Maximum Principle for Mean Field Type Control Problems with General Volatility Functions

In this paper, we study the maximum principle of mean field type control problems when the volatility function depends on the state and its measure and also the control, by using our recently developed method. Our method is to embed the mean field type control problem into a Hilbert space to bypass the evolution in the Wasserstein space. We here give a necessary condition and a sufficient condition for these control problems in Hilbert spaces, and we also derive a system of forward-backward stochastic differential equations

Degenerate Mean Field Type Control with Linear and Unbounded Diffusion, and their Associated Equations

We study the well-posedness of a system of forward-backward stochastic differential equations (FBSDEs) corresponding to a degenerate mean field type control problem, when the diffusion coefficient depends on the state together with its measure and also the control. Degenerate mean field type control problems are rarely studied in the literature. Our method is based on a lifting approach which embeds the control problem and the associated FBSDEs in Wasserstein spaces into certain Hilbert spaces. We use a continuation method to establish the solvability of the FBSDEs and that of the G\^ateaux derivatives of this FBSDEs. We then explore the regularity of the value function in time and in measure argument, and we also show that it is the unique classical solution of the associated Bellman equation. We also study the higher regularity of the linear functional derivative of the value function, by then, we obtain the classical solution of the mean field type master equation

Can we beat the “buy-and-hold” strategy? Analysis on European and American securitized real estate indices

The aim of this paper is to use the Shiryaev-Zhou index to examine the performances of securitized real estate indices of four countries: US, UK, Canada and Germany. The result reveals that the Shiryaev-Zhou index is a leading indicator and can act as a predictor on certain securitized real estate indices. Furthermore, our results show that the trading strategy we constructed according to the Shiryaev-Zhou index generally outperforms the “buy-and-hold” strategy under the assumption of no transaction costs. The stronger the predictive power of the Shiryaev-Zhou index is, the larger extent our trading strategy beats the “buy-and-hold” strategy. This is useful in strategic property management that property practitioners can follow our strategy to trade real estate stocks/funds in order to increase their profits

Feedback Stackelberg-Nash equilibria in mixed leadership games with an application to cooperative advertising

In this paper we characterize the feedback equilibrium of a general infinite-horizon Stackelberg-Nash differential game where the roles of the players are mixed. By mixed we mean that one player is a leader on some decisions and a follower on other decisions. We prove a verification theorem that reduces the task of finding equilibrium strategies in functional spaces to two simple steps: First solving two static Nash games at the Hamiltonian level in a nested version and then solving the associated system of Hamilton-Jacobi-Bellman equations. As an application, we study a novel manufacturer-retailer cooperative advertising game where, in addition to the traditional setup into which the manufacturer subsidizes the retailer's advertising effort, we also allow the reverse support from the retailer to the manufacturer. We find an equilibrium that can be expressed by a solution of a set of quartic algebraic equations. We then conduct an extensive numerical study to assess the impact of model parameters on the equilibrium

Nonparametric Likelihood Ratio Test for Univariate Shape-constrained Densities

We provide a comprehensive study of a nonparametric likelihood ratio test on whether a random sample follows a distribution in a prespecified class of shape-constrained densities. While the conventional definition of likelihood ratio is not well-defined for general nonparametric problems, we consider a working sub-class of alternative densities that leads to test statistics with desirable properties. Under the null, a scaled and centered version of the test statistic is asymptotic normal and distribution-free, which comes from the fact that the asymptotic dominant term under the null depends only on a function of spacings of transformed outcomes that are uniform distributed. The nonparametric maximum likelihood estimator (NPMLE) under the hypothesis class appears only in an average log-density ratio which often converges to zero at a faster rate than the asymptotic normal term under the null, while diverges in general test so that the test is consistent. The main technicality is to show these results for log-density ratio which requires a case-by-case analysis, including new results for k-monotone densities with unbounded support and completely monotone densities that are of independent interest. A bootstrap method by simulating from the NPMLE is shown to have the same limiting distribution as the test statistic

Shiryaev-Zhou index – a noble approach to benchmarking and analysis of real estate stocks

Real estate markets and real estate stocks are interrelated and are important not only to the investors, but also to the academics. Real estate stocks are, in a sense, good measures of performance of the physical real estate market. The objective of this paper is to provide a preliminary study on gauging the performances of real estate stocks in Hong Kong using the Shiryaev-Zhou index. Evidence shows that the Shiryaev-Zhou index can gauge a real estate stock's performance, good or bad, according to the sign of the Shiryaev-Zhou index. Thus a trading strategy can be formulated as follows: buy a stock if its Shiryaev-Zhou index changes from negative to positive, then hold it until its Shiryaev-Zhou index turns negative, when it is time to sell the stock. We examine the Shiryaev-Zhou indices of the real estate stocks in Hong Kong, and from this we deduce the latest best selling dates of the stocks during the period of our study. The Shiryaev-Zhou index could be an indicator of whether the market is bullish or bearish and consequently tells an investor to hold a stock or not, and it naturally leads to an optimal selling strategy that maximize the average ratio of the selling price to the maximum stock price when the underlying coefficients are assumed to be constant over a definite period of time