Flots et series de Taylor stochastiques

Summary We study the expansion of the solution of a stochastic differential equation as an (infinite) sum of iterated stochastic (Stratonovitch) integrals. This enables us to give a universal and explicit formula for any invariant diffusion on a Lie group in terms of Lie brackets, as well as a universal and explicit formula for the brownian motion on a Riemannian manifold in terms of derivatives of the curvature tensor. The first of these formulae contains, and extends to the non nilpotent case, the results of Doss [6], Sussmann [17], Yamato [18], Fliess and Normand-Cyrot [7], Krener and Lobry [19] and Kunita [11] on the representation of solutions of stochastic differential equations.     

Computing conditional expectations of multidimensional diffusion processes

We study multidimensional diffusion processes and give an explicit representation for their conditional expectation. Starting from the solution formula for one dimensional stochastic differential equations found in Lanconelli and Proske [8], we compute the conditional expectation of a certain class of multidimensional diffusions without resorting to the Markov property of the process and therefore without requiring an explicit expression for the semi group associated to it.     

Methods de laplace et de la phase stationnaire sur l'espace de wiener

We obtain, using large deviations principles, full asymptotic expansions of functionals of Laplace type on Wiener space for general (i.e. degenerate) diffusions extending the results of Schilder [12], Azencott [2] and Doss [7, 20]. The variational hypothesis (non degeneracy of the minima) used here is shown to be optimal. The first term of the expansion is explicitly computed. Using the integration by parts of Malliavin calculus the stationary phase method is also developed. The results of this paper are the basic fact used (in [5]) to obtain the asymptotic expansion for small time of the density of a degenerate diffusion, they are also relevant for semi-classical expansions     

Extrapolation methods in Lie groups

Summary. The numerical solution of differential equations on Lie groups by extrapolation methods is investigated. The main principles of extrapolation for ordinary differential equations are extended on the general case of differential equations in noncommutative Lie groups. An asymptotic expansion of the global error is given. A symmetric method is given and quadratic asymptotic expansion of the global error is proved. The theoretical results are verified by numerical experiments. Received September 27, 1999 / Revised version received February 14, 2000 / Published online April 5, 2001     

Almost sure stability of linear stochastic differential equations with jumps

 Under the nondegenerate condition as in the diffusion case, see [14, 21, 6], the linear stochastic jump-diffusion process projected on the unit sphere is a strong Feller process and has a unique invariant measure which is also ergodic using the relation between the transition probabilities of jump-diffusions and the corresponding diffusions due to [22]. The unique deterministic Lyapunov exponent can be represented by the Furstenberg-Khas'minskii formula as an integral over the sphere or the projective space with respect to the ergodic invariant measure so that the almost sure asymptotic stability of linear stochastic systems with jumps depends on its sign. The critical case of zero Lyapunov exponent is discussed and a large deviations result for asymptotically stable systems is further investigated. Several examples are treated for illustration. Received: 22 June 2000 / Revised version: 20 November 2001 / Published online: 13 May 2002     

Semimartingales on rays,Walsh diffusions,and related problems of control and stopping

We introduce a class of continuous planar processes, called “semimartingales on rays”, and develop for them a change-of-variable formula involving quite general classes of test functions. Special cases of such processes are diffusions which choose, once at the origin, the rays for their subsequent voyage according to a fixed probability measure in the manner of Walsh (1978). We develop existence and uniqueness results for these “Walsh diffusions”, study their asymptotic behavior, and develop tests for explosions in finite time. We use these results to find an optimal strategy, in a problem of stochastic control with discretionary stopping involving Walsh diffusions.     

Itô-Wiener Chaos Expansion with Exact Residual and Correlation, Variance Inequalities

We give a formula of expanding the solution of a stochastic differential equation (abbreviated as SDE) into a finite Itô-Wiener chaos with explicit residual. And then we apply this formula to obtain several inequalities for diffusions such as FKG type inequality, variance inequality and a correlation inequality for Gaussian measure. A simple proof for Houdré-Kagan's variance inequality for Gaussian measure is also given.     

