arXiv subjects

Sheung Chi Phillip Yam

Publications and source records attributed to Sheung Chi Phillip Yam.

At least 19 recordsLinked to original sources

arXiv2026

Retirement Planning with Minimum Guarantees for Both Beneficiaries and Fund Managers

This article studies a retirement planning problem from a new perspective in which a retiree delegates an initial lump sum to a professional fund manager. The fund is managed dynamically to deliver lifelong benefits while satisfying a guarantee that, at all times, wealth remains above a prescribed solvency level and the benefit rate remains above a minimum level. We formulate this problem as a continuous-time stochastic control model, in which the controls are the benefit rate and the portfolio allocation, and the objective incorporates general utility functions over benefits, management fees, and terminal bequest. Conventional approaches such as standard viscosity solution and martingale methods are not immediately applicable here due to the simultaneous influence of (i) the control variable nature of the benefit rate; (ii) the guarantee constraints; and (iii) the running wealth-dependent utility. We develop a new approach that deals directly with the associated fully nonlinear Hamilton-Jacobi-Bellman equation and establish the unique existence of a classical solution. One particular trick for resolving this equation is to apply an intricate transform that diverts the former to a semilinear auxiliary equation. A major contribution is to establish the existence of the solution to the auxiliary equation and characterize its precise growth, derivative, and boundary behavior. These results are crucial for establishing the verification theorems and deriving the optimal strategy. Besides, numerical experiments indicate that guarantee constraints induce more conservative policies, particularly at low wealth levels, while providing effective downside protection for both wealth and benefits. The associated well-being loss relative to the unconstrained framework remains small, suggesting that incorporating guarantees into retirement products is practically appealing.

q-fin.MF↗

arXiv2026

Risk-minimizing reinsurance with adverse selection

This paper provides a comprehensive characterization of optimal reinsurance mechanisms under adverse selection when a monopolistic reinsurer faces a continuum of insurers with multi-dimensional private information on loss exposures and Value-at-Risk (VaR) preferences. Moving beyond standard single-dimensional or discrete-type models, we resolve an infinite-dimensional, non-convex screening problem by introducing a novel pre-reinsurance VaR reparameterization and an indirect utility representation. This framework converts global incentive constraints into tractable, typewise conditions. Using a Breeden-Litzenberger representation, we show that optimal indirect utility consistently adopts a hinge form across stop-loss, quota-share, and change-loss contract classes, yielding an endogenous market-exclusion threshold. Among participating types, stop-loss contracts screen via type-dependent deductibles, whereas quota-share contracts induce complete pooling at full coverage. Moreover, under mild conditions, expanding the contract space to change-loss policies yields no extra profit beyond optimal stop-loss designs. Numerical examples further illustrate the fundamental economic trade-off between extracting information rents and excluding low-exposure types. Our approach provides deeper insight into the mechanisms of discrimination and exclusion in reinsurance markets characterized by asymmetric information, enriching theoretical and practical perspectives on contract design in the presence of adverse selection.

q-fin.RM↗

arXiv2026

Likelihood-based Spacings Goodness-of-Fit Statistics for Univariate Shape-constrained Densities

A variety of statistics based on sample spacings have been studied for testing goodness-of-fit to parametric distributions. To test the goodness-of-fit to a nonparametric class of univariate shape-constrained densities, including widely studied classes such as k-monotone and log-concave densities, a likelihood ratio test with a working alternative density estimate based on the spacings of the observations is considered, and is shown to be asymptotically normal and distribution-free under the null, consistent under fixed and certain local alternatives, and admits bootstrap calibration. The distribution-freeness under the null comes from the fact that the asymptotic dominant term depends only on a function of the spacings of transformed outcomes that are uniformly distributed. Applications and extensions of theoretical results in the literature of shape-constrained estimation are required to show that the average log-density ratio converges to zero at a faster rate than the sample spacing term under the null, and diverges under the alternatives. Numerical studies are conducted to demonstrate that the test is applicable to various classes of shape-constrained densities and has a good balance between type-I error control under the null and power under alternative distributions.

math.ST↗

arXiv2026

A Class of Degenerate Mean Field Games, Associated FBSDEs and Master Equations

In this paper, we study a class of degenerate mean field games (MFGs) with state-distribution dependent and unbounded functional diffusion coefficients. With a probabilistic method, we study the well-posedness of the forward-backward stochastic differential equations (FBSDEs) associated with the MFG and arising from the maximum principle, and estimate the corresponding Jacobian and Hessian flows. We further establish the classical regularity of the value functional $V$; in particular, we show that when the cost function is $C^3$ in the spatial and control variables and $C^2$ in the distribution argument, then the value functional is $C^1$ in time and $C^2$ in the spatial and distribution variables. As a consequence, the value functional $V$ is the unique classical solution of the degenerate MFG master equation.

