李娟
博士 教授 博士生导师
山东大学 山东省威海市文化西路180号
主要研究领域为随机分析、随机控制，随机微分对策，倒向随机微分方程与金融数学。
个性化签名
 姓名：李娟
 目前身份：在职研究人员
 担任导师情况：博士生导师
 学位：博士

学术头衔：
博士生导师
 职称：高级教授

学科领域：
随机分析
 研究兴趣：主要研究领域为随机分析、随机控制，随机微分对策，倒向随机微分方程与金融数学。
李娟，1972年生，山东省威海市人。教授、硕士生导师、博士生导师。主要研究领域为随机分析、随机控制，随机微分对策，倒向随机微分方程与金融数学。2012年获得首届国家自然科学基金优秀青年基金和2012年山东省自然科学基金杰出青年基金；入选2013年度教育部新世纪优秀人才支持计划；2016年获得牛顿高级学者基金项目；2017年入选教育部特聘教授奖励计划。
学习经历：
2000/92003/7， 山东大学，概率论与数理统计，博士
1994/91997/7， 山东师范大学，概率论与数理统计，硕士
1990/91994/7， 山东师范大学，数学，学士
学术经历：
2007/9至今，山东大学（威海），数学与统计学院，教授
2004/92007/9，山东大学（威海），数学与统计学院，副教授
2011/6至今，山东大学（威海），数学与统计学院，博士生导师
2006/6至今，山东大学（威海），数学与统计学院，硕士生导师
2005/22007/1，复旦大学数学院博士后；法国西布列塔尼大学数学系博士后
行政职务与学术兼职：
2008/年5月至今任数学与统计学院副院长；
2009年9月至今任美国数学会《数学评论》评论员；
2014年任国内核心学术期刊《系统科学与数学》的编委；
2016年任《Probability, Uncertainty and Quantitative Risk》的执行总编；
（杂志网址：http://probabilityrisk.springeropen.com/）
2017年任国际SCI学术期刊《Mathematical Control and Related Fields (MCRF)》的编委；
（杂志网址：http://aimsciences.org/journals/home.jsp?journalID=23）
2019年任中国工业与应用数学学会系统与控制数学专业委员会主任；
(网址：http://www.csiam.org.cn/index.php?s=/Home/Article/lists/category/8.html)
2019年任国内核心学术期刊《应用概率统计》的编委。
(杂志网址: http://aps.ecnu.edu.cn/CN/volumn/current.shtml)

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37
【期刊论文】Representation of limit values for nonexpansive stochastic differential games
Journal of Differential Equations，2021，276（）：187227
2021年03月05日
A classical problem in ergodic control theory consists in the study of the limit behaviour of λVλ(⋅) as λ↘0, when Vλ is the value function of a deterministic or stochastic control problem with discounted cost functional with infinite time horizon and discount factor λ. We study this problem for the lower value function Vλ of a stochastic differential game with recursive cost, i.e., the cost functional is defined through a backward stochastic differential equation with infinite time horizon. But unlike the ergodic control approach, we are interested in the case where the limit can be a function depending on the initial condition. For this we extend the socalled nonexpansivity assumption from the case of control problems to that of stochastic differential games and we derive that λVλ(⋅) is bounded and Lipschitz uniformly with respect to λ>0. Using PDE methods and assuming radial monotonicity of the Hamiltonian of the associated HamiltonJacobiBellmanIsaacs equation we obtain the monotone convergence of λVλ(.) and we characterize its limit W0 as maximal viscosity subsolution of a limit PDE. Using BSDE methods we prove that W0 satisfies a uniform dynamic programming principle involving the supremum and the infimum with respect to the time, and this is the key for an explicit representation formula for W0.
Stochastic nonexpansivity condition Stochastic differential games Radial monotonicity of Hamiltonians Limit value BSDE
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【期刊论文】Representation Formulas for Limit Values of Long Run Stochastic Optimal Controls
SIAM J. Control Optim.，2020，58（4）：1846–1873&
2020年07月06日
A classical problem in stochastic ergodic control consists of studying the limit behavior of the optimal value of a discounted integral in infinite horizon (the so called Abel mean of an integral cost) as the discount factor $\lambda$ tends to zero or the value defined with a Cesàro mean of an integral cost when the horizon $T$ tends to $+ \infty$. We investigate the possible limits in the norm of uniform convergence topology of values defined through Abel means or Ceàro means when $ \lambda \to 0^+ $ and $T \to + \infty $, respectively. Here we give two types of new representation formulas for the accumulation points of the values when the averaging parameter converges. We show that there is only one possible accumulation point which is the same for Abel means or Cesàro means. The first type of representation formula is based on probability measures on the product of the state space and the control state space, which are limits of occupational measures. The second type of representation formula is based on measures which are the projection of invariant measure on the space of relaxed controls. We also give a result comparing the both sets of measures involved in both classes of representation formulas. An important consequence of the representation formulas is the existence of the limit value when one has the equicontinuity property of Abel or Cesàro mean values. This is the case, for example, for nonexpansive stochastic control systems. In the end some insightful examples are given which help to better understand the results.
