International Conference on Stochastic Analysis and...

International Conference onStochastic Analysis and Related Fields

JUNE 22 - 27, 2019

JIANGSU NORMAL UNIVERSITY, XUZHOU, CHINA

PROGRAM

WeChat

Scientific Committee:Mufa Chen, Zhiming Ma, Michael Rockner

Organizers:Wei Liu, Renming Song, Yingchao Xie

Venues:Hanyuan Hotel (ÇU,), 246 Jiefang South Road,Yunlong District, Xuzhou City, Jiangsu Province, China

Supported by the project funded by Priority Academic Pro-gram Development of Jiangsu Higher Education Institutions

♦ Jun. 23 (Morning) õõU�we(Second floor)

8:30-9:00 Opening Ceremony

9:00-9:20 Conference Photo

ChairµJiagang Ren

9:20-10:00 Mufa Chen New Mathematical View on Quantum Mechanics

10:00-10:30 Tea Break

ChairµWei Liu

10:30-11:10 Xicheng Zhang Stochastic Lagrangian Path for Leray Solutions of 3DNavier-Stokes Equations

11:10-11:50 Jinghai Shao The Existence of Geodesics in Wasserstein Spaces overPath Groups and Loop Groups

12:00-13:30 Lunch ¥ue(First floor)

♦ Jun. 23 (Afternoon) õõU�we(Second floor)

ChairµXicheng Zhang

14:00-14:40 Juan Li Representation formulas for limit values of long run stochas-tic optimal controls

14:40-15:20 Kai Du On Stochastic Parabolicity Conditions for SPDEs

15:20-16:00 Yulin Song Regularity for Mean-field SDEs Driven by Jump Processes

16:00-16:30 Tea Break

ChairµJuan Li

16:30-17:10 Xiangdong Li On Geometric Flows on Wasserstein Space over RiemannianManifolds

17:10-17:50 Jianliang Zhai Well-posedness and Large Deviations for 2-D StochasticNavier-Stokes Equations with Jumps

18:00-20:00 Dinner ¥ue(First floor)

I


ChairµXiangdong Li

8:30-9:10 Elton P. Hsu From Geodesic Flow to Riemannian Brownian Motion

9:10-9:50 Yiming Jiang A Class of SPDE Driven by Fractional Noise

9:50-10:20 Tea Break

ChairµElton P. Hsu

10:20-11:00 Zhao Dong Asymptotic Behavior of Stochastic Evolution Equation

11:00-11:40 Guoli Zhou Global Well-posedness of Stochastic Nematic Liquid Crys-tals with Random Initial and Random Boundary ConditionsDriven by Multiplicative Noise



ChairµLong Jiang

14:00-14:40 Xia Chen Quenched Asymptotics for the Parabolic Anderson Modelsin Time-independent Gaussian Noise

14:40-15:20 Rongchan Zhu Sharp Interface Limit of the Stochastic Cahn-Hilliard Equa-tions

15:20-16:00 Wei Liu (WHU) Large Deviations for Empirical Measures of Mean-fieldGibbs Measures

16:00-16:30 Tea Break

ChairµXia Chen

16:30-17:10 Qi Zhang Robust Consumption Portfolio Optimization with Stochas-tic Differential Utility

17:10-17:50 Fangjun Xu Fractional stochastic wave equation driven by a Gaussiannoise rough in space

18:00-20:00 Banquet ¥ue(First floor)

II


Chairµ Renming Song

8:30-9:10 Michael Rockner Schauder Theorems for Local and Nonlocal Ornstein-Uhlenbeck Operators on Finite and Infinite DimensionalState Spaces

9:10-9:50 Qing Zhou Some Problems about European Vulnerable Option Pricing


ChairµMichael Rockner

10:20-11:00 Fengyu Wang Diffusions and PDEs on Wasserstein Space

11:00-11:40 Weina Wu Stochastic Generalized Porous Media Equations over � -finite Measure Spaces with Non-continuous Nonlinearity



ChairµFengyu Wang

14:00-14:40 Jian Wang Quenched Invariance Principle for Long Range RandomWalks in Balanced Random Environments

14:40-15:20 Siyu Lv Optimal Switching under a Hybrid Diffusion Model andApplications to Stock Trading

15:20-16:00 Huilin Zhang Exponential Mixing for Dissipative PDEs with BoundedNon-degenerate Noise

16:00-16:30 Tea Break

ChairµJian Wang

16:30-17:10 Yingqiu Li Application of Poisson Approach in Occupation Times

17:10-17:50 Yong Ren Stochastic Multi-group Models with Dispersal Driven by G-Brownian Motion


III


ChairµYingqiu Li

8:30-9:10 Jiagang Ren Equivalence of Distribution and Viscosity Solutions of Neu-mann Problem

9:10-9:50 Lidan Wang Stability of Heat Kernel Estimates for Jump Diffusions un-der Feynman-Kac Perturbations


ChairµYu Miao

10:20-11:00 Tusheng Zhang Stochastic Heat Equation with Logarithmic Nonlinearity

11:00-11:40 Chek Hin Choi Accelerated Simulated Annealing with Fast Cooling



ChairµTusheng Zhang

14:00-14:40 Dongsheng Wu Weak Convergence of Martingales and Its Application toNonlinear Cointegrating Regression Model