math.OC↗

arXiv2025

A Variational Approach to Mean Field Type Control

Variational methods have been used to study stochastic control for long, see Bensoussan (1982) and Bensoussan-Lions (1978) for the early works. More precisely, variational approaches apply to the study of Bellman equation as a parabolic quasi-linear equation, when the nonlinearity affects only the gradient of the solution, and the second order derivative term is linear and not degenerate. This corresponds to a stochastic control problem, where the state equation is a diffusion process. The primary objective of this article is to extend this approach to mean field control theory, as an alternative to the current approach, which considers a coupled system of Hamilton-Jacobi (HJ) and Fokker-Planck (FP) equations, since the introduction of the theory by Lasry-Lions (2007). The main novelty lies in that the equation studied here is the HJB equation, neither the HJ-FP system nor the master equation; and our results also provide another perspective for probabilistic approaches; see Chassagneux-Crisan-Delarue (2022), Bensoussan-Wong-Yam-Yuan (2024), Bensoussan-Tai-Yam (2025) and Bensoussan-Huang-Tang-Yam (2025) for instance. Within the scope of the PDE methods, the advantage of this article is to solve a larger class of mean field control problems, with moderate regularity; and this kind of variational methods fairly require few conditions on the regularity of the coefficients.

math.OC↗

arXiv2025

Mean Field Analysis of Mutual Insurance Market

A mutual insurance company (MIC) is a type of consumer cooperative owned by its policyholders. By purchasing insurance from an MIC, policyholders effectively become member-owners of the company and are entitled to a share of the surplus, which is determined by their own collective claims and premium contributions. This sharing mechanism creates an interactive environment in which individual insurance strategies are influenced by the actions of others. Given that mutual insurers account for nearly one-third of the global insurance market, the analysis of members' behavior under such a sharing mechanism is of both practical and theoretical importance. This article presents a first dynamic study of members' behavior in the prevalent mutual insurance market under the large-population limit. With members' wealth processes depending on the law of the insurance strategies, we model the surplus-sharing mechanism using an extended mean field game (MFG) framework and address the fundamental question of how strategic interactions in this setting influence individual decisions. Mathematically, we establish the global-in-time existence and uniqueness of the mean field forward-backward stochastic differential equation (MF-FBSDE) characterizing the Nash equilibrium strategy, employing techniques to accommodate realistic insurance constraints. Computationally, we develop a modified deep BSDE algorithm capable of solving the extended MFG problem with an additional fixed-point structure on the control. Utilizing this scheme, we examine how structural features of the MIC's design, such as the composition of risk classes and surplus-sharing proportions, reshape members' decisions and wealth through collective interactions, underscoring the central role of these mechanisms in MICs.

q-fin.RM↗

arXiv2025

Generalized Taylor's Law for Dependent and Heterogeneous Heavy-Tailed Data

Taylor's law, also known as fluctuation scaling in physics and the power-law variance function in statistics, is an empirical pattern widely observed across fields including ecology, physics, finance, and epidemiology. It states that the variance of a sample scales as a power function of the mean of the sample. We study generalizations of Taylor's law in the context of heavy-tailed distributions with infinite mean and variance. We establish the probabilistic limit and analyze the associated convergence rates. Our results extend the existing literature by relaxing the i.i.d. assumption to accommodate dependence and heterogeneity among the random variables. This generalization enables application to dependent data such as time series and network-structured data. We support the theoretical developments by extensive simulations, and the practical relevance through applications to real network data.

