Sheung Chi Phillip Yam

Linear-Quadratic Mean Field Games

We provide a comprehensive study of a general class of linear-quadratic mean field games. We adopt the adjoint equation approach to investigate the unique existence of their equilibrium strategies. Due to the linearity of the adjoint equations, the optimal mean field term satisfies a forward–backward ordinary differential equation. For the one-dimensional case, we establish the unique existence of the equilibrium strategy. For a dimension greater than one, by applying the Banach fixed point theorem under a suitable norm, a sufficient condition for the unique existence of the equilibrium strategy is provided, which is independent of the coefficients of controls in the underlying dynamics and is always satisfied whenever the coefficients of the mean field term are vanished, and hence, our theories include the classical linear-quadratic stochastic control problems as special cases. As a by-product, we also establish a neat and instructive sufficient condition, which is apparently absent in the literature and only depends on coefficients, for the unique existence of the solution for a class of nonsymmetric Riccati equations. Numerical examples of nonexistence of the equilibrium strategy will also be illustrated. Finally, a similar approach has been adopted to study the linear-quadratic mean field type stochastic control problems and their comparisons with mean field games.

Optimal reinsurance under general law-invariant risk measures

In recent years, general risk measures play an important role in risk management in both finance and insurance industry. As a consequence, there is an increasing number of research on optimal reinsurance decision problems using risk measures beyond the classical expected utility framework. In this paper, we first show that the stop-loss reinsurance is an optimal contract under law-invariant convex risk measures via a new simple geometric argument. A similar approach is then used to tackle the same optimal reinsurance problem under Value at Risk and Conditional Tail Expectation; it is interesting to note that, instead of stop-loss reinsurances, insurance layers serve as the optimal solution. These two results highlight that law-invariant convex risk measure is better and more robust, in the sense that the corresponding optimal reinsurance still provides the protection coverage against extreme loss irrespective to the potential increment of its probability of occurrence, to expected larger claim than Value at Risk and Conditional Tail Expectation which are more commonly used. Several illustrative examples will be provided.

A class of non-zero-sum stochastic differential investment and reinsurance games

Abstract In this article, we provide a systematic study on the non-zero-sum stochastic differential investment and reinsurance game between two insurance companies. Each insurance company’s surplus process consists of a proportional reinsurance protection and an investment in risky and risk-free assets. Each insurance company is assumed to maximize his utility of the difference between his terminal surplus and that of his competitor. The surplus process of each insurance company is modeled by a mixed regime-switching Cramer–Lundberg diffusion approximation process, i.e. the coefficients of the diffusion risk processes are modulated by a continuous-time Markov chain and an independent market-index process. Correlation between the two surplus processes, independent of the risky asset process, is allowed. Despite the complex structure, we manage to solve the resulting non-zero sum game problem by applying the dynamic programming principle. The Nash equilibrium, the optimal reinsurance/investment, and the resulting value processes of the insurance companies are obtained in closed forms, together with sound economic interpretations, for the case of an exponential utility function.

Time-Consistent Portfolio Selection under Short-Selling Prohibition: From Discrete to Continuous Setting ∗

In this paper, we study the time consistent strategies in the mean-variance portfolio selection with short-selling prohibition in both discrete and continuous time settings. Recently, [T. Bjork, A. Murgoci, and X. Y. Zhou, Math. Finance, 24 (2014), pp. 1--24] considered the problem with state dependent risk aversion in the sense that the risk aversion is inversely proportional to the current wealth, and they showed that the time consistent control is linear in wealth. Considering the counterpart of their continuous time equilibrium control in the discrete time framework, the corresponding “optimal” wealth process can take negative values; and this negativity in wealth will lead the investor to a risk seeker which results in an unbounded value function that is economically unsound; even more, the limiting of the discrete solutions has shown to be their obtained continuous solution in [T. Bjork, A. Murgoci, and X. Y. Zhou, Math. Finance, 24 (2014), pp. 1--24]. To deal this limitation, we eliminate the chanc...