14:40-15:20 Cheng Ouyang Density of the Signature Process of an fBM

15:20-16:00 Jing Zhang Dirichlet Form Theory and Probabilistic Representations ofSolutions of Boundary Value Problems

16:00-16:30 Tea Break

Chairµ Dongsheng Wu

16:30-17:10 Yongsheng Song Stein’s Method under Sublinear Expectations

17:10-17:50 Dejian Tian A Generalized Stochastic Differential Utility Driven by G-Brownian Motion


IV


Chairµ Yongsheng Song

8:30-9:10 Fabrice Baudoin BV Functions in Dirichlet Spaces

9:10-9:50 Guoqiang Zheng BSDEs Driven by G-Brownian Motion with UniformlyContinuous Generators


Chairµ Fabrice Baudoin

10:20-11:00 Litan Yan Principal Values of Some Integral Functionals of FractionalBrownian Motion

11:00-11:40 Ran Wang Transportation Inequalities for SPDEs with Time-white andSpace-colored Gaussian Noise or with Levy Noise


♦ Jun. 27 (Afternoon)

Free Discussion


V

| Abstract

BV Functions in Dirichlet SpacesFabrice Baudoin

(University of Connecticut)Abstract In abstract Dirichlet spaces, we present a theory of Besov spaces which is based

on the heat semigroup. This approach offers a new perspective on the class of bounded varia-tion functions in settings including Riemannian manifolds, sub-Riemannian manifolds. In roughspaces like fractals it offers totally new research directions. The key assumption on the underly-ing space is a weak Bakry-Emery type curvature assumption. The talk is based on joint workswith Patricia Alonso-Ruiz, Li Chen, Luke Rogers, Nageswari Shanmugalingam and AlexanderTeplyaev.

New Mathematical View on Quantum MechanicsMufa Chen

(Beijing Normal University)Abstract The quantum mechanics consist of two equivalent subjects, the matrix mechanics

and the wave mechanics. To exihbit the wave property, one needs complex operator; and for thephysical observation, its real spectrum is required. Traditionally, one choose the simplest Hermitematrix (operator). Our first suggestion is replacing the Hermite by Hermitizable. That is theself-adjoint operator on the complex L2.�/ space with explicitly computable positive measure �.The main result says that each Hermitizable matrix is isospectral to a birth–death Q-matrix. Theequivalence consists of three parts. The first and the last ones are explicit, and the second one isprogrammly explicit. For the wave mechanics, we use a gradient term instead of the potential onein the usual Schrodinger operator. This enable to present compelte criteria for discrete spectrumin dimension one.

Quenched Asymptotics for the Parabolic Anderson Modelsin Time-independent Gaussian Noise

Xia Chen(University of Tennessee)

Abstract In this talk, I will discuss the long term quenched asymptotics for the parabolicAnderson models with time-independent Gaussian noise that has been investigated in recent years.In addition, I will explain why this type of behavior only exists in the setting of time-independence.Part of the talk is based on the joint work with Chakrabotry, P., Gao, B. and Tindel, S..

Accelerated Simulated Annealing with Fast CoolingChek Hin Choi

(The Chinese University of Hong Kong, Shenzhen)Abstract Originated from statistical physics, simulated annealing is a popular stochastic op-

timization algorithm that has found extensive empirical success in disciplines ranging from imageprocessing to statistics and combinatorial optimization. At the heart of the algorithm lies in con-structing a non-homogeneous Markov process that converges to the set of global minima as thetemperature cools down. In this talk, we will first review the classical theory for simulated anneal-ing and discuss some of its theoretical limitations. We will then introduce a promising acceleratedvariant of simulated annealing that provably converges faster and does not suffer from the draw-backs of its classical counterpart. This talk is based on http://arxiv.org/abs/1901.10269.

1

Asymptotic Behavior of Stochastic Evolution Equations

Zhao Dong(Academy of Mathematics and Systems Science, CAS)

Abstract The dynamical system determined by ordinary differential equations generally con-tains multiple ergodic states, and the changes in these ergodic states after random perturbationsare the subject of this report. Starting from several special models, we discuss the ergodicity ofstochastic differential equations determined by degenerate and non-degenerate Gaussian noise.The results show that when the noise intensity tends to zero, in the case of non-degenerate noise,although the solution of stochastic differential equations tends to the original deterministic equa-tion, the ergodicity of the solutions of the two equations is essentially different; in the case ofdegraded noise, the ergodicity of the solution becomes more complicated.

On Stochastic Parabolicity Conditions for SPDEs

Kai Du(Fudan University)

Abstract Stochastic parabolicity conditions are a class of structural conditions for stochasticPDEs to ensure wellposedness of the equations, which, in contrast to classical parabolicity con-ditions, involve additionally the coefficients of leading terms in the stochastic integral part. Theproper form of stochastic parabolicity condition for weak solutions of SPDEs were found long agoand also valid in constructing Lp theory and Schauder theory of SPDEs of second-order. How-ever, things may dramatically change for other problems, such as complex-valued SPDEs, systemsof SPDEs, and higher-order SPDEs. More specifically, those conditions for weak solutions (L2

theory) is not sufficient to ensure Lp-integrability of solutions, and certain modified conditionsare required in those cases. The talk will present some p-dependent parabolicity conditions im-posed on systems of SPDEs and on higher-order SPDEs, which ensure us to construct stochasticSchauder theory for those equations. Examples are discussed for necessity of modified parabol-icity conditions and sharpness of our modifications. The talk is based on joint works with JiakunLiu and Fu Zhang and with Yuxing Wang.

