Workshop on Applied Mathematics

June 6-8, 2016, Shandong University

Sponsors

School of Mathematics, Shandong University

Qilu Securities institute for financial studies, Shandong University

Joint Research Center on Financial Mathematics

Joint Research Center Committee

Directors：

Prof. PengShige, Shandong University

Prof. Chen Xiaojun, The Hong Kong Polytechnic University

Prof. Ng Michael Kwok-Po, Hong Kong Baptist University

Deputy Directors：

Prof. Chen Zengjing,Shandong University

Prof. Wu Zhen, Shandong University

Dr. Yiu Ka Fai Cedric, The Hong Kong Polytechnic University

Dr. Yuan Xiaoming, Hong Kong Baptist University

Venue

Zhixin Building B1238, Shandong University, Jinan, Shandong

Participants

Bensoussan Alain, University of Texas at Dallas and City University of Hong Kong

Chen Zengjing, Shandong University

HuMingshang, Shandong University

Huang Jianhui James, The Hong Kong Polytechnic University

Huang Zongyuan, Shandong University

Ji Shaolin, Shandong University

JiaGuangyan, Shandong University

Li Xinpeng, Shandong University

Lin Lu,Shandong University

Luan Yihui, Shandong University

NieTianyang, Shandong University

Peng Shige, Shandong University

Shi Jingtao, Shandong University

Shi Yufeng, Shandong University

Wang Falei, Shandong University

Wang Hanchao, Shandong University

Wang Xin, Shandong University

Wei Gang, Shandong University

Wu Panyu, Shandong University

Wu Zhen, Shandong University

XuZuoquan, The Hong Kong Polytechnic University

Yiu Ka Fai Cedric, The Hong Kong Polytechnic University

Yu Xiang, The Hong Kong Polytechnic University

Zhang Guofeng, The Hong Kong Polytechnic University

Zhang Zaikun, The Hong Kong Polytechnic University

Zhao Weidong, Shandong University

Zhao Xingqiu, The Hong Kong Polytechnic University

Remark

Dinner of June 6:  Buffet in Xueren Hotel

Banquet of June 7: Xueren Hotel

Coffee Break and Lunch: Zhixin Building B1135

Abstracts of Talks

Title: Capital Accumulation and Real Options

Bensoussan Alain

University of Texas at Dallas and City University of Hong Kong

      abensous@cityu.edu.hk and Alain.Bensoussan@utdallas.edu

Abstract: We study here the situation of a firm which exploits an external resource, and decides its investments at appropriate times, in the spirit of real options. However, we are interested in a sequence of projects, and not just a single one. Each project represents a substantial investment, with fixed cost and variable costs measuring the scale of the project. At the same time, the firm is growing and thus accumulates capital, which puts it each time in a more favorable position to exploit the external resource. The problem is to define the sequence of optimal stopping times to invest. We follow the methodology of impulse control, in which the value function is the solution of a Quasi Variational Inequality (QVI). We obtain new types of QVI, which we can solve in some particular cases.

Product space for two processes with independent increments under nonlinear expectations

Hu Mingshang

Shandong University

humingshang@sdu.edu.cn

Abstract: In this paper, we consider the product space for two processes with independent increments under nonlinear expectations. By introducing a discretization method, we construct a nonlinear expectation under which the given two processes can be seen as a new process with independent increments.

Linear-Quadratic Mean-Field Games with Input Constraints

Huang Jianhui James

The Hong Kong Polytechnic University

majhuang@polyu.edu.hk

Abstract:We study the linear-quadratic mean-field games in which the input is subject to constraints characterized by some convex-cone. The consistency condition is established by the monotonicity condition of projection operator and some mean-field type forward-backward stochastic differential equation. The approximate Nash equilibrium is also verified.

Title: Risk, uncertainty and arbitrage

Li Xinpeng

Shandong University

lixinpeng@sdu.edu.cn

Abstract: We discuss the notions of risk and uncertainty in the financial market following Knight (1921). Empirical research of Chinese and French option markets will support our theoretical results.