On a robust version of the integral representation formula of nonlinear filtering

The paper is concerned with completing “unfinished business” on a robust representation formula for the conditional expectation operator of nonlinear filtering. Such a formula, robust in the sense that its dependence on the process of observations is continuous, was stated in [2] without proof. The main purpose of this paper is to repair this deficiency.The formula is “almost obvious” as it can be derived at a formal level by a process of integration-by-parts applied to the stochastic integrals that appear in the integral representation formula. However, the rigorous justification of the formula is quite subtle, as it hinges on a measurability argument the necessity of which is easy to miss at first glance. The continuity of the representation (but not its validity) was proved by Kushner [9] for a class of diffusions.Here we follow the definition given in [11].     

A Small Noise Asymptotic Expansion for Young SDE Driven by Fractional Brownian Motion: A Sharp Error Estimate With Malliavin Calculus

This article shows an analytically tractable small noise asymptotic expansion with a sharp error estimate for the expectation of the solution to Young’s pathwise stochastic differential equations (SDEs) driven by fractional Brownian motions with the Hurst index H > 1/2. In particular, our asymptotic expansion can be regarded as small noise and small time asymptotics by the error estimate with Malliavin culculus. As an application, we give an expansion formula in one-dimensional general Young SDE driven by fractional Brownian motion. We show the validity of the expansion through numerical experiments.     

关于SL(2,R)上速降函数的反演公式

在局部紧可分群的一般理论中，分解正则表示以及获得反演公式（或 Ｐｌａｎ－ｃｈｅｒｅｌ定理的明确表示）是调和分析的基本目标之一．ＳＬ（２，　）是最简单的非交换局部紧么模半单Ｌｉｅ群．Ｈａｒｉｓｈ－Ｃｈａｎｄｒａ在 Ｃ∞ｃ（ＳＬ（２，　））上获得了反演公式，Ｘｉａｏ和ｈｅｎｇ在文［１］中证明了Ｃ３ｃ（ＳＬ（２， ）上的反演公式．在文［２］中Ｚｈｅｎｇ引入了Ｌｉｅ群Ｇ上函数的广义微分（Ａ导数）概念．在本文中，我们利用文［２］中的微分概念来研究ＳＬ（２，　）上可微函数的Ｆｏｕｒｉｅｒ变换的阶，并获得了ＳＬ（２，　）上速降函数的反演公式．     

A BLACK-SCHOLES FORMULA FOR OPTION PRICING WITH DIVIDENDS

Abstract. We obtain a Black-Scholes formula for the arbitrage-free pricing of Eu-ropean Call options with constant coefficients when the underlylng stock generatesdividends. To hedge the Call option, we will always borrow money from bank. We seethe influence of the dividend term on the option pricing via the comparison theoremof BSDE(backward stochastic di~erential equation [5], [7]). We also consider the option pricing problem in terms of the borrowing rate R whichis not equal to the interest rate r. The corresponding Black-Sdxoles formula is given.We notice that it is in fact the borrowing rate that plays the role in the pricing formula.     

Asymptotic behaviors for functionals of random dynamical systems

In this article, we consider asymptotic behaviors for functionals of dynamical systems with small random perturbations. First, we present a deviation inequality for Gaussian approximation of dynamical systems with small random perturbations under Hölder norms and establish the moderate deviation principle and the central limit theorem for the dynamical systems by the deviation inequality. Then, applying these results to forward-backward stochastic differential equations and diffusions in small time intervals, combining the delta method in large deviations, we give a moderate deviation principle for solutions of forward-backward stochastic differential equations with small random perturbations, and obtain the central limit theorem, the moderate deviation principle and the iterated logarithm law for functionals of diffusions in small time intervals.     