Oracle, multiple robust and multipurpose calibration in a missing response problem

In the presence of a missing response, reweighting the complete case subsample by the inverse of nonmissing probability is both intuitive and easy to implement. When the population totals of some auxiliary variables are known and when the inclusion probabilities are known by design, survey statisticians have developed calibration methods for improving efficiencies of the inverse probability weighting estimators and the methods can be applied to missing data analysis. Model-based calibration has been proposed in the survey sampling literature, where multidimensional auxiliary variables are first summarized into a predictor function from a working regression model. Usually, one working model is being proposed for each parameter of interest and results in different sets of calibration weights for estimating different parameters. This paper considers calibration using multiple working regression models for estimating a single or multiple parameters. Contrary to a common belief that overfitting hurts efficiency, we present three rather unexpected results. First, when the missing probability is correctly specified and multiple working regression models for the conditional mean are posited, calibration enjoys an oracle property: the same semiparametric efficiency bound is attained as if the true outcome model is known in advance. Second, when the missing data mechanism is misspecified, calibration can still be a consistent estimator when any one of the outcome regression models is correctly specified. Third, a common set of calibration weights can be used to improve efficiency in estimating multiple parameters of interest and can simultaneously attain semiparametric efficiency bounds for all parameters of interest. We provide connections of a wide class of calibration estimators, constructed based on generalized empirical likelihood, to many existing estimators in biostatistics, econometrics and survey sampling and perform simulation studies to show that the finite sample properties of calibration estimators conform well with the theoretical results being studied.

Universal Repetitive Learning Control for Nonparametric Uncertainty and Unknown State-Dependent Control Direction Matrix

We propose a continuous universal repetitive learning control to track periodic trajectory for a class of nonlinear dynamical systems with nonparametric uncertainty and unknown state-dependent control direction matrix. The proposed controller is an integration of high-gain feedback, repetitive learning and Nussbaum gain matrix selector. The control signal is always continuous, thus it avoids the potential chattering effect caused by discontinuity. Asymptotic convergence of the tracking error is achieved by the controller, and the control performance is illustrated by simulation. Although the proposed method is derived for input-state systems, it can be readily extended to multi-input-multi-output systems under appropriate assumption.

Game Call Options Revisited

In this paper, having been inspired by the work of Kunita and Seko, we study the pricing of δ‐penalty game call options on a stock with a dividend payment. For the perpetual case, our result reveals that the optimal stopping region for the option seller depends crucially on the dividend rate d. More precisely, we show that when the penalty δ is small, there are two critical dividends 0 d. When d∈ [d, d], the value function is not continuously differentiable at the optimal stopping boundary for the option seller, therefore our result in the perpetual case cannot be established by the free boundary approach with smooth‐fit conditions imposed on both free boundaries. For the finite time horizon case, the dependence of the optimal stopping region for the option seller on the time to maturity is exhibited; more precisely, when both δ and d are small, we show that there are two critical times 0 T. In summary, for both the perpetual and the finite horizon cases, we characterize in terms of model parameters how the optimal stopping region for the option seller shrinks when the dividend rate d increases and the time to maturity decreases; these results complete the original work of Emmerling for the perpetual case and Kunita and Seko for the finite maturity case. In addition, for the finite time horizon case, we also extend the probabilistic method for the establishment of existence and regularity results of the classical American option pricing problem to the game option setting. Finally, we characterize the pair of optimal stopping boundaries for both the seller and the buyer as the unique pair of solutions to a couple of integral equations and provide numerical illustrations.

MEAN FIELD STACKELBERG GAMES: AGGREGATION OF DELAYED INSTRUCTIONS ∗

In this paper, we consider an

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

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A Test for the Equality of Multiple Sharpe Ratios

-player interacting strategic game in the presence of a (endogenous) dominating player, who gives direct influence on individual agents, through its impact on their control in the sense of Stackelberg game, and then on the whole community. Each individual agent is subject to a delay effect on collecting information, specifically at a delay time, from the dominating player. The size of his delay is completely known by the agent, while to others, including the dominating player, his delay plays as a hidden random variable coming from a common fixed distribution. By invoking a noncanonical fixed point property, we show that for a general class of finite