From Geodesic Flow to Riemannian Brownian Motion

Elton P. Hsu(Northwestern University)

Abstract We will discuss a natural family of diffusion processes with continuously differ-entiable paths on the tangent bundle over a compact Riemannian manifold that interpolates be-tween Brownian motion and the geodesic flow. We will show that they converge respectivel tothe geodesic flow and Riemannian Brownian motion at the two ends of the parameter interval inthe strong sense in the path space. In the simplest case of the standard Brownian motion this idealeads to an interesting proof of the classical Ito’s formula without discrete approximation.

A Class of SPDE Driven by Fractional Noise

Yiming Jiang(Nankai University)

Abstract In this talk, we mainly consider a class of SPDE with gradient driven by fractionalnoise. We first recall a few definitions about the fractional noise and their stochastic integrals.Then we study existence, uniqueness, and regularity of mild solutions to these equations on somehypotheses. The main challenges lie in the Holder regularity estimations of the stochastic integralof Green kernels w.r.t. the fractional noise.

2

Representation Formulas for Limit Values of Long Run Stochastic

Juan Li(Shandong University)

Abstract A classical problem in stochastic ergodic control consists of studying the limitbehavior of the optimal value of a discounted integral in infinite horizon (the so called Abel meanof an integral cost) as the discount factor � tends to zero or the value defined with a Cesaro mean ofan integral cost when the horizon T tends to C1. We investigate the possible limits in the normof uniform convergence topology of values defined through Abel mean or Cesaro means when�! 0C and T !C1, respectively. Here we give two types of new representation formulas forthe ccumulation points of the values when the averaging parameter converges. Based on a jointwork with Rainer Buckdahn (Brest, France), Marc Quincampoix (Brest,France), Jerome Renault(Toulouse, France).

On Geometric Flows on Wasserstein Space over Riemannian Manifolds

Xiangdong Li(Chinese Academy of Sciences)

Abstract We introduce a family of Langevin flows on the Wasserstein space over over Rie-mannian manifolds, which interpolate the geodesic flow on the Wasserstein space and the heatflow over the underlying manifolds. We prove that the Langevin flow converges to the heat flowand the geodesic flow when the parameter in the flow tends to zero and to infinity respectively.Joint work with Songzi Li.

Application of Poisson Approach in Occupation Times

Yingqiu Li(Changsha University of Science and Technology)

Abstract We adopt the Poisson approach of Li and Zhou (2014) to consider the joint Laplacetransform of occupation times for SNLP and diffusion processes. We discuss the occupation timesrelated first exiting time and last exiting time, and obtain some results: 1) Occupation times overintervals .a; r/ and .r; b/ before it first exits from either a or b. 2) Potential measures that arediscounted by joint occupation times over semi-infinite intervals .�1; a/ and .a;C1/. 3) JointLaplace transforms involving the last exit time (from a semi-infinite interval), the value of theprocess at the last exit time and associate occupation time.

Large Deviations for Empirical Measures of Mean-Field Gibbs Measures

Wei Liu(Wuhan University)

Abstract We will show that the empirical measure of mean-field model satisfies the largedeviation principle with respect to the weak convergence topology or the stronger Wassersteinmetric, under the strong exponential integrability condition on the negative part of the interactionpotentials. In contrast to the known results we prove this without any continuity or boundednesscondition on the interaction potentials. Joint work with Liming Wu.

3

Optimal Switching under a Hybrid Diffusion Modeland Applications to Stock Trading

Siyu Lv(Southeast University)

Abstract This talk is concerned with the optimal switching problem under a hybrid diffu-sion (or, regime switching) model in an infinite horizon. The state of the system consists of anumber of diffusions coupled by a finite-state continuous-time Markov chain. Based on the dy-namic programming principle, the value function of our optimal switching problem is proved tobe the unique viscosity solution to the associated system of variational inequalities. The opti-mal switching strategy, indicating when and where it is optimal to switch, is given in terms of theswitching and continuation regions. In many applications, the underlying Markov chain has a largestate space and exhibits two-time-scale structure. In this case, a singular perturbation approach isemployed to reduce the computational complexity involved. It is shown that as the time-scaleparameter " goes to zero, the value function of the original problem converges to that of a limitproblem. The limit problem is much easier to solve, and its optimal switching solution leads to anapproximate solution to the original problem. Finally, as an application of our theoretical results,an example concerning the stock trading problem in a regime switching market is provided. It isemphasized that, this paper is the first time to introduce the optimal switching as a general.

Density of the Signature Process of an fBM

Cheng Ouyang(University of Illinois at Chicago)

Abstract We study the density of the signature of a fr actional Brownian motions with pa-rameterH > 1=4. In particular, we prove existence, positivity, global Gaussian upper bounds andVaradhan’s type asymptotics for this density. A key result is that the estimates on the density weobtain are controlled by the Carnot-Caratheodory distance of the group.