Title:A BSDE approach to fair bilateral pricing under funding costs and collateralization

NieTianyang

Shandong University

nietianyang@sdu.edu.cn

Abstract: Bielecki and Rutkowski (2015) introduced and studied a generic non-linear market model,which includes several risky assets, multiple funding accounts and margin accounts. In this talk, we examine the pricing and hedging of contract both from the perspective of the hedgerand the counterparty with arbitrary initial endowments. We derive inequalities for unilateralprices and we study the range of fair bilateral prices andwe study the positive homogeneityand monotonicity of unilateral prices with respect to the initial endowments. We alsoextend the results to the case of an endogenous marginaccount depending on the contract’s value for the hedger and/or the counterparty by using the backward stochasticviability property. Our results generalized in several respects theoption pricing results from Bergman, Mercurio and Piterbarg, respectively, byconsidering contracts with cash stream flows and allowing for idiosyncraticfunding costs for risky assets.

This talk is based on joint works with Prof. MarekRutkowski (University of Sydney, Australia).

Title: Connection between MP and DPP for Stochastic Recursive Optimal Control Problems: Viscosity Solution Framework in General Case

Shi Jingtao

Shandong University

shijingtao@sdu.edu.cn

Abstract: This talk deals with a stochastic recursive optimal control problem, where the diffusion coefficient depends on the control variable and the control domain is not necessarily convex.We focus on the connection between the general maximum principle and the dynamic programming principle for such control problem without the assumption that the value is smooth enough, the set inclusions among the sub- and super-jets of the value function and the first-order and second-order adjoint processes as well as the generalized Hamiltonian function are established. Moreover, bycomparing this results with the ones in Yong and Zhou [Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999], it is natural to obtain the first-order andsecond-order adjoint equations of Hu [Diret method on stochastic maximum principle for optimization with recursive utilities, arXiv:1507.03567v1 [math.OC], 13 Jul. 2015]. (Joint work with Dr. TianyangNie and Prof. Zhen Wu.)

Title: Stochastic optimal control with infinite horizon and Hamilton- Jacobi-Bellman equations in the G-expectation framework

Wang falei

Shandong University

flwang@sdu.edu.cn

Abstract: The present paper considers a stochastic optimal control problem,  in which the cost function is defined through  a backward stochastic differential equation with infinite horizon driven by G-Brownian motion. Then we study the regularities of the value function and establish the dynamic programming principle. Moreover, we prove that the value function is the uniqueness viscosity solution of the related HJB equation.

Title:Weak convergence to stochastic integrals for econometric applications

Wang Hanchao

Shandong University

wanghanchao@sdu.edu.cn

Abstract: In cointegration model, endogeneity and nonlinearity play major roles and complicate the limit theory.  In this talk, we explore weak convergence limit theory to overcome endogeneity and nonlinearity, and obtain some weak convergence theorems on stochastic integrals.  An nonlinear extension of FM regression is used to illustrate practical application of our results.

Title:Strong laws of large numbers for non-additive probabilities

Wu Panyu

Shandong University

wupanyu@sdu.edu.cn

Abstract: In this talk, we give the strong laws of large numbers for non-additive probabilities with the notion of independence for random variables under upper expectations. These results are natural extensions of the classical Kolmogorov’s strong law of large numbers to the case where the probability is no longer additive.

Title:Investment model with intractable claims

XuZuoquan

The Hong Kong Polytechnic University

zuoquan.xu@polyu.edu.hk

Abstract: We will present a Markowitz mean-variance models with intractable claim involvedin the terminal wealth. The term “intractable claim” refers to claims (rewardsor losses) that are irrelevant to the underlying market. The payoffs of such claimscannot be predicted or hedged based on the underlying financial market even if theinformation of the financial market is increasingly available to the investor over time.The target of the investor is to minimize the variance in the worst scenario over allthe possible realizations of the underlying intractable claim. Because of the timeinconsistentnature of the problem, both the standard penalization approach andthe duality method used to tackle robust stochastic control problems fail for thesemodels. Instead, the quantile formulation and martingale approaches are adoptedto tackle the problems and closed-form solutions are derived. The properties of themean-variance frontiers will also be discussed.

The presentation is based on jointworks with DanlinHou (The Hong Kong Polytechnic University) and Xun Yu Zhou(Columbia University and University of Oxford).