math.ST↗

arXiv2025

Mean Field Type Control Problems Driven by Jump-diffusions

In this article, we apply a probabilistic approach to study general mean field type control (MFTC) problems with jump-diffusions, and give the first global-in-time solution. We allow the drift coefficient $b$ and the diffusion coefficient $σ$ to nonlinearly depend on the state, distribution and control variables, and both can be unbounded and possibly degenerate; besides, the jump coefficient $γ$ is allowed to be non-constant. To tackle the non-linear and control-dependent diffusion $σ$, we further formulate a joint cone property and estimates for both processes $P$ and $Q$ of the corresponding adjoint process (where $(P,Q,R)$ is the solution triple of the associated adjoint process as a backward stochastic differential equation with jump), in contrast to our previous single cone property of the only process $P$. We study first the system of forward-backward stochastic differential equations (FBSDEs) with jumps arising from the maximum principle, and then the related Jacobian flows, which altogether yield the classical regularity of the value function and thus allow us to show that the value function is the unique classical solution of the HJB integro-partial differential equation. Most importantly, our proposed probabilistic approach can apparently handle the MFTC problem driven by a fairly general process far beyond Brownian motion, in a relatively easier manner than the existing analytic approach.

math.PR↗

arXiv2025

On Convergence Rates of General $N$-Player Stackelberg Games to their Mean Field Limits

In this article, we establish precise convergence rates of a general class of $N$-Player Stackelberg games to their mean field limits, which allows the response time delay of information, empirical distribution based interactions, and the control-dependent diffusion coefficients. All these features makes our problem nonstandard, barely been touched in the literature, and they complicate the analysis and therefore reduce the convergence rate. We first justify the same convergence rate for both the followers and the leader. Specifically, for the most general case, the convergence rate is shown to be $\mathcal{O}\left(N^{-\frac{2(q-2)}{n_1(3q-4)}}\right)$ when $n_1>4$ where $n_1$ is the dimension of the follower's state, and $q$ is the order of the integration of the initial; and this rate has yet been shown in the literature, to the best of our knowledge. Moreover, by classifying cases according to the state dimension $n_1$, the nature of the delay, and the assumptions of the coefficients, we provide several subcases where faster convergence rates can be obtained; for instance the $\mathcal{O}\left(N^{-\frac{2}{3n_1}}\right)$-convergence when the diffusion coefficients are independent of control variable. Our result extends the standard $o(1)$-convergence result for the mean field Stackelberg games in the literature, together with the $\mathcal{O}(N^{-\frac{1}{n_1+4}})$-convergence for the mean field games with major and minor players. We also discuss the special case where our coefficients are linear in distribution argument while nonlinear in state and control arguments, and we establish an $\mathcal{O}(1/\sqrt{N})$ convergence rate, which extends the linear quadratic cases in the literature.

math.OC↗

arXiv2025

Second Order Fully Nonlinear Mean Field Games with Degenerate Diffusions

In this article, we study the global-in-time well-posedness of second order mean field games (MFGs) with both nonlinear drift functions simultaneously depending on the state, distribution and control variables, and the diffusion term depending on both state and distribution. Besides, the diffusion term is allowed to be degenerate, unbounded and even nonlinear in the distribution, but it does not depend on the control. First, we establish the global well-posedness of the corresponding forward-backward stochastic differential equations (FBSDEs), which arise from the maximum principle under a so-called $β$-monotonicity commonly used in the optimal control theory. The $β$-monotonicity admits more interesting cases, as representative examples including but not limited to the displacement monotonicity, the small mean field effect condition or the Lasry-Lions monotonicity; and ensures the well-posedness result in diverse non-convex examples. In our settings, we pose assumptions directly on the drift and diffusion coefficients and the cost functionals, rather than indirectly on the Hamiltonian, to make the conditions more visible. Our probabilistic method tackles the nonlinear dynamics with a linear but infinite dimensional version, and together with our recently proposed cone property for the adjoint processes, following in an almost straightforward way the conventional approach to the classical stochastic control problem, we derive a sufficiently good regularity of the value functional, and finally show that it is the unique classical solution to the MFG master equation. Our results require fairly few conditions on the functional coefficients for solution of the MFG, and a bit more conditions -- which are least stringent in the contemporary literature -- for classical solution of the MFG master equation.

math.OC↗

arXiv2025

A Control Theoretical Approach to Mean Field Games and Associated Master Equations