Equivalence of Distribution and Viscosity Solutions of Neumann Problem

Jiagang Ren(Sun Yat-sen University)

Abstract We prove that for linear elliptic or parabolic equations in a bounded smooth do-main, the two widely used notions of weak solutions, the distribution ones and viscosity ones, areequivalent.

Stochastic Multi-group Models with Dispersal Driven by G-Brownian Motion

Yong Ren(Anhui Normal University)

Abstract In this talk, I will introduce a class of stochastic multi-group models with dispersaldriven by G-Brownian motion. The talk mainly include three topics. In the first part, I will give thestability analysis for stochastic multi-group models with dispersal driven by G-Brownian motion.In the second part, I will show the stabilization for stochastic multi-group models with dispersalby feedback control based on discrete-time observations in diffusion part. In the last part, I willestablish the exponential synchronization for stochastic multi-group models with dispersal drivenby G-Brownian motion. This work is joint with Huijing Yang, Kai Wang and Lanying Hu.

4

Schauder theorems for local and nonlocal Ornstein-Uhlenbeck operatorson finite and infinite dimensional state spaces

Michael Rockner(Bielefeld University)

Abstract We prove maximal regularity results in Hoolder and Zygmund spaces for linearstationary and evolution equations driven by a large class of differential and pseudo-differentialoperators L, both in finite and in infinite dimension. The assumptions are given in terms of thesemigroup generated by L. We cover the cases of fractional Laplacians and Ornstein-Uhlenbeckoperators with fractional diffusion in finite dimension, and several types of local and nonlocalOrnstein-Uhlenbeck operators, as well as the Gross Laplacian and its negative powers, in infinitedimension.

The Existence of Geodesics in Wasserstein Spacesover Path Groups and Loop Groups

Jinghai Shao(Tianjin University)

Abstract In this talk, we introduce a result on the existence and uniqueness of the optimaltransport map for Lp-Wasserstein distance with p > 1, and particularly present an explicit expres-sion of the optimal transport map for the case p D 2. As an application, we show the existence ofgeodesics connecting probability measures satisfying suitable condition on path groups and loopgroups.

Stein’s Method under Sublinear Expectations

Yongsheng Song(Chinese Academy of Sciences)

Abstract As is known, Stein’s method is a powerful tool to study normal approximation. Weestablish Stein’s method under sublinear expectations, through which we obtain error estimatesfor the central limit theorem as well as the law of large numbers under sublinear expectations.

Regularity for Mean-field SDEs Driven by Jump Processes

Yulin Song(Nanjing University)

Abstract In this talk, by Malliavin calculus for Poisson functional, sharp gradient estimatesfor Mean-field SDEs driven by jump processes are established in non-degenerate case. When thedriven noises are additive degenerate LWvy processes, smoothness of the density functions arederived.

5

A Generalized Stochastic Differential Utility Driven by G-Brownian Motion

Dejian Tian(China University of Mining and Technology)

Abstract This paper introduces a class of generalized stochastic differential utility model(GSDU) in a continuous-time framework that captures aversion ambiguity about both expectedreturn and volatility of asset payoffs. We demonstrate the continuity, monotonicity, time consis-tency, concavity, and homotheticity of this generalized stochastic differential utility model. Wealso discuss comparative ambiguity aversion and direction aversion under sufficient conditions forthis class of GSDU model. This class of GSDU model is sufficiently general to include many clas-sical approaches to ambiguity aversion. As an illustration, we also consider an optimal portfoliochoice problem under this generalized stochastic differential utility model. This is a joined workwith Qian Lin and Weidong Tian.

Diffusions and PDEs on Wasserstein Space

Fengyu Wang(Tianjin University)

Abstract By solving image dependent SDEs, diffusion processes are constructed on theWasserstein space. The exponenential ergodicity and the Feyman-Kac formula are investigatedfor the diffusion processes, where the latter provides probabilistic representations of solutions toparabolic PDEs involving the second order intrinsic derivative on the Wasserstein space.

Quenched Invariance Principle for Lang Range Random Walksin Balanced Random Environments

Jian Wang(Fujian Normal University)

Abstract We adopt a pure probabilistic approach to establish the quenched invariance prin-ciple for a class of long range random walks, with the transition probability from x to y is onaverage comparable to jx � yj�.dC˛/ with ˛ 2 .0; 2�, on independent (not necessarily identicallydistribution) balanced random environments. We use the martingale property to control distribu-tions of exit times and then to derive tightness of the scaled processes, and apply the uniquenessof the martingale problem to identify the limit. When ˛ 2 .0; 1/, the approach still works even fornon-balanced cases. When ˛ D 2, under the diffusive with the logarithmic perturbation scalingthe limit of scaled processes is a diffusion.

Stability of Heat Kernel Estimates for Jump Diffusionsunder Feynman-Kac Perturbations

Lidan Wang(Nankai University)

Abstract In this talk, we consider any strong Markov process whose transition density func-tion enjoys two-sided estimates consisting of both Gaussian and power components. We will showthat, under certain Kato class conditions, the heat kernel of a non-local Feynman-Kac semigrouphas similar two-sided estimates, but with a set of possibly different Gaussian coefficients. This isa joint work with Prof. Zhen-Qing Chen.

