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Mathematical Control & Related Fields

2015 , Volume 5 , Issue 3

Special issue in Memory of Xunjing Li

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A biographical note and tribute to xunjing li on his 80th birthday
Jin Ma , Shige Peng , Jiongmin Yong and  Xunyu Zhou
2015, 5(3): i-iii doi: 10.3934/mcrf.2015.5.3i +[Abstract](48) +[PDF](87.4KB)
Abstract:
Professor Xunjing Li was born in Qingdao, Shandong Province, China, on the 13th June 1935. Shandong is a province with a rich culture that has nurtured a great number of influential intellectuals during its long history, including Confucius. Immediately after his graduation from the Department of Mathematics at Shandong University in 1956, Professor Li was enrolled into the master program at Fudan University specializing in function approximation theory, supervised by Professor Jiangong Chen, one of the most prominent Chinese mathematicians in modern history. He stayed at Fudan as an Assistant Lecturer upon graduation in 1959, and was promoted to Lecturer, Associate Professor and Professor in 1962, 1980, and 1984, respectively. He became a Chair Professor in 1997, before retiring in 2001. He died of cancer in February 2003 at the age of 68.

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Sparse initial data identification for parabolic PDE and its finite element approximations
Eduardo Casas , Boris Vexler and  Enrique Zuazua
2015, 5(3): 377-399 doi: 10.3934/mcrf.2015.5.377 +[Abstract](78) +[PDF](455.0KB)
Abstract:
We address the problem of inverse source identification for parabolic equations from the optimal control viewpoint employing measures of minimal norm as initial data. We adopt the point of view of approximate controllability so that the target is not required to be achieved exactly but only in an approximate sense. We prove an approximate inversion result and derive a characterization of the optimal initial measures by means of duality and the minimization of a suitable quadratic functional on the solutions of the adjoint system. We prove the sparsity of the optimal initial measures showing that they are supported in sets of null Lebesgue measure. As a consequence, approximate controllability can be achieved efficiently by means of controls that are activated in a finite number of pointwise locations. Moreover, we discuss the finite element numerical approximation of the control problem providing a convergence result of the corresponding optimal measures and states as the discretization parameters tend to zero.
Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process
Shaokuan Chen and  Shanjian Tang
2015, 5(3): 401-434 doi: 10.3934/mcrf.2015.5.401 +[Abstract](43) +[PDF](599.0KB)
Abstract:
In this paper we investigate classical solution of a semi-linear system of backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. By proving an Itô-Wentzell formula for jump diffusions as well as an abstract result of stochastic evolution equations, we obtain the stochastic integral partial differential equation for the inverse of the stochastic flow generated by a stochastic differential equation driven by a Brownian motion and a Poisson point process. By composing the random field generated by the solution of a backward stochastic differential equation with the inverse of the stochastic flow, we construct the classical solution of the system of backward stochastic integral partial differential equations. As a result, we establish a stochastic Feynman-Kac formula.
Strong law of large numbers for upper set-valued and fuzzy-set valued probability
Zengjing Chen , Yuting Lan and  Gaofeng Zong
2015, 5(3): 435-452 doi: 10.3934/mcrf.2015.5.435 +[Abstract](74) +[PDF](430.7KB)
Abstract:
In this paper, we introduce the concepts of upper-lower set-valued probabilities and related upper-lower expectations for random variables. With a new concept of independence for random variables, we show a strong law of large numbers for upper-lower set-valued probabilities. Furthermore, we extend those concepts and theorem to the case of fuzzy-set.
BMO martingales and positive solutions of heat equations
Ying Hu and  Zhongmin Qian
2015, 5(3): 453-473 doi: 10.3934/mcrf.2015.5.453 +[Abstract](32) +[PDF](421.9KB)
Abstract:
In this paper, we develop a new approach to establish gradient estimates for positive solutions to the heat equation of elliptic or subelliptic operators on Euclidean spaces or on Riemannian manifolds. More precisely, we give some estimates of the gradient of logarithm of a positive solution via the uniform bound of the logarithm of the solution. Moreover, we give a generalized version of Li-Yau's estimate. Our proof is based on the link between PDE and quadratic BSDE. Our method might be useful to study some (nonlinear) PDEs.
Continuous-time portfolio selection under ambiguity
Hanqing Jin and  Xun Yu Zhou
2015, 5(3): 475-488 doi: 10.3934/mcrf.2015.5.475 +[Abstract](91) +[PDF](424.4KB)
Abstract:
In a financial market, the appreciation rates of stocks are statistically difficult to estimate, and typically only some confidence intervals in which the rates reside can be estimated. In this paper we study continuous-time portfolio selection under ambiguity, in the sense that the appreciation rates are only known to be in a certain convex closed set and the portfolios are allowed to be based on only the historical stocks prices. We formulate the problem in both the expected utility and the mean--variance frameworks, and derive robust portfolios explicitly for both models.
Pairs trading: An optimal selling rule
Kevin Kuo , Phong Luu , Duy Nguyen , Eric Perkerson , Katherine Thompson and  Qing Zhang
2015, 5(3): 489-499 doi: 10.3934/mcrf.2015.5.489 +[Abstract](147) +[PDF](335.5KB)
Abstract:
Pairs trading involves two cointegrated securities. When divergence is underway, i.e., one stock moves up while the other moves down, a pairs trade is entered consisting of a short position in the outperforming stock and a long position in the underperforming one. Such a strategy bets the ``spread'' between the two would eventually converge. This paper is concerned with an optimal pairs-trade selling rule. In this paper, a difference of the pair is governed by a mean-reverting model. The trade will be closed whenever the difference reaches a target level or a cutloss limit. Given a fixed cutloss level, the objective is to determine the optimal target so as to maximize an overall return. This optimization problem is related to an optimal stopping problem as the cutloss level vanishes. Expected holding time and profit probability are also obtained. Numerical examples are reported to demonstrate the results.
Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs
Juan Li and  Wenqiang Li
2015, 5(3): 501-516 doi: 10.3934/mcrf.2015.5.501 +[Abstract](93) +[PDF](401.9KB)
Abstract:
In this paper, we consider a new type of reflected mean-field backward stochastic differential equations (reflected MFBSDEs, for short), namely, controlled reflected MFBSDEs involving their value function. The existence and the uniqueness of the solution of such equation are proved by using an approximation method. We also adapt this method to give a comparison theorem for our reflected MFBSDEs. The related dynamic programming principle is obtained by extending the approach of stochastic backward semigroups introduced by Peng [11] in 1997. Finally, we show that the value function which our reflected MFBSDE is coupled with is the unique viscosity solution of the related nonlocal parabolic partial differential equation with obstacle.
Optimal blowup/quenching time for controlled autonomous ordinary differential equations
Hongwei Lou and  Weihan Wang
2015, 5(3): 517-527 doi: 10.3934/mcrf.2015.5.517 +[Abstract](33) +[PDF](374.2KB)
Abstract:
Blowup/Quenching time optimal control problems for controlled autonomous ordinary differential equations are considered. The main results are maximum principles for these time optimal control problems, including the transversality conditions.
Transposition method for backward stochastic evolution equations revisited, and its application
Qi Lü and  Xu Zhang
2015, 5(3): 529-555 doi: 10.3934/mcrf.2015.5.529 +[Abstract](35) +[PDF](544.1KB)
Abstract:
The main purpose of this paper is to improve our transposition method to solve both vector-valued and operator-valued backward stochastic evolution equations with a general filtration. As its application, we obtain a general Pontryagin-type maximum principle for optimal controls of stochastic evolution equations in infinite dimensions. In particular, we drop the technical assumption appeared in [12, Theorem 9.1]. We also establish a Pontryagin-type maximum principle for a stochastic linear quadratic problems.
Dynamic equilibrium limit order book model and optimal execution problem
Jin Ma , Xinyang Wang and  Jianfeng Zhang
2015, 5(3): 557-583 doi: 10.3934/mcrf.2015.5.557 +[Abstract](25) +[PDF](570.0KB)
Abstract:
In this paper we propose a dynamic model of Limit Order Book (LOB). The main feature of our model is that the shape of the LOB is determined endogenously by an expected utility function via a competitive equilibrium argument. Assuming zero resilience, the resulting equilibrium density of the LOB is random, nonlinear, and time inhomogeneous. Consequently, the liquidity cost can be defined dynamically in a natural way.
    We next study an optimal execution problem in our model. We verify that the value function satisfies the Dynamic Programming Principle, and is a viscosity solution to the corresponding Hamilton-Jacobi-Bellman equation which is in the form of an integro-partial-differential quasi-variational inequality. We also prove the existence and analyze the structure of the optimal strategy via a verification theorem argument, assuming that the PDE has a classical solution.
Generalized homogeneous systems with applications to nonlinear control: A survey
Chunjiang Qian , Wei Lin and  Wenting Zha
2015, 5(3): 585-611 doi: 10.3934/mcrf.2015.5.585 +[Abstract](50) +[PDF](662.1KB)
Abstract:
