Search

In our present article, we follow our way of developing mean field type control theory in our earlier works [Bensoussan et al., Mean Field Games and Mean Field Type Control Theory. Springer, New York (2013)], by first introducing the Bellman and then master equations, the system of Hamilton-Jacobi-Bellman (HJB) and Fokker-Planck (FP) equations, and then tackling them by looking for the semi-explicit solution for the linear quadratic case, especially with an arbitrary initial distribution; such a problem, being left open for long, has not been specifically dealt with in the earlier literature, such as Bensoussan [Stochastic Control of Partially Observable Systems. Cambridge University Press, (1992)] and Nisio [Stochastic control theory: Dynamic programming principle. Springer (2014)], which only tackled the linear quadratic setting with Gaussian initial distributions. Thanks to the effective mean-field theory, we propose a solution to this long standing problem of the general non-Gaussian case. Besides, our problem considered here can be reduced to the model in Bandini et al. [Stochastic Process. Appl.129 (2019) 674–711], which is fundamentally different from our present proposed framework. [ABSTRACT FROM AUTHOR]

In this article, we study the applicability of Benford's law and Zipf's law to national COVID-19 case figures with the aim of establishing guidelines upon which methods of fraud detection in epidemiology, based on formal statistical analysis, can be developed. Moreover, these approaches may also be used in evaluating the performance of public health surveillance systems. We provide theoretical arguments for why the empirical laws should hold in the early stages of an epidemic, along with preliminary empirical evidence in support of these claims. Based on data published by the World Health Organization and various national governments, we find empirical evidence that suggests that both Benford's law and Zipf's law largely hold across countries, and deviations can be readily explained. To the best of our knowledge, this paper is among the first to present a practical application of Zipf's law to fraud detection. [ABSTRACT FROM AUTHOR]

Being inspired by the work of Imkeller et al., who first developed a measured-valued version of Malliavin calculus to study the enlargement of filtration by an ‘exogenous’ random variable on the Wiener space, in this article we consider the enlargement of filtration on the Poisson space. We construct a measure-valued Malliavin calculus on the Poisson space with the Malliavin type calculus defined by Mensi and Privault. Without resorting to the traditional Jacod’s condition nor its local version, the measure-valued Poisson Malliavin calculus can be applied to compute the ‘information drift’. Our result applies for a general -integrable ‘exogenous’ random variable, while previous works require a more restrictive Malliavin differentiability. Finally, we illustrate a concrete application of our results with an example from Mensi and Privault. [ABSTRACT FROM AUTHOR]

Recently, there have been numerous insightful applications of zero-sum stochastic differential games in insurance, as discussed in Liu et al. [Liu, J., Yiu, C. K.-F. & Siu, T. K. (2014). Optimal investment of an insurer with regime-switching and risk constraint.Scandinavian Actuarial Journal2014(7), 583–601]. While there could be some practical situations under which nonzero-sum game approach is more appropriate, the development of such approach within actuarial contexts remains rare in the existing literature. In this article, we study a class of nonzero-sum reinsurance-investment stochastic differential games between two competitive insurers subject to systematic risks described by a general compound Poisson risk model. Each insurer can purchase the excess-of-loss reinsurance to mitigate both systematic and idiosyncratic jump risks of the inter-arrival claims; and can invest in one risk-free asset and one risky asset whose price dynamics follows the famous Heston stochastic volatility model [Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options.Review of Financial Studies6, 327–343]. The main objective of each insurer is to maximize the expected utility of his terminal surplus relative to that of his competitor. Dynamic programming principle for this class of nonzero-sum game problems leads to a non-canonical fixed-point problem of coupled non-linear integral-typed equations. Despite the complex structure, we establish the unique existence of the Nash equilibrium reinsurance-investment strategies and the corresponding value functions of the insurers in a representative example of the constant absolute risk aversion insurers under a mild, time-independent condition. Furthermore, Nash equilibrium strategies and value functions admit closed forms. Numerical studies are also provided to illustrate the impact of the systematic risks on the Nash equilibrium strategies. Finally, we connect our results to that under the diffusion-approximated model by proving explicitly that the Nash equilibrium under the diffusion-approximated model is an-Nash equilibrium under the general Poisson risk model, thereby establishing that the analogous Nash equilibrium in Bensoussan et al. [Bensoussan, A., Siu, C. C., Yam, S. C. P. & Yang, H. (2014). A class of nonzero-sum stochastic differential investment and reinsurance games.Automatica50(8), 2025–2037] serves as an interesting complementary case of the present framework. [ABSTRACT FROM AUTHOR]

Optimal control problem is typically solved by first finding the value function through the Hamilton-Jacobi equation (HJE) and then taking the minimizer of the Hamiltonian to obtain the control. In this work, instead of focusing on the value function, we propose a new formulation for the gradient of the value function (value-gradient) as a decoupled system of partial differential equations in the context of a continuous-time deterministic discounted optimal control problem. We develop an efficient iterative scheme for this system of equations in parallel by utilizing the fact that they share the same characteristic curves as the HJE for the value function. For the theoretical part, we prove that this iterative scheme converges linearly in L²α sense for some suitable exponent α in a weight function. For the numerical method, we combine a characteristic line method with machine learning techniques. Specifically, we generate multiple characteristic curves at each policy iteration from an ensemble of initial states and compute both the value function and its gradient simultaneously on each curve as the labeled data. Then supervised machine learning is applied to minimize the weighted squared loss for both the value function and its gradients. Experimental results demonstrate that this new method not only significantly increases the accuracy but also improves the efficiency and robustness of the numerical estimates, particularly with less characteristics data or fewer training steps. [ABSTRACT FROM AUTHOR]

