Time consistent policy of multi-period mean-variance problem in stochastic markets
Zhiping Chen Jia Liu Gang Li
Journal of Industrial & Management Optimization 2016, 12(1): 229-249 doi: 10.3934/jimo.2016.12.229
Due to the non-separability of the variance operator, the optimal investment policy of the multi-period mean-variance model in Markovian markets doesn't satisfy the time consistency. We propose a new weak time consistency in stochastic markets and show that the pre-commitment optimal policy satisfies the weak time consistency at any intermediate period as long as the investor's wealth is no more than a specific threshold. When the investor's wealth exceeds the threshold, the weak time consistency no longer holds. In this case, by modifying the pre-commitment optimal policy, we derive a wealth interval, from which we determine a more efficient revised policy. The terminal wealth obtained under this revised policy can achieve the same mean as, but not greater variance than those of the terminal wealth obtained under the pre-commitment optimal policy; a series of superior investment policies can be obtained depending on the degree the investor wants the conditional variance to decrease. It is shown that, in the above revising process, a positive cash flow can be taken out of the market. Finally, an empirical example illustrates our theoretical results. Our results generalize existing conclusions for the multi-period mean-variance model in deterministic markets.
keywords: policy revision. Markovian markets Bellman's optimality principle mean-variance Time inconsistency
The initial-boundary value problem on a strip for the equation of time-like extremal surfaces
Yi Zhou Jianli Liu
Discrete & Continuous Dynamical Systems - A 2009, 23(1&2): 381-397 doi: 10.3934/dcds.2009.23.381
In this paper we study the mixed initial-boundary value problem for the equation of time-like extremal surfaces in Minkowski space $R^{1+(1+n)}$ on the strip $R^{+}\times[0,1]$. Under the assumptions that the boundary data are small and decaying, we get the global existence and uniqueness of classical solutions.
keywords: global classical solution initial-boundary value problem The equation of time-like extremal surface
On the classical solvability of near field reflector problems
Jiakun Liu Neil S. Trudinger
Discrete & Continuous Dynamical Systems - A 2016, 36(2): 895-916 doi: 10.3934/dcds.2016.36.895
In this paper we prove the existence of classical solutions to near field reflector problems, both for a point light source and for a parallel light source, with planar receivers. These problems involve Monge-Ampère type equations, subject to nonlinear oblique boundary conditions. Our approach builds on earlier work in the optimal transportation case by Trudinger and Wang and makes use of a recent extension of degree theory to oblique boundary conditions by Li, Liu and Nguyen.
keywords: existence obliqueness Monge-Ampère regularity. Reflector
Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems
Cunming Liu Jianli Liu
Discrete & Continuous Dynamical Systems - A 2014, 34(11): 4735-4749 doi: 10.3934/dcds.2014.34.4735
In this paper we consider the existence and stability of traveling wave solutions to Cauchy problem of diagonalizable quasilinear hyperbolic systems. Under the appropriate small oscillation assumptions on the initial traveling waves, we derive the stability result of the traveling wave solutions, especially for intermediate traveling waves. As the important examples, we will apply the results to some systems arising in fluid dynamics and elementary particle physics.
keywords: linearly degenerate stability. traveling wave solution classical solution Diagonalizable quasilinear hyperbolic system
Global existence for a thin film equation with subcritical mass
Jian-Guo Liu Jinhuan Wang
Discrete & Continuous Dynamical Systems - B 2017, 22(4): 1461-1492 doi: 10.3934/dcdsb.2017070
In this paper, we study existence of global entropy weak solutions to a critical-case unstable thin film equation in one-dimensional case
$h_t+\partial_x (h^n\,\partial_{xxx} h)+\partial_x (h^{n+2}\partial_{x} h)=0,$
$n≥q 1$
. There exists a critical mass
found by Witelski et al.(2004 Euro. J. of Appl. Math. 15,223-256) for
. We obtain global existence of a non-negative entropy weak solution if initial mass is less than
. For
$n≥q 4$
, entropy weak solutions are positive and unique. For
, a finite time blow-up occurs for solutions with initial mass larger than
. For the Cauchy problem with
and initial mass less than
, we show that at least one of the following long-time behavior holds:the second moment goes to infinity as the time goes to infinity or
$ h(·, t_k)\rightharpoonup 0$
for some subsequence
${t_k} \to \infty $
keywords: Long-wave instability free-surface evolution equilibrium the Sz. Nagy inequality long-time behavior
Ultra-contractivity for Keller-Segel model with diffusion exponent $m>1-2/d$
Shen Bian Jian-Guo Liu Chen Zou
Kinetic & Related Models 2014, 7(1): 9-28 doi: 10.3934/krm.2014.7.9
This paper establishes the hyper-contractivity in $L^\infty(\mathbb{R}^d)$ (it's known as ultra-contractivity) for the multi-dimensional Keller-Segel systems with the diffusion exponent $m>1-2/d$. The results show that for the supercritical and critical case $1-2/d < m ≤ 2-2/d$, if $||U_0||_{d(2-m)/2} < C_{d,m}$ where $C_{d,m}$ is a universal constant, then for any $t>0$, $||u(\cdot,t)||_{L^\infty(\mathbb{R}^d)}$ is bounded and decays as $t$ goes to infinity. For the subcritical case $m>2-2/d$, the solution $u(\cdot,t) \in L^\infty(\mathbb{R}^d)$ with any initial data $U_0 \in L_+^1(\mathbb{R}^d)$ for any positive time.
keywords: ultra-contractive nonlocal aggregation degenerate diffusion. chemotaxis Hyper-contractive
Uniform $L^{∞}$ boundedness for a degenerate parabolic-parabolic Keller-Segel model
Wenting Cong Jian-Guo Liu
Discrete & Continuous Dynamical Systems - B 2017, 22(2): 307-338 doi: 10.3934/dcdsb.2017015