6

Transportation Inequalities for SPDEs with Time-white and Space-coloredGaussian Noise or with Levy Noise

Ran Wang

Abstract Recently, the problem of Talagrand.s transportation inequalities to stochastic (par-tial) differential equations has been widely studied. We study the W1H transportation inequalityon the continuous paths space w.r.t. the uniform distance, for the law of a stochastic heat equationdefined on Œ0; T � � Œ0; 1�d . This equation is driven by a Gaussian noise, white in time and corre-lated in space. For SPDE with jumps, we prove the W1H transportation inequality on the rightcontinuous paths space w.r.t. the L1-metric. This talk is based the joint works with Shijie Shangand Yutao Ma, respectively.

Weak Convergence of Martingales and Its Application toNonlinear Cointegrating Regression Model

Dongsheng Wu(University of Alabama in Huntsville)

Abstract In this talk, we provide a weak convergence result for a class of martingales. As anapplication, using the marked empirical processes, we develop a test of parametric specification ina nonlinear cointegrating regression model. This talk is based on a joint work with Qiying Wangand Ke Zhu.

Stochastic Generalized Porous Media Equations over � -finiteMeasure Spaces with Non-continuous Nonlinearity

Weina Wu(Nanjing University of Finance and Economics)

Abstract In this paper, we prove that the stochastic porous media equations over a � -finitemeasure spaces .E;B; �/, driven by time-dependent multiplicative noise, with the Laplacian re-placed by a self-adjoint transient Dirichlet operator L and the nonlinearity given by a maximalmonotone multi-valued function of polynomial growth, has a unique solution. This generalizesprevious results in two directions: First, we don’t restrict ourselves to open domains E � Rd ,and second, we can drop the coercivity assumption on . The result in particular applies to caseswhere E is a manifold or a fractal. This talk is based on a joint work with Michael Roeckner andYingchao Xie.

Fractional Stochastic Wave Equation Driven by a Gaussian Noise Rough inSpace

Fangjun Xu(East China Normal University)

Abstract In this article, we consider fractional stochastic wave equations on R driven by amultiplicative Gaussian noise which is white/colored in time and has the covariance of a fractionalBrownian motion with Hurst parameter H 2 .1=4; 1=2/ in space. We prove the existence anduniqueness of the mild Skorohod solution, establish lower and upper bounds for the p-th momentof the solution for all p � 2, and obtain the Holder continuity in time and space variables for thesolution. This is a joint work with Jian Song and Xiaoming Song.

7

Pricipal Values of Some Integral Functionals of Fractional Brownian Motion

Litan Yan(Donghua University)

Abstract Let BH be a fractional Brownian motion with Hurst index 0 < H < 1 and theweighted local time L H .x; t/. In this talk, we consider the existence of the limit

KH;ft .a/ WD lim"#0

�Z t

0

f .BHs � a/ds2HC �Ht .a; "/

�; a 2 R; t � 0

in L2 and almost surely, where f is not locally integrable and

�Ht .a; "/ WD L H .aC "; t/g.aC "/ �L H .a � "; t/g.a � "/:

with g0 D f . This limit (if it exists) can be called the principal value of the integralR t0f .BHs �

a/ds2H . By using the obtained results we give the Ito and occupation type formulas including theprincipal value.

Exponential Mixing for Dissipative PDEs with Bounded Non-degenerate Noise

Huilin Zhang(Fudan University)

Abstract We prove that well-posed quasilinear equations of parabolic type, perturbed bybounded nondegenerate random forces, are exponentially mixing for a large class of randomforces.

Dirichlet Form Theory and Probabilistic Representationsof Solutions of Boundary Value Problems

Jing Zhang(Hainan Normal University)

Abstract Using probabilistic approaches to solve boundary value problems has a long his-tory. In 1944, Kakutani used Brownian motion to represent the solution of the classical Dirichletboundary value problem with the Laplacian operator. In this talk, we will introduce the probabilis-tic representations of solutions of boundary value problems with different operators by using theDirichlet form theory.

Robust Consumption Portfolio Optimization with Stochastic DifferentialUtility

Qi Zhang(Fudan University)

Abstract We study a continuous time intertemporal consumption and portfolio choice prob-lem with a stochastic differential utility preference of Epstein-Zin type for a robust investor, whoworries about model misspecification and seeks robust decision rules. The verification theoremwhich formulates the Hamilton-Jacobi-Bellman-Isaacs equation under a non-Lipschitz conditionis provided. Then with the verification theorem, the explicit closed-form optimal robust consump-tion and portfolio solutions to an Heston model are given. This is a jonit work with JiangyanPu.

8

Stochastic Heat Equation with Logarithmic Nonlinearity

Tusheng Zhang(University of Manchester and USTC)

Abstract In this talk, I will present recent results on the existence and uniqueness of solutionsto stochastic heat equations with logarithmic nonlinearity on a bounded domain in the setting ofL2 spaces.

Stochastic Lagrangian Path for Leray’s solutionsof 3D Navier-Stokes Equations

Xicheng Zhang(Wuhan University)

Abstract In this paper we show the existence of stochastic Lagrangian particle trajectory forLeray’s solution of 3D Navier-Stokes equations. More precisely, for any Leray’s solution u of3D-NSE and each .s; x/ 2 RC � R3, we show the existence of weak solutions to the followingSDE, which has a density �s;x.t; y/ belonging to H1;pq provided p; q 2 Œ1; 2/ with 3

pC

2q> 4:

dXs;t D u.s; Xs;t/dt Cp2�dWt ; Xs;s D x; t � s;

where W is a three dimensional standard Brownian motion, � > 0 is the viscosity constant.Moreover,we also show that for Lebesgue almost all .s; x/, the solution Xn

s;�.x/ of the above SDEassociated with the mollifying velocity field un weakly converges toXs;�.x/ so that X is a Markovprocess in almost sure sense. This is a joint work with Guohuan Zhao.