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【期刊论文】Representation of asymptotic values for nonexpansive stochastic control systems
Stochastic Processes and their Applications，2019，129（2）：634673
2019年02月01日
In ergodic stochastic problems the limit of the value function Vλ of the associated discounted cost functional with infinite time horizon is studied, when the discounted factor λ tends to zero. These problems have been well studied in the literature and the used assumptions guarantee that the value function λVλ converges uniformly to a constant as λ→0. The objective of this work consists in studying these problems under the assumption, namely, the nonexpansivity assumption, under which the limit function is not necessarily constant. Our discussion goes beyond the case of the stochastic control problem with infinite time horizon and discusses also Vλ given by a Hamilton–Jacobi–Bellman equation of second order which is not necessarily associated with a stochastic control problem. On the other hand, the stochastic control case generalizes considerably earlier works by considering cost functionals defined through a backward stochastic differential equation with infinite time horizon and we give an explicit representation formula for the limit of λVλ, as λ→0.
Stochastic nonexpansivity condition Limit value BSDE
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Stochastics ，2018，91（1）：136
2018年07月22日
We mainly investigate the existence of the Nash equilibrium payoffs for nonzerosum stochastic differential games without assuming Isaacs condition in this paper. Along the partition π of the time interval , we choose a suitable random nonanticipative strategy with delay to study our nonzerosum stochastic differential game. We prove for the corresponding both zerosum stochastic differential games without Isaacs condition the existence of the value functions. With the help of these value functions we give the characterization of the Nash equilibrium payoffs. This characterization allows to prove the existence of Nash equilibrium payoffs.
Nash equilibrium payoffs， Isaacs condition， value function， viscosity solution
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【期刊论文】General meanfield BSDEs with continuous coefficients
Journal of Mathematical Analysis and Applications，2018，466（1）：264280
2018年10月01日
In this paper we consider one dimensional general meanfield backward stochastic differential equations (BSDEs), i.e., the generator of our meanfield BSDEs depends not only on the solution but also on the law of the solution. We first give a totally new comparison theorem for such type of BSDEs under Lipschitz condition. Furthermore, we study the existence of the solution of such meanfield BSDEs when the coefficients are only continuous and with a linear growth.
Backward stochastic differential equations Meanfield Wasserstein metric
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Stochastic Processes and their Applications，2018，128（9）：31183180
2018年09月01日
In this paper we consider a meanfield backward stochastic differential equation (BSDE) driven by a Brownian motion and an independent Poisson random measure. Translating the splitting method introduced by Buckdahn et al. (2014) to BSDEs, the existence and the uniqueness of the solution (Yt,ξ,Zt,ξ,Ht,ξ), (Yt,x,Pξ,Zt,x,Pξ,Ht,x,Pξ) of the split equations are proved. The first and the second order derivatives of the process (Yt,x,Pξ,Zt,x,Pξ,Ht,x,Pξ) with respect to x, the derivative of the process (Yt,x,Pξ,Zt,x,Pξ,Ht,x,Pξ) with respect to the measure Pξ, and the derivative of the process (∂μYt,x,Pξ(y),∂μZt,x,Pξ(y),∂μHt,x,Pξ(y)) with respect to y are studied under appropriate regularity assumptions on the coefficients, respectively. These derivatives turn out to be bounded and continuous in L2. The proof of the continuity of the second order derivatives is particularly involved and requires subtle estimates. This regularity ensures that the value function V(t,x,Pξ)≔Ytt,x,Pξ is regular and allows to show with the help of a new Itô formula that it is the unique classical solution of the related nonlocal quasilinear integralpartial differential equation (PDE) of meanfield type.
BSDEs with jump Meanfield BSDEs with jump IntegralPDE of meanfield type Itô’s formula Value function
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【期刊论文】Meanfield stochastic differential equations and associated PDEs
Ann. Probab.，2017，45（2）：824878
2017年03月01日
In this paper, we consider a meanfield stochastic differential equation, also called the McKean–Vlasov equation, with initial data (t,x)∈[0,T]×Rd, whose coefficients depend on both the solution Xt,xs and its law. By considering square integrable random variables ξ as initial condition for this equation, we can easily show the flow property of the solution Xt,ξs of this new equation. Associating it with a process Xt,x,Pξs which coincides with Xt,ξs, when one substitutes ξ for x, but which has the advantage to depend on ξ only through its law Pξ, we characterize the function V(t,x,Pξ)=E[Φ(Xt,x,PξT,PXt,ξT)] under appropriate regularity conditions on the coefficients of the stochastic differential equation as the unique classical solution of a nonlocal partial differential equation of meanfield type, involving the first and the secondorder derivatives of V with respect to its space variable and the probability law. The proof bases heavily on a preliminary study of the first and secondorder derivatives of the solution of the meanfield stochastic differential equation with respect to the probability law and a corresponding Itô formula. In our approach, we use the notion of derivative with respect to a probability measure with finite second moment, introduced by Lions in [Cours au Collège de France: Théorie des jeu à champs moyens (2013)], and we extend it in a direct way to the secondorder derivatives.
