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Portfolio selection problem with Value-at-Risk constraints under non-extensive statistical mechanics

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2016年 298卷 64-71页

作者： Zhao, Pan Xiao, Qingxian Univ Shanghai Sci & Technol Sch Business Shanghai Peoples R China West Anhui Univ Coll Finance & Math Luan Anhui Peoples R China West Anhui Univ Financial Risk Intelligent Control & Prevent Inst Luan Anhui Peoples R China

The optimal portfolio selection problem is a major issue in the financial field in which the process of asset prices is usually modeled by a Wiener process. That is, the return distribution of the asset is normal. However, several empirical results have shown that the return distribution of the asset has the characteristics of fat tails and aiguilles and is not normal. In this work, we propose an optimal portfolio selection model with a Value-at-Risk (VaR) constraint in which the process of asset prices is modeled by the non-extensive statistical mechanics instead of the classical Wiener process. The model can describe the characteristics of fat tails and aiguilles of returns. Using the dynamic programming principle, we derive a Hamilton-Jacobi-Bellman (HJB) equation. Then, employing the Lagrange multiplier method, we obtain closed-form solutions for the case of logarithmic utility. Moreover, the empirical results show that the price process can more accurately fit the empirical distribution of returns than the familiar Wiener process. In addition, as the time increases, the constraint becomes binding. That is, to control the risk the agent reduces the proportion of the wealth invested in the risky asset. Furthermore, at the same confidence level, the agent reduces the proportion of the wealth invested in the risky asset more quickly under our model than under the model based on the Wiener process. This can give investors a good decision-making reference. (C) 2015 Elsevier B.V. All rights reserved.

关键词： Non-extensive statistics Portfolio selection Value-at-Risk Utility dynamic programming principle

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CONVERGENCE OF DISCONTINUOUS GALERKIN SCHEMES FOR FRONT PROPAGATION WITH OBSTACLES

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MATHEMATICS OF COMPUTATION 2016年第301期85卷 2131-2159页

作者： Bokanowski, Olivier Cheng, Yingda Shu, Chi-Wang Univ Paris Diderot Sorbonne Paris Cite Lab Jacques Louis Lions UPMCCNRSUMR 7598 F-75205 Paris France Michigan State Univ Dept Math E Lansing MI 48824 USA Brown Univ Div Appl Math Providence RI 02912 USA

We study semi-Lagrangian discontinuous Galerkin (SLDG) and Runge-Kutta discontinuous Galerkin (RKDG) schemes for some front propagation problems in the presence of an obstacle term, modeled by a nonlinear Hamilton-Jacobi equation of the form min(u(t) vertical bar cu(x), u - g(x)) = 0, in one space dimension. New convergence results and error bounds are obtained for Lipschitz regular data. These "low regularity" assumptions are the natural ones for the solutions of the studied equations. Numerical tests are given to illustrate the behavior of our schemes.

关键词： Hamilton-Jacobi-Bellman equations discontinuous Galerkin methods level sets front propagation obstacle problems dynamic programming principle convergence

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Optimal reinsurance policies with two reinsurers in continuous time

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ECONOMIC MODELLING 2016年第0期59卷 182-195页

作者： Meng, Hui Zhou, Ming Siu, Talc Kuen Cent Univ Finance & Econ China Inst Actuarial Sci Beijing 100081 Peoples R China Univ China Acad Sci Sch Management Beijing 100190 Peoples R China Macquarie Univ Fac Business & Econ Dept Appl Finance & Actuarial Studies Sydney NSW 2109 Australia

An optimal reinsurance problem of an insurer is studied in a continuous-time model, where insurance risk is partly transferred to two reinsurers, one adopting the expected-value premium principle and another one using the variance premium principle. The insurer aims to select an optimal reinsurance arrangement to minimize the probability of ruin. To provide an easy-to-implement solution to the problem, (semi)-explicit expressions for the optimal reinsurance strategies as well as the minimal ruin probabilities are derived for several claims distributions. Numerical studies including a real-data example based on the Danish fire insurance losses are provided to illustrate the solution of the problem. Our empirical results based on the Danish data reveal that the heavy-right-tailedness of claims distributions has a significant impact on the optimal reinsurance strategies and has a quite pronounced impact on the residual risk described by the minimal ruin probability. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Expected value premium principle Variance premium principle Ruin probability Proportional reinsurance Excess of loss reinsurance dynamic programming principle Heavy-tailed claims

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Continuous-time Martingale Optimal Transport and Optimal Skorokhod Embedding

Continuous-time Martingale Optimal Transport and Optimal Sko...