We prove the global-in-time well-posedness for a broad class of mean field game problems, which is beyond the special linear-quadratic setting, as long as the mean field sensitivity is not too large. Through the stochastic maximum principle, we adopt the FBSDE approach to investigate the unique existence of the corresponding equilibrium strategies. The corresponding FBSDEs are first solved locally in time, then by controlling the sensitivity of the backward solutions with respect to the initial condition via some suitable apriori estimates for the corresponding Jacobian flows, the global-in-time solution is warranted. Further analysis on these Jacobian flows will be discussed to establish the regularities, such as linear functional differentiability, of the respective value functions that leads to the ultimate classical well-posedness of the master equation on $\mathbb{R}^d$. To the best of our knowledge, it is the first article to deal with the mean field game problem, as well as its associated master equation, with general cost functionals having quadratic growth under the small mean field effect. In this current approach, we directly impose the structural conditions on the cost functionals, rather than conditions on the Hamiltonian. The advantages of this are threefold: (i) compared with imposing conditions on Hamiltonian, the structural conditions imposed in this work are easily verified, and less demanding on the regularity requirements of the cost functionals while solving the master equation; (ii) the displacement monotonicity is basically just a direct consequence of small mean field effect in the structural conditions; and (iii) when the mean field effect is not that small, we can still provide an accurate lifespan for the local existence. The method in this work can be readily extended to the case with nonlinear drift and non-separable cost functionals.

math.OC↗

arXiv2024

On Mean Field Monotonicity Conditions from Control Theoretical Perspective

In this article, from the viewpoint of control theory, we discuss the relationships among the commonly used monotonicity conditions that ensure the well-posedness of the solutions arising from problems of mean field games (MFGs) and mean field type control (MFTC). We first introduce the well-posedness of general forward-backward stochastic differential equations (FBSDEs) defined on some suitably chosen Hilbert spaces under the $β$-monotonicity. We then propose a monotonicity condition for the MFG, namely partitioning the running cost functional into two parts, so that both parts still depend on the control and the state distribution, yet one satisfies a strong convexity and a small mean field effect condition, while the other has a newly introduced displacement quasi-monotonicity. To the best of our knowledge, the latter quasi type condition has not yet been discussed in the contemporary literature, and it can be considered as a bit more general monotonicity condition than those commonly used. Besides, for the MFG, we show that convexity and small mean field effect condition for the first part of running cost functional and the quasi-monotonicity condition for the second part together imply the $β$-monotonicity and thus the well-posedness for the associated FBSDEs. For the MFTC problem, we show that the $β$-monotonicity for the corresponding FBSDEs is simply the convexity assumption on the cost functional. Finally, we consider a more general setting where the drift functional is allowed to be non-linear for both MFG and MFTC problems.

math.OC↗

arXiv2024

Optimal Savings and Value of Population in A Stochastic Environment: Transient Behavior

We extend the work on optimal investment and consumption of a population considered in [2] to a general stochastic setting over a finite time horizon. We incorporate the Cobb-Douglas production function in the capital dynamics while the consumption utility function and the drift rate in the population dynamics can be general, in contrast with [2, 30, 31]. The dynamic programming formulation yields an unconventional nonlinear Hamilton-Jacobi-Bellman (HJB) equation, in which the Cobb-Douglas production function as the coefficient of the gradient of the value function induces the mismatching of power rates between capital and population. Moreover, the equation has a very singular term, essentially a very negative power of the partial derivative of the value function with respect to the capital, coming from the optimization of control, and their resolution turns out to be a complex problem not amenable to classical analysis. To show that this singular term, which has not been studied in any physical systems yet, does not actually blow up, we establish new pointwise generalized power laws for the partial derivative of the value function. Our contribution lies in providing a theoretical treatment that combines both the probabilistic approach and theory of partial differential equations to derive the pointwise upper and lower bounds as well as energy estimates in weighted Sobolev spaces. By then, we accomplish showing the well-posedness of classical solutions to a non-canonical parabolic equation arising from a long-lasting problem in macroeconomics.

math.AP↗

arXiv2024

Mean Field Analysis of Two-Party Governance: Competition versus Cooperation among Leaders

This article studies linear-quadratic Stackelberg games between two dominating players (or equivalently, leaders) and a large group of followers, each of whom interacts under a mean field game (MFG) framework. Unlike the conventional major-minor player game, the mean field term herein is endogenously affected by the two leaders simultaneously. These homogeneous followers are non-cooperative, whereas the two leaders can either compete or cooperate with each other, which are respectively formulated as a Nash and a Pareto game. The complete solutions of the leader-follower game can be expressed in terms of the solutions of some non-symmetric Riccati equations. Notably, our analysis suggests that both modes of interactions between leaders has their own merits and neither of them is always more favourable to the community of followers. In our knowledge, a comparative study of the effect of different modes of governance on the society is relatively rare in the existing literature, we here provide its first preliminary quantitative analysis; under a broad class of practically relevant models, we provide sufficient conditions to decide whether cooperation or competition between leaders is more favourable to the followers. Being in common with modern folklore, the relative merits of the two Stackelberg games depend on whether the interests between the two leaders and the followers align among themselves. Representative numerical examples are also supplemented.