Well-posedness and Large Deviations for 2-D StochasticNavier-Stokes Equations with Jumps

Jianliang Zhai(University of Science and Technology of China)

Abstract Under the classical Lipschitz and one sided linear growth assumptions on the coeffi-cients of the stochastic perturbations, we first establish the existence and the uniqueness of a strong(in both the probabilistic and PDEs sense) solution to the 2-D Stochastic Navier-Stokes equationsdriven by multiplicative Levy noise. Applying the weak convergence method for the case of thePoisson random measures, we establish a Freidlin-Wentzell type large deviation principle for thestrong solution in PDE sense.

BSDEs Driven by G-Brownian Motion withUniformly Continuous Generators

Guoqiang Zheng(Southeast University)

Abstract We investigate the existence and uniqueness of solutions to a class of non-Lipschitzscalar valued backward stochastic differential equations driven by G-Brownian motion (G-BSDEs).In fact, when the generators of G-BSDEs are Lipschitz continuous in y and uniformly continuousin z, we construct the unique solution to such equations by monotone convergence argument. Thecomparison theorem and related Feynman-Kac formula are stated as well.

9

Global Well-posedness of Stochastic Nematic Liquid Crystals withRandom Initial and Random Boundary Conditions

Driven by Multiplicative NoiseGuoli Zhou

(Chongqing University)Abstract The flow of nematic liquid crystals can be described by a highly nonlinear stochas-

tic hydrodynamical model, thus is often influenced by random fluctuations, such as uncertaintyin specifying initial conditions and boundary conditions. In this article, we consider the 2-Dstochastic nematic liquid crystals with the velocity field perturbed by affine-linear multiplicativewhite noise, with random initial data and random boundary conditions. Our main objective is toestablish the global well-posedness of the stochastic equations under certain sufficient Malliavinregularity of the initial conditions and the boundary conditions. The Malliavin calculus techniquesplay important roles in proving the global existence of the solutions to the stochastic nematic liq-uid crystal models with random initial and random boundary conditions. It should be pointed outthat the global well-posedness is also true when the stochastic system is perturbed by the noise onthe boundary.

Some Problems about European Vulnerable Option PricingQing Zhou

(Beijing University of Posts and Telecomunications)Abstract For the pricing of vulnerable options, firstly, we improve the results of Klein and

Inglis [Journal of Banking and Finance] and Tian et al. [The Journal of Futures and Markets],considering the circumstances in which the writers of options face financial crisis. Our pricingmodel faces the risks of default and the occasional impact experienced by the underlying assetsand counterparty’s assets. The correlation between the option’s underlying assets and the optionwriter’s assets is clearly modeled. Asset prices are driven by the jump-diffusion processes oftwo related assets. Furthermore, we consider a variable default boundary (VDB) based on theoption’s potential debt and the option writer’s other liabilities. In case financial distress happens,the payout rate is connected to the option writer’s assets. Through the Taylor expansion, we derivean approximate explicit valuation for vulnerable options. Secondly, we assume that the dynamicof the corporate liability is a geometric Brownian motion that is related to the underlying assetand the counterparty asset. Under this new framework, we give an explicit pricing formula of thevulnerable European options. Thirdly, we study the pricing of European vulnerable call optionsunder ambiguity. Our research introduces ambiguity into option pricing by two different methods.In both cases, we derive the partial differential equation, by solving which we obtain an explicitsolution of European vulnerable options with ambiguity.

Sharp Interface Limit of the Stochastic Cahn-Hilliard EquationsRongchan Zhu

(Beijing Institute of Technology)Abstract We study the asymptotic limit, as " & 0, of solutions to the stochastic Cahn-

Hilliard equation:

@tu"D �

��"�u" C

1

"f .u"/

�C PW"

t ;

where W" D "�W or W" D "�W ", W is a Q-Wiener process and W " is smooth in time andconverges to W as "& 0. In the case that W" D "�W , we prove that for all � > 1

2, the solution

u" converges to a weak solution to an appropriately defined limit of the deterministic Cahn-Hilliardequation. In radial symmetric case we prove that for all � � 1

2, u" converges to the deterministic

Hele-Shaw model. In the case that W" D "�W ", we prove that for all � > 0, u" converges to theweak solution to the deterministic limit Cahn-Hilliard equation. In radial symmetric case we provethat u" converges to deterministic Hele-Shaw model when � > 0 and converges to a stochasticmodel related to stochastic Hele-Shaw model when � D 0.