This survey provides a unified homogeneous perspective on recent advances in the global stabilization of various nonlinear systems with uncertainty. We first review definitions and properties of homogeneous systems and illustrate how the homogeneous system theory can yield elegant feedback stabilizers for certain homogeneous systems. By taking advantage of homogeneity, we then present the so-called Adding a Power Integrator (AAPI) technique and discuss how it can be employed to recursively construct smooth state feedback stabilizers for uncertain nonlinear systems with uncontrollable linearizations. Based on the AAPI technique, a non-smooth version as well as a generalized version of AAPI approaches can be further developed from a homogeneous viewpoint, resulting in solutions to the global stabilization of genuinely nonlinear systems that may not be controlled, even locally, by any smooth state feedback. In the case of output feedback control, we demonstrate in this survey why the homogeneity is the key in developing a homogeneous domination approach, which has been successful in solving some difficult nonlinear control problems including, for instance, the global stabilization of systems with higher-order nonlinearities via output feedback. Finally, we show how the notion of Homogeneity with Monotone Degrees (HWMD) plays a decisive role in unifying smooth and non-smooth AAPI methods under one framework. Other applications of HWMD will be also summarized and discussed in this paper, along the directions of constructing smooth stabilizers for nonlinear systems in special forms and ``low-gain'' controllers for a class of general upper-triangular systems.
Optimal control problems of forward-backward stochastic Volterra integral equations
Yufeng Shi , Tianxiao Wang and  Jiongmin Yong
2015, 5(3): 613-649 doi: 10.3934/mcrf.2015.5.613 +[Abstract](56) +[PDF](523.3KB)
Abstract:
Optimal control problems of forward-backward stochastic Volterra integral equations (FBSVIEs, in short) are formulated and studied. A general duality principle is established for linear backward stochastic integral equation and linear stochastic Fredholm-Volterra integral equation with mean-field. With the help of such a duality principle, together with some other new delicate and subtle skills, Pontryagin type maximum principles are proved for two optimal control problems of FBSVIEs.
Time-inconsistent optimal control problem with random coefficients and stochastic equilibrium HJB equation
Haiyang Wang and  Zhen Wu
2015, 5(3): 651-678 doi: 10.3934/mcrf.2015.5.651 +[Abstract](174) +[PDF](438.3KB)
Abstract:
In this paper, we study a class of time-inconsistent optimal control problems with random coefficients. By the method of multi-person differential games, a family of parameterized backward stochastic partial differential equations, called the stochastic equilibrium Hamilton-Jacobi-Bellman equation, is derived for the equilibrium value function of this problem. Under appropriate conditions, we obtain the wellposedness of such an equation and construct the time-consistent equilibrium strategy of closed-loop. Besides, we investigate the linear-quadratic problem as a special and important case.
Constrained nonsmooth utility maximization on the positive real line
Nicholas Westray and  Harry Zheng
2015, 5(3): 679-695 doi: 10.3934/mcrf.2015.5.679 +[Abstract](27) +[PDF](414.5KB)
Abstract:
We maximize the expected utility of terminal wealth in an incomplete market where there are cone constraints on the investor's portfolio process and the utility function is not assumed to be strictly concave or differentiable. We establish the existence of the optimal solutions to the primal and dual problems and their dual relationship. We simplify the present proofs in this area and extend the existing duality theory to the constrained nonsmooth setting.
Razumikhin-type theorems on moment exponential stability of functional differential equations involving two-time-scale Markovian switching
Fuke Wu , George Yin and  Le Yi Wang
2015, 5(3): 697-719 doi: 10.3934/mcrf.2015.5.697 +[Abstract](69) +[PDF](504.6KB)
Abstract:
This work develops moment exponential stability of functional differential equations (FDEs) with Markovian switching, in which a two-time-scale (real time $t$ and fast time $t/\epsilon$ with a small parameter $\epsilon>0$) continuous-time and finite-state Markov chain is used to represent the switching process. The essence is that there is a time-scale separation, which is motivated by the consideration of networked control systems and manufacturing systems. Under suitable conditions, we establish a Razumikhin-type theorem on the $p$th moment exponential $\epsilon$-stability for the small parameter $\epsilon$. By virtue of the Razumikhin-type theorem, we further deduce mean-square exponential stability results for delay differential equations (DDEs) and ordinary differential equations (ODEs) with two-time-scale Markovian switching. These stability results show that the overall system may be stabilized by the Markov switching even when some of the underlying subsystems are unstable. It is noted that in the presence of the Markovian switching, the stationary distribution of the fast changing part of the Markov chain plays an important role. Explicit conditions for the mean-square exponential stability of linear equations are derived and illustrative examples are provided to demonstrate our results.

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