Density analysis of non-Markovian BSDEs and applications to biology and finance

In this paper, we provide conditions which ensure that stochastic Lipschitz BSDEs admit Malliavin differentiable solutions. We investigate the problem of existence of densities for the first components of solutions to general path-dependent stochastic Lipschitz BSDEs and obtain results for the second components in particular cases. We apply these results to both the study of a gene expression model in biology and to the classical pricing problems in mathematical finance.     

Max-Plus Stochastic Control and Risk-Sensitivity

In the Maslov idempotent probability calculus, expectations of random variables are defined so as to be linear with respect to max-plus addition and scalar multiplication. This paper considers control problems in which the objective is to minimize the max-plus expectation of some max-plus additive running cost. Such problems arise naturally as limits of some types of risk sensitive stochastic control problems. The value function is a viscosity solution to a quasivariational inequality (QVI) of dynamic programming. Equivalence of this QVI to a nonlinear parabolic PDE with discontinuous Hamiltonian is used to prove a comparison theorem for viscosity sub- and super-solutions. An example from mathematical finance is given, and an application in nonlinear H-infinity control is sketched.     

Lipschitz properties of the scalarization function and applications

The scalarization functions were used in vector optimization for a long period. Similar functions were introduced and used in economics under the name of shortage function or in mathematical finance under the name of (convex or coherent) measures of risk. The main aim of this article is to study Lipschitz continuity properties of such functions and to give some applications for deriving necessary optimality conditions for vector optimization problems using the Mordukhovich subdifferential.     

Lagrange necessary conditions for Pareto minimizers in Asplund spaces and applications

In this paper, new necessary conditions for Pareto minimal points to sets and Pareto minimizers for constrained multiobjective optimization problems are established without the sequentially normal compactness property and the asymptotical compactness condition imposed on closed and convex ordering cones in Bao and Mordukhovich [10] and Durea and Dutta [5], respectively. Our approach is based on a version of the separation theorem for nonconvex sets and the subdifferentials of vector-valued and set-valued mappings. Furthermore, applications in mathematical finance and approximation theory are discussed.     

Econophysics and Quantum Statistics

In this paper, we associate notions of entropy, temperature, free energy, and Hamiltonian occuring in problems in probability theory with a system of identical objects so that modern methods of quantum statistics can be applied to problems in mathematical finance.     

Use of stochastic and mathematical programming in?portfolio theory and practice

Standard finance portfolio theory draws graphs and writes equations usually with no constraints and frequently in the univariate case. However, in reality, there are multivariate random variables and multivariate asset weights to determine with constraints. Also there are the effects of transaction costs on asset prices in the theory and calculation of optimal portfolios in the static and dynamic cases. There we use various stochastic programming, linear complementary, quadratic programming and nonlinear programming problems. This paper begins with the simplest problems and builds the theory to the more complex cases and then applies it to real financial asset allocation problems, hedge funds and professional racetrack betting. This paper is based on a keynote lecture at the APMOD conference in Madrid in June 2006. It was also presented at the London Business School. Many thanks are due to APMOD organizers Antonio Alonso-Ayuso, Laureano Escudero, and Andres Ramos for inviting me and for excellent hospitality in Madrid. Thanks are also due to my teachers at Berkeley who got me on the right track on stochastic and mathematical programming, especially Olvi Mangasarian, Roger Wets and Willard Zangwill, and my colleagues and co-authors on portfolio theory in finance and horseracing, especially Chanaka Edirishinge, Donald Hausch, Jarl Kallberg, Victor Lo, Leonard MacLean, Raymond Vickson and Yonggan Zhao.     

Fair dynamic valuation of insurance liabilities: Merging actuarial judgement with market- and time-consistency

In this paper, we investigate the fair valuation of insurance liabilities in a dynamic multi-period setting. We define a fair dynamic valuation as a valuation which is actuarial (mark-to-model for claims independent of financial market evolutions), market-consistent (mark-to-market for any hedgeable part of a claim) and time-consistent, extending the work of Dhaene et al. (2017) and Barigou and Dhaene (2019). We provide a complete hedging characterization for fair dynamic valuations. Moreover, we show how to implement fair dynamic valuations through a backward iterations scheme combining risk minimization methods from mathematical finance with standard actuarial techniques based on risk measures.     

On analytical solutions of the Black–Scholes equation

This work presents a theoretical analysis for the Black–Scholes equation. Given a terminal condition, the analytical solution of the Black–Scholes equation is obtained by using the Adomian approximate decomposition technique. The mathematical technique employed in this work also has significance in studying some other problems in finance theory.     

中国数学规划学科发展概述

数学规划又称数学优化, 是运筹学的一个重要分支. 它主要研究在一定约束条件下, 如何求一个实数或者整数变量的实函数的最大值或者最小值. 它是运筹学和管理科学中最常用的一种建模工具和求解问题的方法, 在工程、经济和金融等领域有非常广泛的应用. 首先简单介绍数学规划的发展历史、应用领域及其主要研究方向; 然后简述数学规划的发展现状和在中国的发展进程; 最后, 讨论数学规划若干研究前沿问题与研究展望.     