10

| Contacts

Name Institute Email

Luigi Accardi University of Roma [email protected]

Yongqiang Bai Jiangsu Normal University

Fabrice Baudoin University of Connecticut [email protected]

Dirk Bloemker University of Augsburg [email protected]

Zdzislaw Brzez-niak

University of York [email protected]

Chek Hin Choi The Chinese University of Hong Kong [email protected]

Bin Chen Jiangsu Normal University [email protected]

Lei Chen Jiangsu Normal University [email protected]

Li Chen University of Connecticut [email protected]

Mufa Chen Beijing Normal University [email protected]

Xia Chen University of Tennessee [email protected]

Zhenqing Chen University of Washington [email protected]

MichaelCranston

University of California at Irvine [email protected]

David Nualart University of Kansas [email protected]

Zhao Dong Chinese Academy of Sciences [email protected]

Weiyong Ding Jiangsu Normal University

Fengjiao Du China University of Mining and Technol-ogy

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Kai Du Fudan University [email protected]

Jie Fang China University of Mining and Technol-ogy

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Shizan Fang University of Burgundy [email protected]

Shui Feng McMaster University [email protected]

Peng Gao China University of Mining and Technol-ogy

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Benjamin Gess MPI Leipzig [email protected]

Zimo Hao Wuhan University

Elton P. Hsu Northwestern University [email protected]

Eryan Hu Tianjin University [email protected]

Mengdie Hu China University of Mining and Technol-ogy

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Yaozhong Hu University of Alberta [email protected]

11

Yueyun Hu University of Paris 13 [email protected]

Xing Huang Tianjin University

Long Jiang China University of Mining and Technol-ogy

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Yiming Jiang Nankai University [email protected]

Panki Kim Seoul National University [email protected]

Chen Li Shandong University [email protected]

Chenggang Li Southwest Jiaotong University [email protected]

Huaiqian Li Tianjin University Tianjin University

Huiqin Li Jiangsu Normal University [email protected]

Jianbo Li Jiangsu Normal University [email protected]

Juan Li Shandong University [email protected]

Ruinan Li Shanghai University of InternationalBusiness and Economics

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Shihu Li Nankai University [email protected]

Xianbin Li Jiangsu Normal University [email protected]

Xiangdong Li Chinese Academy of Sciences [email protected]

Yingqiu Li Changsha University of Science andTechnology

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Yueling Li Jiangsu Normal University [email protected]

Qian Lin Wuhan University

Yiwei Lin Shandong University lin [email protected]

Jiang Liu Jiangsu Normal University

Shuhui Liu Shandong University [email protected]

Wei Liu Jiangsu Normal University [email protected]

Wei Liu Wuhan University [email protected]

Wenjie Liu China University of Mining and Technol-ogy

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Xiaoying Liu Jiangsu Normal University

Liangbing Luo [email protected] University of Connecticut

Linxuan Lv China University of Mining and Technol-ogy

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Siyu Lv Southeast University [email protected]

Zhongxue Lv Jiangsu Normal University

Hanmin Ma China University of Mining and Technol-ogy

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Wei Mao Jiangsu Second Normal University

Yu Miao Henan Normal University [email protected]

Zhengke Miao Jiangsu Normal University [email protected]

Cheng Ouyang University of Illinois at Chicago [email protected]

Lei Pan Shandong University [email protected]

Pierre Patie Cornell University

Huimin Pei Jiangsu Normal University [email protected]

Zhu Peng Jiangsu Normal University

Bin Qian Changshu Institut of Technology [email protected]

Qianyun Qian China University of Mining and Technol-ogy

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Zhongmin Qian Oxford University [email protected]

Huijie Qiao Southeast University [email protected]

Chongyang Ren Wuhan University

Jiagang Ren Sun Yat-sen University [email protected]

Yong Ren Anhui Normal University [email protected]

Michael Rockner Bielefeld University [email protected]

Rene Schilling University of Dresden [email protected]

Jinghai Shao Tianjin University [email protected]

Qiman Shao University of Hong Kong [email protected]

Guangjun Shen Anhui Normal University [email protected]

Xiaohui Shen Xuzhou Medical University [email protected]

Yinghui Shi Jiangsu Normal University [email protected]

Zhan Shi University of Paris 6

Renming Song University of Illinois [email protected]

Yongsheng Song Chinese Academy of Sciences [email protected]

Yulin Song Nanjing University [email protected]

Wilhelm Stannat Technische Universitat Berlin [email protected]

Jianbing Su Jiangsu Normal University

Xiaoyan Su Institute of Applied Physics and Compu-tational Mathematics

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Jie Sun Hubei University of Arts and Science sunjie [email protected]

Shiliang Sun Jiangsu Normal University

Libin Sun Jiangsu Normal University

Wei Sun Concordia University [email protected]

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Xiaobin Sun Jiangsu Normal University

Dejian Tian China University of Mining and Technol-ogy

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Samy Tindel Purdue University [email protected]

Zoran Vondracek University of Zagreb [email protected]

ChenguangWang

China University of Mining and Technol-ogy

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Fengyu Wang Tianjin University [email protected]

Jian Wang Fujian Normal University [email protected]

Kai Wang Anhui University of Finance and Eco-nomics

Lidan Wang Nankai University [email protected]

Ran Wang Wuhan University [email protected]

Tao Wang Huaiyin Normal University

Dongsheng Wu University of Alabama in Huntsville [email protected]

Jianglun Wu Swansea University [email protected]

Mingyan Wu Wuhan University

Shangri Wu China University of Mining and Technol-ogy

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Weina Wu Nanjing University of Finance and Eco-nomics