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作者： Gaoyue Guo ECOLE POLYTECHNIQUE

学位级别：博士

This PhD dissertation presents three research topics, the first two being independent and the last one relating the first two issues in a concrete case. In the first part we focus on the martingale optimal transport problem on the Skorokhod space, which aims at studying systematically the tightness of martingale transport plans. Using the S≠topology introduced by Jakubowski [67], we obtain the desired tightness which yields the upper semicontinuity of the primal problem with respect to the marginal distributions, and further the first duality. Then, we provide also two dual formulations that are related to the robust superhedging in financial mathematics, and we establish the corresponding dualities by adapting the dynamic programming principle and the discretization argument initiated by Dolinsky & Soner [37]. The second part of this dissertation addresses the optimal Skorokhod embedding problem under finitely-many marginal constraints. We formulate first this optimi- zation problem by means of probability measures on an enlarged space as well as its dual problems. Using the classical convex duality approach together with the optimal stopping theory, we obtain the duality results. We also relate these results to the martingale optimal transport on the space of continuous functions, where the corresponding dualities are derived for a special class of reward functions. Next, We provide an alternative proof of the monotonicity principle established in Beiglbock, Cox & Huesmann [5], which characterizes the optimizers by their geometric support. Finally, we show a stability result that is twofold: the stability of the optimization problem with respect to target marginals and the relation with another optimal embedding problem. The last part concerns the application of stochastic control to the martingale optimal transport with a payoff depending on the local time, and the Skorokhod embedding problem. For the one-marginal case, we recover the optimizers for both primal a

关键词： martingale optimal transport robust (super)hedging S≠topology dynamic programming principle optimal Skorokhod embedding optimal stopping monotonicity principle Levy-Prokhorov/Wasserstein metric local time stochastic control Vallois' solution

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An Explicit Solution for a Portfolio Selection Problem with Stochastic Volatility

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Journal of Mathematical Finance 2016年 7卷 199-218页

作者： Albert N. Sandjo Fabrice Colin Salissou Moutari Department of Mathematics & Computer Science Laurentian University Sudbury Canada School of Mathematics and Physics Queen’s University Belfast Belfast UK

In this paper, we revisit the optimal consumption and portfolio selection problem for an investor who has access to a risk-free asset (e.g. bank account) with constant return and a risky asset (e.g. stocks) with constant expected return and stochastic volatility. The main contribution of this study is twofold. Our first objective is to provide an explicit solution for dynamic portfolio choice problems, when the volatility of the risky asset returns is driven by the Ornstein-Uhlenbeck process, for an investor with a constant relative risk aversion (CRRA). The second objective is to carry out some numerical experiments using the derived solution in order to analyze the sensitivity of the optimal weight and consumption with respect to some parameters of the model, including the expected return on risky asset, the aversion risk of the investor, the mean-reverting speed, the long-term mean of the process and the diffusion coefficient of the stochastic factor of the standard Brownian motion.

关键词： Portfolio Selection Stochastic Volatility Explicit Solution Ornstein-Uhlenbeck Model dynamic programming principle

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Limit Order Strategic Placement with Adverse Selection Risk and the Role of Latency

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MARKET MICROSTRUCTURE AND LIQUIDITY 2017年第1期3卷

作者： Lehalle, Charles-Albert Mounjid, Othmane Capital Fund Management Paris France Imperial Coll London England Univ Paris 06 Paris France

This paper is split in three parts: first, we use labeled trade data to exhibit how market participants' decisions depend on liquidity imbalance;then, we develop a stochastic control framework where agents monitor limit orders, by exploiting liquidity imbalance, to reduce adverse selection. For limit orders, we need optimal strategies essentially to find a balance between fast execution and avoiding adverse selection: if the price has chances to go down, the probability to be filled is high, but it is better to wait a little more to get a better price. In a third part, we show how the added value of exploiting liquidity imbalance is eroded by latency: being able to predict future liquidity consuming owsis of less use if you do not have enough time to cancel and reinsert your limit orders. There is thus a rationale for market makers to be as fast as possible to reduce adverse selection. Latency costs of our limit order driven strategy can be measured numerically. To authors' knowledge, this paper is the first to make the connection between empirical evidences, a stochastic framework for limit orders including adverse selection, and the cost of latency. Our work is a first step to shed light on the role played by latency and adverse selection in optimal limit order placement.