Optional decomposition of optional supermartingales and applications to filtering and finance

ABSTRACT

The classical Doob–Meyer decomposition and its uniform version the optional decomposition are stated on probability spaces with filtrations satisfying the usual conditions. However, the comprehensive needs of filtering theory and mathematical finance call for their generalizations to more abstract spaces without such technical restrictions. The main result of this paper states that there exists a uniform Doob–Meyer decomposition of optional supermartingales on unusual probability spaces. This paper also demonstrates how this decomposition works in the construction of optimal filters in the very general setting of the filtering problem for optional semimartingales. Finally, the application of these optimal filters of optional semimartingales to mathematical finance is presented.     

最优投资消费策略

投资消费问题是数理金融中的一个主要问题,Merton在假设股票价格过程为扩散过程的情形下,给出了最优投资消费策略的显式解,本在股份价格过程为跳－扩散过程的情形下,讨论了最优投资消费策略问题,得到了最优投资消费策略的偏微分方程。     

A simple characterization of tightness for convex solid sets of positive random variables

We show that for a convex solid set of positive random variables to be tight, or equivalently bounded in probability, it is necessary and sufficient to be is radially bounded, i.e. that every ray passing through one of its elements eventually leaves the set. The result is motivated by problems arising in mathematical finance.     

Large time asymptotic problems for optimal stochastic control with superlinear cost

The paper is concerned with stochastic control problems of finite time horizon whose running cost function is of superlinear growth with respect to the control variable. We prove that, as the time horizon tends to infinity, the value function converges to a function of variable separation type which is characterized by an ergodic stochastic control problem. Asymptotic problems of this type arise in utility maximization problems in mathematical finance. From the PDE viewpoint, our results concern the large time behavior of solutions to semilinear parabolic equations with superlinear nonlinearity in gradients.     

Multiplicative Programming Problems: Analysis and Efficient Point Search Heuristic

Multiplicative programming problems are difficult global optimization problems known to be NP-hard. At the same time, these problems have some important applications in engineering, finance, economics, and other fields. This article has two purposes. The first is to present an analysis that shows several relationships between concave multiplicative programs and concave minimization problems, and between concave multiplicative programs and certain multiple-objective mathematical programs. The second purpose is to propose and report computational results for a heuristic efficient-point search algorithm that we have designed for use on linear multiplicative programming problems. To our knowledge, this is the first heuristic algorithm of its type. The theoretical and algorithmic results given in the article offer some potentially important new avenues for analyzing and solving multiplicative programming problems of various types.     

Characterization of Stochastic Viability of Any Nonsmooth Set Involving Its Generalized Contingent Curvature

Abstract

Thanks to the Stroock and Varadhan “Support Theorem” and under convenient regularity assumptions, stochastic viability problems are equivalent to invariance problems for control systems (also called tychastic viability), as it has been singled out by Doss in 1977 for instance. By the way, it is in this framework of invariance under control systems that problems of stochastic viability in mathematical finance are studied. The Invariance Theorem for control systems characterizes invariance through first‐order tangential and/or normal conditions whereas the stochastic invariance theorem characterizes invariance under second‐order tangential conditions. Doss's Theorem states that these first‐order normal conditions are equivalent to second‐order normal conditions that we expect for invariance under stochastic differential equations for smooth subsets. We extend this result to any subset by defining in an adequate way the concept of contingent curvature of a set and contingent epi‐Hessian of a function, related to the contingent curvature of its epigraph. This allows us to go one step further by characterizing functions the epigraphs of which are invariant under systems of stochastic differential equations. We shall show that they are (generalized) solutions to either a system of first‐order Hamilton‐Jacobi equations or to an equivalent system of second‐order Hamilton‐Jacobi equations.     

动态随机弹性在期权定价中的应用

给出动态随机弹性的概念及运算性质,讨论了动态随机弹性在期权定价模型中的应用.主要结果有:(1)在波动率为常数时,期权价格对的弹性,得到了动态随机弹性服从运动,并给出了相应的经济解释;(2)由于波动率一般不是常数,也是随机过程,因此本文进一步研究了期权价格对波动率的弹性,就股票价格的波动情况给出了数学描述和金融意义上的解释.     

Niveloids and their extensions: Risk measures on small domains

Given a functional defined on a nonempty subset of an Archimedean Riesz space with unit, necessary and sufficient conditions are obtained for the existence of a (convex or concave) niveloid that extends the functional to the entire space. In the language of mathematical finance, this problem is equivalent to the one of verifying if the policy adopted by a regulator is consistent with monetary risk measurement, when only partial information is available.     

On a class of dynamic programming problems whose optimal controls and states are independent of the future

When applying dynamic programming for optimal decision making one usually needs considerable knowledge about the future. This knowledge, e.g. about future functions and parameters, necessary to determine optimal control policies, however, is often not available and thus precludes the application of dynamic programming.In the present paper it is shown that for a certain class of dynamic programming problems the optimal control policy is independent of the future. To illustrate the results an application in inventory control is given and further applications in the theories of economic growth and corporate finance are listed in the references.     

Hemivariational inequality approach to the dynamic viscoelastic contact problem with nonmonotone normal compliance and slip-dependent friction

We examine a mathematical model which describes dynamic viscoelastic contact problems with nonmonotone normal compliance condition and the slip displacement dependent friction. First, we derive a weak formulation of the model in the form of a hemivariational inequality. Then we embed the hemivariational inequality into a class of second-order evolution inclusions for which we provide a result on the existence of a solution. We conclude with examples of the subdifferential boundary conditions for contact with normal compliance and the slip dependent friction.