Aihua Xia University of Melbourne [email protected]

Yimin Xiao Michigan State University [email protected]

Yingchao Xie Jiangsu Normal University [email protected]

Fangjun Xu East China Normal University [email protected]

Jie Xu Henan Normal University [email protected]

Maochao Xu Jiangsu Normal University [email protected]

Litan Yan Donghua University [email protected]

Guang Yang University of Connecticut [email protected]

Guang Yang Jiangsu Normal University

Ting Yang Jiangsu Normal University [email protected]

Xu Yang China University of Mining and Technol-ogy

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Nian Yao Shenzhen University [email protected]

Yanqing Yin Jiangsu Normal University [email protected]

Jiangang Ying Fudan University [email protected]

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Qingpei Zang Huaiyin Normal University [email protected]

Yanchao Zang Henan University of Science and Tech-nology

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Jianliang Zhai University of Science and Technology ofChina

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Chun Zhang Huaiyin Normal University

Guodong Zhang Shandong University [email protected]

Huilin Zhang Fudan University [email protected]

Jichen Zhang Shandong University [email protected]

Jing Zhang Fudan University zhang [email protected]

Jing Zhang Hainan Normal University zh [email protected]

Qi Zhang Fudan University [email protected]

Shuxia Zhang Jiangsu Normal University [email protected]

Tusheng Zhang University of Manchester [email protected]

Wei Zhang China University of Mining and Technol-ogy

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Chao Zhang Jiangsu Normal University

Xicheng Zhang Wuhan University [email protected]

Zhaoang Zhang Shandong University [email protected]

ZhengliangZhang

Wuhan University [email protected]

Ziwu Zhang Shandong University [email protected]

Huaizhong Zhao Laughborough University [email protected]

Peng Zhao Jiangsu Normal University [email protected]

Yongqian Zhao Jiangsu Normal University

Guoqiang Zheng Southeast University [email protected]

Guoli Zhou Chongqing University [email protected]

Qin Zhou Jiangsu Normal University

Qing Zhou Beijing University of Posts and Teleco-munications

Xiaowen Zhou Concordia University [email protected]

Fubo Zhu Huaiyin Normal University

Rongchan Zhu Beijing Institute of Technology [email protected]

Yuanze Zhu Jiangsu Normal University

Yan Zhuang Jiangsu Normal University

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Gaofeng Zong Shandong University of Finance and Eco-nomics

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Xiangfeng Zuo China University of Mining and Technol-ogy

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Yifan Ding Jiangsu Normal University [email protected]

Yue Ding Jiangsu Normal University [email protected]

Tingting Dong Jiangsu Normal University [email protected]

Qiuxiang Du Jiangsu Normal University [email protected]

Lulu Fan Jiangsu Normal University [email protected]

Junrong Gao Jiangsu Normal University [email protected]

Min Gao Jiangsu Normal University [email protected]

Xinyu Gao Jiangsu Normal University [email protected]

Kai Gu Jiangsu Normal University [email protected]

Wei Hong Jiangsu Normal University [email protected]

Jinlong Huang Jiangsu Normal University [email protected]

Nan Jiang Jiangsu Normal University [email protected]

Chao Ling Jiangsu Normal University [email protected]

Qianyi Li Jiangsu Normal University [email protected]

Yu Li Jiangsu Normal University [email protected]

Changyan Liu Jiangsu Normal University [email protected]

Ruonan Shen Jiangsu Normal University [email protected]

Tingting Shen Jiangsu Normal University [email protected]

Haokun Shi Jiangsu Normal University [email protected]

Hong Sun Jiangsu Normal University [email protected]

Jiahuan Sun Jiangsu Normal University [email protected]

Hui Wan Jiangsu Normal University [email protected]

Chuchu Wang Jiangsu Normal University [email protected]

Hairu Wang Jiangsu Normal University [email protected]

Qiuxia Wang Jiangsu Normal University [email protected]

Wei Wang Jiangsu Normal University [email protected]

Yuzhen Wang Jiangsu Normal University [email protected]

Xiao Xiao Jiangsu Normal University [email protected]

Juan Xu Jiangsu Normal University [email protected]

Shuwei Xu Jiangsu Normal University [email protected]

Lu Yang Jiangsu Normal University [email protected]

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Kang You Jiangsu Normal University [email protected]

Yufang Yuan Jiangsu Normal University [email protected]

Jiayao Zhang Jiangsu Normal University [email protected]

Xiaolong Zhang Jiangsu Normal University [email protected]

Xiaoyu Zhang Jiangsu Normal University [email protected]

Xuemei Zhang Jiangsu Normal University [email protected]

Ying Zhao Jiangsu Normal University [email protected]

Jie Wang Jiangsu Normal University [email protected]

Huijie Lu Jiangsu Normal University [email protected]

Ting Jiang Jiangsu Normal University [email protected]

Tianhui Liu Jiangsu Normal University [email protected]

Xingcheng Zhou Jiangsu Normal University [email protected]

Yi Ge Jiangsu Normal University [email protected]

Huilian Xia Jiangsu Normal University [email protected]

Wenshu Yan Jiangsu Normal University [email protected]

Xiaomin Huang Jiangsu Normal University [email protected]

Junyan Wu Jiangsu Normal University [email protected]

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International Conference on Stochastic Analysis and...

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