ItemFinite Horizon Optimal Execution with Bounded Rate of Transaction(Stochastic Models, 2019) Yipeng, YangIn this article, we consider an optimal execution problem with fixed time horizon and bounded transaction rate, which is more natural in practice. We show that, different from traditional stochastic control or singular control problems, this problem is of the stochastic bang-bang control type. Under some parameter settings we show that the optimal control does not involve buy action, and the optimal value function is the viscosity solution to the associated Hamilton-Jacobi-Bellman equation. We further show that the optimal policy is unique, and provide a numerical example to explore the form of the optimal control. ItemStability Analysis of Networked Control Systems with Bounded Random Delay and State Compensation: How Large is the Actual System Scale?(AIMS Electronic Engineering, 2019) Yipeng, YangIn this paper, a level set analysis is proposed which aims to analyze the S&P 500 return with a certain magnitude. It is found that the process of large jumps/drops of return tend to have negative serial correlation, and volatility clustering phenomenon can be easily seen. Then, a nonparametric analysis is performed and new patterns are discovered. An ARCH model is constructed based on the patterns we discovered and it is capable of manifesting the volatility skew in option pricing. A comparison of our model with the GARCH(1,1) model is carried out. The explanation of the validity on our model through prospect theory is provided, and, as a novelty, we linked the volatility skew phenomenon to the prospect theory in behavioral finance. ItemA Level Set Analysis and a Nonparametric Regression on S&P 500 Daily Return(International Journal of Financial Studies, 2016) Yipeng, YangIn this paper, a level set analysis is proposed which aims to analyze the S&P 500 return with a certain magnitude. It is found that the process of large jumps/drops of return tend to have negative serial correlation, and volatility clustering phenomenon can be easily seen. Then, a nonparametric analysis is performed and new patterns are discovered. An ARCH model is constructed based on the patterns we discovered and it is capable of manifesting the volatility skew in option pricing. A comparison of our model with the GARCH(1,1) model is carried out. The explanation of the validity on our model through prospect theory is provided, and, as a novelty, we linked the volatility skew phenomenon to the prospect theory in behavioral finance. ItemOn the Paradox of Pesticides(Communications in Nonlinear Science and Numerical Simulation, 2015) Yipeng, YangThe paradox of pesticides was observed experimentally, which says that pesticides may dramatically increase the population of a pest when the pest has a natural predator. Here we use a mathematical model to study the paradox. We find that the timing for the application of pesticides is crucial for the resurgence or non-resurgence of the pests. In particular, regularly applying pesticides is not a good idea as also observed in experiments [3,7]. In fact, the best time to apply pesticides is when the pest population is reasonably high. ItemConvergence Studies on Monte Carlo Methods for Pricing Mortgage-Backed Securities(International Journal of Financial Studies, 2015) Yipeng, YangMonte Carlo methods are widely-used simulation tools for market practitioners from trading to risk management. When pricing complex instruments, like mortgage-backed securities (MBS), strong path-dependency and high dimensionality make the Monte Carlo method the most suitable, if not the only, numerical method. In practice, while simulation processes in option-adjusted valuation can be relatively easy to implement, it is a well-known challenge that the convergence and the desired accuracy can only be achieved at the cost of lengthy computational times. In this paper, we study the convergence of Monte Carlo methods in calculating the option-adjusted spread (OAS), effective duration (DUR) and effective convexity (CNVX) of MBS instruments. We further define two new concepts, absolute convergence and relative convergence, and show that while the convergence of OAS requires thousands of simulation paths (absolute convergence), only hundreds of paths may be needed to obtain the desired accuracy for effective duration and effective convexity (relative convergence). These results suggest that practitioners can reduce the computational time substantially without sacrificing simulation accuracy. ItemA Multi-dimensional Stochastic Singular Control Problem Via Dynkin Game and Dirichlet Form(SIAM Journal on Control and Optimization, 2014) Yipeng, YangThe traditional difficulty with stochastic singular control is characterizing the regularities of the value function and the optimal control policy. In this paper, a multidimensional singular control problem is considered. We found the optimal value function and the optimal control policy of this problem via a Dynkin game, whose solution is given by the saddle point of the cost function. The existence and uniqueness of the solution to this Dynkin game are proved through an associated variational inequality problem involving Dirichlet form. As a consequence, the properties of the value function of this Dynkin game imply the smoothness of the value function of the stochastic singular control problem. In this way, we are able to show the existence of a classical solution to this multidimensional singular control problem, which was traditionally solved in the sense of viscosity solutions, and this enables the application of the verification theorem to prove optimality. ItemRefined Solutions of Time Inhomogeneous Optimal Stopping Problem and Zero-sum Game via Dirichlet Form(Probability and Mathematical Statistics, 2014) Yipeng, YangThe properties of value functions of time inhomogeneous optimal stopping problem and zero-sum game (Dynkin game) are studied through time dependent Dirichlet form. Under the absolute continuity condition on the transition function of the underlying diffusion process and some other assumptions, the refined solutions without exceptional starting points are proved to exist, and the value functions of the optimal stopping and zero-sum game, which are finely and cofinely continuous, are characterized as the solutions of some variational inequalities, respectively. ItemThe Effect of Synchronous Firing on the Clustering Dynamics of Social Amoebae(Complexity, 2014) Yipeng, YangA discrete model for computer simulations of the clustering dynamics of social amoebae is presented. This model incorporates the wavelike propagation of extracellular signaling of 3′–5′‐cyclic adenosine monophosphate (cAMP), the sporadic firing of cells at early stage of aggregation, the signal relaying as a response to stimulus, and the inertia and purposeful random walk of the cell movement. It is found that the sporadic firing below the threshold of cAMP concentration plays an important role because it allows time for the cells to form synchronous firing right before the stage of aggregation, and the synchronous firing is critical for the onset of clustering behavior of social amoebae. A Monte‐Carlo simulation was also run which showed the existence of potential equilibriums of mean and variance of aggregation time. The simulation result of this model could well reproduce many phenomena observed by actual experiments. ItemA Stochastic Portfolio Optimization Model with Bounded Memory(Mathematics of Operations Research, 2011) Yipeng, YangThis paper considers a portfolio management problem of Merton's type in which the risky asset return is related to the return history. The problem is modeled by a stochastic system with delay. The investor's goal is to choose the investment control as well as the consumption control to maximize his total expected, discounted utility. Under certain situations, we derive the explicit solutions in a finite dimensional space. ItemConstant Elasticity of Variance model for Pro-portional Reinsurance and Investment(Insurance: Mathematics and Economics, 2010) Yipeng, YangIn our model, the insurer is allowed to buy reinsurance and invest in a risk-free asset and a risky asset. The claim process is assumed to follow a Brownian motion with drift, while the price process of the risky asset is described by the constant elasticity of variance (CEV) model. The Hamilton–Jacobi–Bellman (HJB) equation associated with the optimal reinsurance and investment strategies is established, and solutions are found for insurers with CRRA or CARRA utility.