PARAMETER ESTIMATION OF PATH-DEPENDENT MCKEAN-VLASOV STOCHASTIC DIFFERENTIAL EQUATIONS

This work concerns a class of path-dependent McKean-Vlasov stochastic differential equations with unknown parameters.First,we prove the existence and uniqueness of these equations under non-Lipschitz conditions.Second,we construct maximum likelihood estimators of these parameters and then discuss their strong consistency.Third,a numerical simulation method for the class of path-dependent McKean-Vlasov stochastic differential equations is offered.Finally,we estimate the errors between solutions of these equations and that of their numerical equations.

Survey on Path-Dependent PDEs

In this paper,the authors provide a brief introduction of the path-dependent partial differential equations(PDEs for short)in the space of continuous paths,where the path derivatives are in the Dupire(rather than Fréchet)sense.They present the connections between Wiener expectation,backward stochastic differential equations(BSDEs for short)and path-dependent PDEs.They also consider the well-posedness of path-dependent PDEs,including classical solutions,Sobolev solutions and viscosity solutions.

BSDE,path-dependent PDE and nonlinear Feynman-Kac formula

We introduce a new type of path-dependent quasi-linear parabolic PDEs in which the continuous paths on an interval [0, t] become the basic variables in the place of classical variables （t, x） ∈[0, T]× R^d. This new type of PDEs are formulated through a classical BSDE in which the terminal values and the generators are allowed to be general function of Brownian motion paths. In this way, we establish the nonlinear Feynman- Kac formula for a general non-Markoviau BSDE. Some main properties of solutions of this new PDEs are also obtained.

BSDEs with Jumps and Path-Dependent Parabolic Integro-differential Equations

This paper deals with backward stochastic differential equations with jumps,whose data(the terminal condition and coefficient) are given functions of jump-diffusion process paths. The author introduces a type of nonlinear path-dependent parabolic integrodifferential equations, and then obtains a new type of nonlinear Feynman-Kac formula related to such BSDEs with jumps under some regularity conditions.

Path-dependent backward stochastic Volterra integral equations with jumps,differentiability and duality principle

We study the existence and uniqueness of a solution to path-dependent backward stochastic Volterra integral equations(BSVIEs)with jumps,where path-dependence means the dependence of the free term and generator of a path of a c`adl`ag process.Furthermore,we prove path-differentiability of such a solution and establish the duality principle between a linear path-dependent forward stochastic Volterra integral equation(FSVIE)with jumps and a linear path-dependent BSVIE with jumps.As a result of the duality principle we get a comparison theorem and derive a class of dynamic coherent risk measures based on path-dependent BSVIEs with jumps.

Continuous Estimation of Wrist Torque from Surface EMG Signals Using Path-dependent Model

Continuous estimation of wrist torque from surface electromyography(EMG) signals has been studied by some research institutes. Hysteresis effect is a phenomenon in EMG force relationship. In this work, a path-dependent model based on hysteresis effect was used for continuously estimating wrist torque from surface EMG signals. The surface EMG signals of the flexor carpi ulnaris(FCU) and extensor carpi radialis(ECR) were collected along with wrist torque of flexion/extension degree-of-freedom. EMG signal of FCU was used to estimate the torque of wrist flexion and EMG signal of ECR to estimate the torque of wrist extension. The existence of hysteresis effect has been proven either during wrist flexion or extension on all subjects.And the estimation performance of path-dependent model is much better than the overall model. Thus, the path-dependent model is suitable to improve the wrist torque's estimation accuracy.

Optimal stopping investment in a logarithmic utility-based portfolio selection problem

Background:In this paper,we study the right time for an investor to stop the investment over a given investment horizon so as to obtain as close to the highest possible wealth as possible,according to a Logarithmic utility-maximization objective involving the portfolio in the drift and volatility terms.The problem is formulated as an optimal stopping problem,although it is non-standard in the sense that the maximum wealth involved is not adapted to the information generated over time.Methods:By delicate stochastic analysis,the problem is converted to a standard optimal stopping one involving adapted processes.Results:Numerical examples shed light on the efficiency of the theoretical results.Conclusion:Our investment problem,which includes the portfolio in the drift and volatility terms of the dynamic systems,makes the problem including multi-dimensional financial assets more realistic and meaningful.

Coordination game model of co-opetition relationship on cluster supply chains

The research of cluster supply chains is a new direction and a hotspot of the industrial cluster theory. On the condition of the coordination game, the enterprises may be stuck on the non-efficient equilibrium status, which becomes an important problem that must be considered on cluster supply chains. A symmetrical coordination game model is constituted to describe the competition and cooperation relationship of the same-quality manufacturers on cluster supply chains. The methods of the non-cooperation game theory and the evolutionary game theory are respectively used to analyze the model, whose parameters＇ influences under each method are then compared. It can be concluded that the analysis of the evolutionary game theory is more realistic and practical. Finally, three approaches are considered to break away from being path-dependence locked-in non-efficient status during this coordination game evolutionary process, which provide the development of cluster supply chains with an effective forecasting and Pareto optimizing method.

轴心时代与路径依赖

“轴心时代”产生的儒家思想 ,决定了我国古代封建制度的初始选择 ,封建制度的自我强化这一正反馈机制 ,使得封建社会长期停滞。

ON PRICING MODEL OF THE RESET OPTION WITH N PREDETERMINED LEVELS

Motivated by the reset option with n predetermined dates analyzed by W.Cheng, we consider a kind of reset option with uncertain dates by introducing N pie-specifiedbarrier levels. We claim this reset option consists of some standard knock-in and knock-out barrieroptions. The closed-form pricing formula is derived by means of a PDE's approach.

Fully nonlinear stochastic and rough PDEs:Classical and viscosity solutions

We study fully nonlinear second-order(forward)stochastic PDEs.They can also be viewed as forward path-dependent PDEs and will be treated as rough PDEs under a unified framework.For the most general fully nonlinear case,we develop a local theory of classical solutions and then define viscosity solutions through smooth test functions.Our notion of viscosity solutions is equivalent to the alternative using semi-jets.Next,we prove basic properties such as consistency,stability,and a partial comparison principle in the general setting.If the diffusion coefficient is semilinear(i.e,linear in the gradient of the solution and nonlinear in the solution;the drift can still be fully nonlinear),we establish a complete theory,including global existence and a comparison principle.

Backward stochastic differential equations and backward stochastic Volterra integral equations with anticipating generators

For a backward stochastic differential equation(BSDE,for short),when the generator is not progressively measurable,it might not admit adapted solutions,shown by an example.However,for backward stochastic Volterra integral equations(BSVIEs,for short),the generators are allowed to be anticipating.This gives,among other things,an essential difference between BSDEs and BSVIEs.Under some proper conditions,the well-posedness of such BSVIEs is established.Further,the results are extended to path-dependent BSVIEs,in which the generators can depend on the future paths of unknown processes.An additional finding is that for path-dependent BSVIEs,in general,the situation of anticipating generators is not avoidable,and the adaptedness condition similar to that imposed for anticipated BSDEs by Peng−Yang[22]is not necessary.