The objective of this paper is two-fold. The first objective is to complete the former work of Bensoussan and Frehse [2]. One big limitation of this paper was the fact that they are systems of PDE. on a bounded domain. One can expect solutions to be bounded, since one looks for smooth solutions. This is a very important property for the development of the method. It is true also that solutions which exist in a bounded domain may fail to exist on R n , because of the lack of bounds. We give conditions so that the results of [2] can be extended to R n. The second objective is to consider the BSDE (Backward stochastic differential equations) version of the system of PDE. This is the objective of a more recent work of Xing and Z̆itković [8]. They consider systems of BSDE with quadratic growth, which is a well-known open problem in the BSDE literature. Since the BSDE are Markovian, the problem is equivalent to the analytic one. However, because of this motivation the analytic problem is in R n and not on a bounded domain. Xing and Z̆itković developed a probabilistic approach. The connection between the analytic problem and the BSDE is not apparent. Our objective is to show that the analytic approach can be completely translated into a probabilistic one. Nevertheless probabilistic concepts are also useful, after their conversion into the analytic framework. This is in particular true for the uniqueness result. [ABSTRACT FROM AUTHOR]

In this paper, we solve a continuous-time portfolio choice problem of an investor under a Markov jump linear system that effectively captures stochasticity in asset returns, price impacts, and market resilience. Specifically, the investor chooses his portfolio to maximize the expected excess returns netting risks and trading costs due to the price impacts in a Markov switching market. We show that the value function and the optimal feedback trading strategy can be expressed in terms of the solution of a coupled matrix Riccati differential system. The presence of the market resilience and the Markov switching results in the coupled Riccati system being non-canonical. We manage to establish a set of transparent sufficient conditions to ensure the well-posedness of the coupled system. Our numerical results indicate that the investor trades more (less) aggressively in a liquidity regime where the temporary price impact is low (high) and permanent price impact is high (low) with low (high) market resilience. [ABSTRACT FROM AUTHOR]

In this article, we provide the first systematic study on the unique existence of the solution of backward stochastic dynamical variational inequalities on a general complete filtered probability space. We also build up a comprehensive analysis of the correspondence between these stochastic variational inequalities (resp. backward stochastic dynamics) and the weak solutions (instead of viscosity ones due to the intrinsic non-local nature of the integral of the gradient involved) of a class of non-local parabolic variational inequalities (resp. parabolic partial differential equations), which is barely touched in the existing literature due to its unconventional setting. [ABSTRACT FROM AUTHOR]

In this paper, we propose a portmanteau test for whether a graph-structured network dataset without replicates exhibits autocorrelation across units connected by edges. Specifically, the well known Ljung-Box test for serial autocorrelation of time series data is generalized to the network setting using a specially derived central limit theorem for a weakly stationary random field. The asymptotic distribution of the test statistic under the null hypothesis of no autocorrelation is shown to be chi-squared, yielding a simple and easy-to-implement procedure for testing graph-structured autocorrelation, including spatial and spatial-temporal autocorrelation as special cases. Numerical simulations are carried out to demonstrate and confirm the derived asymptotic results. Convergence is found to occur quickly depending on the number of lags included in the test statistic, and a significant increase in statistical power is also observed relative to some recently proposed permutation tests. An example application is presented by fitting spatial autoregressive models to the distribution of COVID-19 cases across counties in New York state. [ABSTRACT FROM AUTHOR]

In his lectures at College de France , P.L. Lions introduced the concept of Master equation, see [8] for Mean Field Games. It is introduced in a heuristic fashion, from the prospective as a system of partial differential equations, that the equation is associated to a Nash equilibrium for a large, but finite, number of players. The method, also explained in [3] , composed of a formalism of derivations. The interest of this equation is that it contains interesting particular cases, which can be studied directly, in particular the system of HJB–FP (Hamilton–Jacobi–Bellman, Fokker–Planck) equations obtained as the limit of the finite Nash equilibrium game, when the trajectories are independent, see [6] . Usually, in mean field theory, one can bypass the large Nash equilibrium, by introducing the concept of representative agent, whose action is influenced by a distribution of similar agents, and obtains directly the system of HJB–FP equations of interest, see for instance [1] . Apparently, there is no such approach for the Master equation. We show here that it is possible. We first do it for the Mean Field type control problem, for which we interpret completely the Master equation. For the Mean Field Games itself, we solve a related problem, and obtain again the Master equation. [ABSTRACT FROM AUTHOR]

As perceived from daily experience together with numerous empirical studies, the multivariate risks demonstrate a strong coherence in the extremal dependence structure especially over the course of financial turmoil or industrial accidents and outbreaks. Under this motivating paradigm, we show the universal asymptotic additivity under upper tail comonotonicity, as the probability level approaching to 1, for Value-at-Risk and Conditional Tail Expectation for a portfolio of fixed number of risks, in which each marginal risk could be any one having a finite endpoint or belonging to one of the three max domains of attraction. Our obtained results do not require the tail equivalence assumption as needed in the existing literature. This resolves a lasting problem in quantitative risk management and covers most distributions commonly encountered in practice. [ABSTRACT FROM AUTHOR]

We revisit the general problem of minimizing a separable convex function with both a budget constraint and a set of box constraints. This optimization problem arises naturally in many resource allocation problems in engineering, economics, finance and insurance. Existing literature tackles this problem by using the traditional Kuhn–Tucker theory, which leads to either iterative schemes or yields explicit solutions only under some special classes of convex functions owe to the presence of box constraints. This paper presents a novel approach of solving this constrained minimization problem by using the theory of comonotonicity. The key step is to apply an integral representation result to express each convex function as the stop-loss transform of some suitable random variable. By using this approach, we can derive and characterize not only the explicit solution, but also obtain its geometric meaning and some other qualitative properties. [ABSTRACT FROM AUTHOR]