关键词： Limit order book market microstructure adverse selection optimal trading strategies stochastic control dynamic programming principle jump markov process queuing model

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Stochastic Optimal Control of Stayed Cable Vibrations with Wide-Band Random Wind Excitation using Axial Support Motion

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ADVANCES IN STRUCTURAL ENGINEERING 2015年第9期18卷 1535-1550页

作者： Fu, Bing Wang, Zhenyu Zhao, Yan Yang, Liu Shenyang Jianzhu Univ Sch Civil Engn Shenyang 110168 Peoples R China Zhejiang Univ Coll Civil Engn & Architecture Hangzhou 310058 Zhejiang Peoples R China

This paper proposes an optimal control strategy and corresponding method, to use the advantageous characteristics of the axial support motion to control the stayed cable vibrations in plane by wind excitation. Through a thorough investigation of wide-band random wind excitation and spatial relativity of the wind load along the cable, this study derives realistic and accurate motion equations of the control system. These motion equations are simplified into the form of the first four modes using the Galerkin method. The Ito stochastic differential equations of partial average for the system energy are derived using the stochastic averaging method of the quasi-integrable Hamiltonian systems. Furthermore, the dynamic programming equations with performance index are established by the stochastic dynamic programming principle. The random optimal control law is obtained by solving the dynamic programming equations using the approximate method. The numerical results show that, when standard deviations of the axial support motion are the same, the reduction of the standard deviation of the cable acceleration, and the cable displacement under the stochastic optimal control (SOC) law proposed by this paper, is larger than that under bilinear (BL) control. The SOC method proposed in this paper provides better results than the BL method.

关键词： stayed cable stochastic optimal control dynamic programming principle stochastic averaging method Hamiltonian system

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OPTIMAL STOCHASTIC CONTROL WITH RECURSIVE COST FUNCTIONALS OF STOCHASTIC DIFFERENTIAL SYSTEMS REFLECTED IN A DOMAIN

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ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 2015年第4期21卷 1150-1177页

作者： Li, Juan Tang, Shanjian Shandong Univ Sch Math & Stat Weihai 264200 Weihai Peoples R China Fudan Univ Sch Math Sci Inst Math Shanghai 200433 Peoples R China Fudan Univ Sch Math Sci Dept Finance & Control Sci Shanghai 200433 Peoples R China

The paper is concerned with optimal control of a stochastic differential system reflected in a domain. The cost functional is implicitly defined via a generalized backward stochastic differential equation developed by Pardoux and Zhang [Probab. Theory Relat. Fields 110 (1998) 535-558]. The value function is shown to be the unique viscosity solution to the associated Hamilton-Jacobi-Bellman equation, which is a fully nonlinear parabolic partial differential equation with a nonlinear Neumann boundary condition. The proof requires new estimates for the reflected stochastic differential system.

关键词： Hamilton-Jacobi-Bellman equation nonlinear Neumann boundary value function backward stochastic differential equations dynamic programming principle viscosity solution

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A stochastic control model of investment, production, and consumption on a finite horizon

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2015年第6期38卷 1070-1080页

作者： Han, Xiaoru Yi, Fahuai Foshan Univ Dept Math Foshan 528000 Guangdong Peoples R China S China Normal Univ Sch Math Sci Guangzhou 510631 Guangdong Peoples R China

In this paper, we consider a stochastic control problem on a finite time horizon. The unit price of capital obeys a logarithmic Brownian motion, and the income from production is also subject to the random Brownian fluctuations. The goal is to choose optimal investment and consumption policies to maximize the finite horizon expected discounted hyperbolic absolute risk aversion utility of consumption. A dynamic programming principle is used to derive a time-dependent Hamilton-Jacobi-Bellman equation. The Leray-Schauder fixed point theorem is used to obtain existence of solution of the HJB equation. At last, we derive the optimal investment and consumption policies by the verification theorem. The main contribution in this paper is the use of PDE technique to the finite time problem for obtaining optimal polices. Copyright (c) 2014 John Wiley & Sons, Ltd.

关键词： PDE technique dynamic programming principle optimal policies

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Stochastic Differential Games for Fully Coupled FBSDEs with Jumps

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APPLIED MATHEMATICS AND OPTIMIZATION 2015年第3期71卷 411-448页

作者： Li Juan Wei Qingmeng Shandong Univ Sch Math & Stat Weihai 264209 Weihai Peoples R China NE Normal Univ Sch Math & Stat Changchun 130024 Peoples R China Shandong Univ Sch Math Jinan 250100 Peoples R China

This paper is concerned with stochastic differential games (SDGs) defined through fully coupled forward-backward stochastic differential equations (FBSDEs) which are governed by Brownian motion and Poisson random measure. The upper and the lower value functions are defined by the doubly controlled fully coupled FBSDEs with jumps. Using a new transformation introduced in Buchdahn (Stocha Process Appl 121:2715-2750, 2011), we prove that the upper and the lower value functions are deterministic. Then, after establishing the dynamic programming principle for the upper and the lower value functions of this SDGs, we prove that the upper and the lower value functions are the viscosity solutions to the associated upper and the lower second order integral-partial differential equations of Isaacs' type combined with an algebraic equation, respectively. Furthermore, for a special case (when and do not depend on ), under the Isaacs' condition, we get the existence of the value of the game.

关键词： Fully coupled FBSDEs with jumps Stochastic differential game Hamilton-Jacobi-Bellman-Isaacs equation Value function Stochastic backward semigroup dynamic programming principle Viscosity solution

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