math.OC↗

arXiv2023

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 Gâteaux 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.

math.PR↗

arXiv2023

Global Well-Posedness of First-Order Mean Field Games and Master Equations with Nonlinear Dynamics

This article presents the variant of the approach introduced in the recent work of Bensoussan, Wong, Yam and Yuan [13] to the generic first-order mean field game problem. A major contribution here is the provision of new crucial a priori estimates, whose establishment is fundamentally different from the mentioned work since the associated forward-backward ordinary differential equation (FBODE) system is notably different. In addition, we require monotonicity conditions intimately on the coefficient functions but not on the Hamiltonians to handle their non-separable nature and nonlinear dynamics; as tackling Hamiltonians directly, it potentially dissolves much useful information. Compared with the assumptions used in [13], we introduce an additional requirement that the first-order derivative of the drift function in the measure variable cannot be too large relative to the convexity of the running cost function; this requirement only arises when the Hamiltonian is non-separable, and this phenomenon can also be seen in the existing literature. On the other hand, we require less here for the second-order differentiability of the coefficient functions in comparison to that in [13]. Our approach involves first demonstrating the local existence of a solution over small time interval, followed by the provision of new crucial a priori estimates for the sensitivity of the backward equation with respect to the initial condition of forward dynamics; and finally, smoothly gluing the local solutions together to form a global solution. In addition, we establish the local and global existence and uniqueness of classical solutions for the mean field game and its master equation.

math.OC↗

arXiv2023

Linear Quadratic Extended Mean Field Games and Control Problems

We provide a thorough study of a general class of linear-quadratic extended mean field games and control problems in any dimensions where the mean field terms are allowed to be unbounded and there are also presence of cross terms in the objective functionals. Our investigation focuses on the unique existence of equilibrium strategies for the extended mean field problems by employing the stochastic maximum principle approach and the appropriate fixed point argument. We provide two distinct proofs, accompanied by two sufficient conditions, that establish the unique existence of the equilibrium strategy over a global time horizon. Both conditions emphasize the importance of sufficiently small coefficients of sensitivity for the cross term, of state and control, and mean field term. To determine the required magnitude of these coefficients, we utilize the singular values of appropriate matrices and Weyl's inequalities. The present proposed theory is consistent with the classical one, namely, our theoretical framework encompasses classical linear-quadratic stochastic control problems as particular cases. Additionally, we establish sufficient conditions for the unique existence of solutions to a particular class of non-symmetric Riccati equations, and we illustrate a counterexample to the existence of equilibrium strategies. Furthermore, we also apply the stochastic maximum principle approach to examine linear-quadratic extended mean field type stochastic control problems. Finally, we conduct a comparative analysis between our method and the alternative master equation approach, specifically addressing the efficacy of the present proposed approach in solving common practical problems, for which the explicit forms of the equilibrium strategies can be obtained directly, even over any global time horizon.

math.OC↗

arXiv2023

A Theory of First Order Mean Field Type Control Problems and their Equations

In this article, by using several new crucial {\it a priori} estimates which are still absent in the literature, we provide a comprehensive resolution of the first order generic mean field type control problems and also establish the global-in-time classical solutions of their Bellman and master equations. Rather than developing the analytical approach via tackling the Bellman and master equation directly, we apply the maximum principle approach by considering the induced forward-backward ordinary differential equation (FBODE) system; indeed, we first show the local-in-time unique existence of the solution of the FBODE system for a variety of terminal data by Banach fixed point argument, and then provide crucial a priori estimates of bounding the sensitivity of the terminal data for the backward equation by utilizing a monotonicity condition that can be deduced from the positive definiteness of the Schur complement of the Hessian matrix of the Lagrangian in the lifted version and manipulating first order condition appropriately; this uniform bound over the whole planning horizon $[0,T]$ allows us to partition $[0,T]$ into a number of sub-intervals with a common small length and then glue the consecutive local-in-time solutions together to form the unique global-in-time solution of the FBODE system. The regularity of the global-in-time solution follows from that of the local ones due to the regularity assumptions on the coefficient functions. Moreover, the regularity of the value function will also be shown with the aid of the regularity of the solution couple of the FBODE system and the regularity assumptions on the coefficient functions, with which we can further deduce that this value function and its linear functional derivative satisfy the Bellman and master equations, respectively.

math.OC↗