Optimal exit probabilities and differential games

The problem considered is to control the drift of a Markov diffusion process in such a way that the probability that the process exits from a given regionD during a given finite time interval is minimum. An asymptotic formula for the minimum exit probability when the process is nearly deterministic is given. This formula involves the lower value of an associated differential game. It is related to a result of Ventsel and Freidlin for nearly deterministic, uncontrolled diffusions.This research was supported in part by the Air Force Office of Scientific Research under AF-AFOSR 76-3063C and in part by the National Science Foundation, NSF-76-07261.     

有强化弹塑性含圆孔单向受拉板的渐近解

本文运用有强化弹塑性平面问题的一般渐近解,以解决有强化弹塑性含圆孔的无限大板在单向受拉下的应力分布问题.文中求得了二次近似解的应力分量的解析表达式,并与其他作者的数值计算[4]和实验结果[5]进行了比较,结果颇为符合.最后,作者还就Neuber公式的正确性进行了考查.     

On Positive Linear Volterra-Stieltjes Differential Systems

We first introduce the notion of positive linear Volterra-Stieltjes differential systems. Then, we give some characterizations of positive systems. An explicit criterion and a Perron-Frobenius type theorem for positive linear Volterra-Stieltjes differential systems are given. Next, we offer a new criterion for uniformly asymptotic stability of positive systems. Finally, we study stability radii of positive linear Volterra-Stieltjes differential systems. It is proved that complex, real and positive stability radius of positive linear Volterra-Stieltjes differential systems under structured perturbations coincide and can be computed by an explicit formula. The obtained results in this paper include ones established recently for positive linear Volterra integro-differential systems [36] and for positive linear functional differential systems [32]-[35] as particular cases. Moreover, to the best of our knowledge, most of them are new. The first author is supported by the Alexander von Humboldt Foundation.     

Random creation and dispersion of mass

Consider evolution of density of a mass or a population, geographically situated in a compact region of space, assuming random creation-annihilation and migration, or dispersion of mass, so the evolution is a random measure. When the creation-annihilation and dispersion are diffusions the situation is described formally by a stochastic partial differential equation; ignoring dispersion make approximations to the initial density by atomic measures and if the corresponding discrete random measures converge “in law” to a unique random measure call it a solution. To account for dispersion Trotter's product formula is applied to semiflows corresponding to dispersion and creation-annihilation. Existence of solutions has been a conjecture for several years despite a claim in ([2], J. Multivariate Anal. 5, 1–52). We show that solutions exist and that non-deterministic solutions are “smeared” continuous-state branching diffusions.     

Quadratic Control for Stochastic Systems Defined by Evolution Operators and Square Integrable Martingales

The infinite dimensional version of the linear quadratic cost control problem is studied by Curtain and Pritchard [2], Gibson [5] by using Riccati integral equations, instead of differential equations. In the present paper the corresponding stochastic case over a finite horizon is considered. The stochastic perturbations are given by Hilbert valued square integrable martingales and it is shown that the deterministic optimal feedback control is also optimal in the stochastic case. Sufficient conditions are given for the convergence of approximate solutions of optimal control problems.     

The differential mean value of divided differences

The paper deals with the asymptotic behaviour of the differential mean value of divided differences as the interval shrinks to zero by presenting an asymptotic expansion. The coefficients are given by a recurrence formula. For a wide class of analytic functions the differential mean value can be represented by a convergent sum. Our results generalize two recent theorems by Powers, Riedel and Sahoo [R.C. Powers, T. Riedel, P.K. Sahoo, Limit properties of differential mean values, J. Math. Anal. Appl. 227 (1998) 216-226].     

n阶中立型时滞方程衰减正解的存在性

研究了ｎ阶中立型方程（ｘ（ｔ）－ｃｘ（ｔ－τ）＾（ｎ）＋ｐ（ｔ）ｘ（ｇ（ｔ））＝０，ｔ≥ｔ０，ｎ≥１（＊）正解的存在性，在ｐ（ｔ）常号和变号的情况下，给出了（＊）存在衰减正解的充分条件，特别，这偏离 文「２」，「４－６」和「８－１２」的有关结果。