McKean–Vlasov equation,， Meanfield stochastic differential equation,， PDE of meanfield type,， value function
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【期刊论文】Zerosum and nonzerosum differential games without Isaacs condition
ESAIM: Control, Optimisation and Calculus of Variations (ESAIM: COCV)，2017，23（3）：12171252
2017年05月12日
In this paper we study differential games without Isaacs condition. The objective is to investigate on one hand zerosum games with asymmetric information on the payoff, and on the other hand, for the case of symmetric information but now for a nonzero sum differential game, the existence of a Nash equilibrium payoff. Our results extend those by Buckdahn, Cardaliaguet and Rainer [SIAM J. Control Optim. 43 (2004) 624–642], to the case without Isaacs condition. To overcome the absence of Isaacs condition, randomization of the nonanticipative strategies with delay of the both players are considered. They differ from those in Buckdahn, Quincampoix, Rainer and Xu [Int. J. Game Theory 45 (2016) 795–816]. Unlike in [Int. J. Game Theory 45 (2016) 795–816], our definition of NAD strategies for a game over the time interval [ t,T ] (0 ≤ t ≤ T) guarantees that a randomized strategy along a partition π of [ 0,T ] remains a randomized NAD strategy with respect to any finer partition π′ (π ⊂ π′). This allows to study the limit behavior of upper and lower value functions defined for games in which the both players use also different partitions.
Zerosum and nonzerosum differential game /， asymmetric information /， Isaacs condition /， Nash equilibrium payoffs /， Fenchel transformation
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SIAM J. Control Optim.，2016，54（3）：1826–1858&
2016年06月30日
This work is devoted to the study of stochastic differential equations (SDEs) whose diffusion coefficient $\sigma(s,X_{\cdot\wedge s})$ is Lipschitz continuous with respect to the path of the solution process $X$, while its drift coefficient $b(s,X_{\cdot\wedge s},Q_{X_s})$ is only measurable with respect to $X$ and depends continuously (in the sense of the 1Wasserstein metric) on the law of the solution process. Embedded in a meanfield game, the weak existence for such SDEs with meanfield term was recently studied by Lacker [Stochastic Process. Appl., 125 (2015), pp. 28562894] and Carmona and Lacker [Ann. Appl. Probab., 25 (2015), pp. 11891231] under only sequential continuity of $b(s,X_{\cdot\wedge s},Q_{X_s})$ in $Q_{X}$ with respect to a weak topology, but for uniqueness, Carmona and Lacker supposed that $b$ is independent of the meanfield term. We prove the uniqueness in law for $b$ depending on the meanfield, and the proof of the existence of a weak solution, relatively short in comparison with Carmona and Lacker's work, is extended in section 5 of this paper to the study of 2person zerosum stochastic differential games described by doubly controlled coupled meanfield forwardbackward SDEs with dynamics whose drift coefficient is only measurable with respect to the state process.
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【期刊论文】BSDES in games, coupled with the value functions. associated nonlocal BellmanIsaacs equations
Acta Mathematica Scientia，2017，37（5）：14971518
2017年09月01日
We establish a new type of backward stochastic differential equations (BSDEs) connected with stochastic differential games (SDGs), namely, BSDEs strongly coupled with the lower and the upper value functions of SDGs, where the lower and the upper value functions are defined through this BSDE. The existence and the uniqueness theorem and comparison theorem are proved for such equations with the help of an iteration method. We also show that the lower and the upper value functions satisfy the dynamic programming principle. Moreover, we study the associated HamiltonJacobiBellmanIsaacs (HJBIsaacs) equations, which are nonlocal, and strongly coupled with the lower and the upper value functions. Using a new method, we characterize the pair (W,U) consisting of the lower and the upper value functions as the unique viscosity solution of our nonlocal HJBIsaacs equation. Furthermore, the game has a value under the Isaacs' condition.
McKeanVlasov SDE BSDE coupled with the lower and the upper value functions dynamic programming principle meanfield BSDE viscosity solution coupled nonlocal HJBIsaacs equation Isaacs'， condition
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