difference inclusion x(k+1) is an element of Ax(k), where A is a compact set of matrices, is asymptotically stable if and only if A can be extended to a set B of nonnegative matrices B with \ \B \ \ (1) < 1 or \ \B \ \ (infinity) < 1. Similar results are derived for differential inclusions.

Stiffly accurate Runge-Kutta methods for stiff stochastic differential equations

In this paper we discuss implicit methods based on stiffly accurate Runge-Kutta methods and splitting techniques for solving Stratonovich stochastic differential equations (SDEs). Two splitting techniques: the balanced splitting technique and the deterministic splitting technique, are used in this paper. We construct a two-stage implicit Runge-Kutta method with strong order 1.0 which is corrected twice and no update is needed. The stability properties and numerical results show that this approach is suitable for solving stiff SDEs. (C) 2001 Elsevier Science B.V. All rights reserved.

Tissue-specific induction of SOCS gene expression by PRL

The mechanisms whereby tissue sensitivity to PRL is controlled are not well understood. Here we report that expression of mRNA and protein for members of the SOCS/CIS/JAB family of cytokine signaling inhibitors is increased by PRL administration in ovary and adrenal gland of the lactating rat deprived of circulating PRL and pups for 24 h but not in mammary gland. Moreover, suckling increases SOCS mRNA in the ovary but not in the mammary gland of pup-deprived rats. Deprivation of PRL and pups for 48 h allows the mammary gland to induce SOCS genes in response to PRL administration, and this is associated with a decrease in basal SOCS-3 mRNA and protein expression to the level seen in other tissues, suggesting that SOCS-3 induced refractoriness related to filling of the gland. In reporter assays, SOCS-1, SOCS-3, and CIS, but not SOCS-2, are able to inhibit transactivation of the STAT 5-responsive beta -lactoglobulin promoter in transient transfection assays. Moreover, suckling results in loss of ovarian and adrenal responsiveness to PRL administered 2 h after commencement of suckling, as determined by STAT 5 gel shift assay. Immunohistochemistry was used to localize the cellular sites of SOCS-3 and CIS protein expression in the ovary and adrenal gland. We propose that induced SOCS-1, SOCS-3, and CIS are actively involved in the cellular inhibitory feedback response to physiological PRL surges in the corpus luteum and adrenal cortex during lactation, but after pup withdrawal, the mammary gland is rendered unresponsive to PRL by increased levels of SOCS-3.