Title:Optimization with engineering and financial applications

YiuK.F.C.

The Hong Kong Polytechnic University,

macyiu@polyu.edu.hk

Abstract: Optimization has been an essential tools for many practical problems, including engineering and financial applications. In this talk, we will discuss some advances in optimal design of filters, as well as broadband beamforming system. We review on various approaches and discuss some of the performance issues. Different optimization models will be considered. In particular, we found that the geometric configuration of the array is important for the accuracy of the designs.

In financial risk management, optimization is applied extensively for portfolio selection. Here, we consider the risk-constrained portfolio selection problems arising from an ordinary investor or an insurer who can invest her surplus into financial market. The goal is to maximize the expected utility of terminal wealth. We will examine a few scenarios with different stochastic processes and discuss how to solve the resulting HJB equation. Furthermore, we will investigate the impacts of the risk constraint on the optimal strategies.

Title:On the Market Viability under Proportional Transaction Costs

Yu Xiang

The Hong Kong Polytechnic University,

xiang.yu@polyu.edu.hk

 Abstract: This project studies the market viability with proportional transaction costs. Instead of requiring the existence of strictly consistent price systems (SCPS) as in the literature, we show that strictly consistent local martingale systems (SCLMS) can successfully serve as the dual elements such that the market viability can be verified. We introduce two weaker notions of no arbitrage conditions on market models named no unbounded profit with bounded risk (NUPBR) and no local arbitrage with bounded portfolios (NLABP). In particular, we show that the NUPBR and NLABP conditions in the robust sense for the smaller bid-ask spreads is the equivalent characterization of the existence of SCLMS for general market models. We also discuss the implications for the utility maximization problem.

Title:Continuous-time multi-photon filtering

Zhang Guofeng

The Hong Kong Polytechnic University,

guofeng.zhang@polyu.edu.hk

Abstract: In this talk we discuss filtering for an arbitrary open quantumsystem driven by a light wavepacket in a continuous-modemulti-photon state.  A continuous-mode multi-photon state is a state of atravelling wavepacket that contains a definite number of photons and ischaracterised by a temporal (or spectral) profile.  After the interactionwith the system, the output light is measured by means of homodynedetection or photodetection. Filters for both cases are derived in thispaper.  As illustrated by an example --- a two-level atom driven by acontinuous-mode two-photon state, multi-photon filters  can revealinteresting optical phenomena absent in either  the single-photon filtercase or the continuous-mode Fock state case.

Title:A Subspace Decomposition Framework for Nonlinear Optimization

Zhang Zaikun

The Hong Kong Polytechnic University,

zaikun.zhang@polyu.edu.hk

Abstract: We present a parallel subspace decomposition framework for nonlinear optimization, which can be regarded as an extension of the domain decomposition method for PDEs. A feature of the framework is that it incorporates the restricted additive Schwarz methodology into the synchronization phase of the algorithm. We establish the global convergence and worst case iteration complexity of the framework, and illustrate how this framework can be applied to design parallel algorithms for optimization problems with or without derivatives.

This is a joint work with S. Gratton (IRIT/ENSEEIHT/INPT, France) and L. N. Vicente (University of Coimbra, Portugal).

Title:Sieve Estimation of Cox Models with Latent Structures

Zhao Xingqiu

The Hong Kong Polytechnic University,

xingqiu.zhao@polyu.edu.hk

Abstract: In this talk we considers sieve estimation of Cox models with unknown structures based on right censored data, which often occur in survival studies. For the problem, we propose a semiparametric pursuit method to simultaneouslyidentify and estimate linear and nonlinear covariate effects on the log hazards function through a penalized groupselection method with folded concave penalties. Both the parametric and nonparametric estimators are consistent, and the parametric estimator is asymptotically normal. To compute the proposed estimators, we develop a modified blockwisemajorization descent algorithm that is easy to implement and has a fast convergence rate. Both simulation studies and real data analysis results indicate that the proposed method works well.

Shi Chunmei15866727100  Email: meisdu@sdu.edu.cn

Shi Jingtao  13583117831  Email: shijingtao@sdu.edu.cn

NieTianyang15288861361  Email: nietianyang@sdu.edu.cn

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