This paper investigates the existence of a uniform in time $L^{∞}$ bounded weak entropy solution for the quasilinear parabolic-parabolic Keller-Segel model with the supercritical diffusion exponent $0<m<2-\frac{2}{d}$ in the multi-dimensional space ${\mathbb{R}}^d$ under the condition that the $L^{\frac{d(2-m)}{2}}$ norm of initial data is smaller than a universal constant. Moreover, the weak entropy solution $u(x,t)$ satisfies mass conservation when $m>1-\frac{2}{d}$. We also prove the local existence of weak entropy solutions and a blow-up criterion for general $L^1\cap L^{∞}$ initial data.

keywords: Chemotaxis fast diffusion critical space semi-group theory global existence
On Pogorelov estimates for Monge-Ampère type equations
Jiakun Liu Neil S. Trudinger
Discrete & Continuous Dynamical Systems - A 2010, 28(3): 1121-1135 doi: 10.3934/dcds.2010.28.1121
In this paper, we prove interior second derivative estimates of Pogorelov type for a general form of Monge-Ampère equation which includes the optimal transportation equation. The estimate extends that in a previous work with Xu-Jia Wang and assumes only that the matrix function in the equation is regular with respect to the gradient variables, that is it satisfies a weak form of the condition introduced previously by Ma,Trudinger and Wang for regularity of optimal transport mappings. We also indicate briefly an application to optimal transportation.
keywords: optimal transportation. Pogorelov estimates
Error estimates of the aggregation-diffusion splitting algorithms for the Keller-Segel equations
Hui Huang Jian-Guo Liu
Discrete & Continuous Dynamical Systems - B 2016, 21(10): 3463-3478 doi: 10.3934/dcdsb.2016107
In this paper, we discuss error estimates associated with three different aggregation-diffusion splitting schemes for the Keller-Segel equations. We start with one algorithm based on the Trotter product formula, and we show that the convergence rate is $C\Delta t$, where $\Delta t$ is the time-step size. Secondly, we prove the convergence rate $C\Delta t^2$ for the Strang's splitting. Lastly, we study a splitting scheme with the linear transport approximation, and prove the convergence rate $C\Delta t$.
keywords: positivity preserving. chemotaxis random particle method Newtonian aggregation
Quasi-periodic solutions for derivative nonlinear Schrödinger equation
Meina Gao Jianjun Liu
Discrete & Continuous Dynamical Systems - A 2012, 32(6): 2101-2123 doi: 10.3934/dcds.2012.32.2101
In this paper, we discuss the existence of time quasi-periodic solutions for the derivative nonlinear Schrödinger equation $$\label{1.1}\mathbf{i} u_t+u_{xx}+\mathbf{i} f(x,u,\bar{u})u_x+g(x,u,\bar{u})=0 $$ subject to Dirichlet boundary conditions. Using an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vector-field and Birkhoff normal form, we will prove that there exist a Cantorian branch of KAM tori and thus many time quasi-periodic solutions for the above equation.
keywords: KAM theory Derivative nonlinear Schrödinger equation Normal form.

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