国家天元数学东南中心2022年随机分析及其应用研讨会

：2022-11-26 08:00 —— 2022-11-28 17:00
：腾讯会议ID：486-5807-5163 （密码：2022）

一、 日程表

二、学术报告题目与摘要

 

Delta Family Approach to Risk-neutral Density Estimation, Quantile Sensitivity, and Stochastic Control

崔振嵛  美国斯蒂文斯理工学院

In this talk, I shall discuss some of my recent research together with collaborators on using the Delta family approximation to the Dirac Delta function to represent the probability density function of a possibly high-dimensional stochastic process. Applications include estimating the model-free risk-neutral density function from options prices, determining the sensitivity of quantiles (e.g. VaR), and solving high-dimensional stochastic control problems (e.g. optimal investment and reinsurance under (rough) stochastic volatility models).

 

Optimal reinsurance and investment with a common shock and a random exit time

陈志平  西安交通大学

Under the mean-variance framework, we study the continuous-time optimal reinsurance and investment problem with a common shock and a random exit time. To describe the influence of the common shock, we propose a new interdependence mechanism between the insurance market and the financial market. It can reflect both the impact of the occurrence of a common shock and its influence degree on the two markets. Both the termination times of reinsurance and investment are random, and the random exit time is affected simultaneously by exogenous and endogenous random events. The insurer's objective is to minimize the variance of her terminal wealth under a given level of expected terminal wealth. We derive the explicit optimal reinsurance-investment strategy by employing stochastic optimal control and Lagrange duality techniques. The influences of the market interdependence and the random exit time on the optimal strategy are demonstrated through numerical experiments. The results reveal some meaningful phenomena and provide insightful guidance for reinsurance and investment practice in reality.

 

Stability of rarefaction for stochastic viscous conservation law

董昭  中科院数学与系统科学研究院

It was proved in [9] that the rarefaction wave for the stochastic Burgers equation with transport noise [14] is time- asymptotically stable. This paper is concerned with more general flux, viscosity and conservative noise. By manipulating the weakly monotone methods, we prove the global well-possedness of strong solutions for general H^1 initial data. Furthermore, we show that the rarefaction wave is still time-asymptotically stable for general stochastic viscous conservation laws with L^p time.  This is the joint work with Fei min Huang and Houqi Su.

 

The optimal merging and branching problems for the first-line insurers

郭军义  南开大学

We consider the branching type risk model and study the optimal starting time of the new business. It is proved that the optimal starting time is the first hitting time into some set and the structure of the stopping region is studied. The optimal time of merger for two first-line insurers is also studied and the corresponding optimal strategies are presented.

 

Zero-sum finite-horizon semi-Markov games under the probability criterion

郭先平  中山大学

This talk is on a two-person zero-sum game for finite horizon semi-Markov processes, where the concerned criterion is the probability that the total payoff produced by a system during a finite horizon exceeds a prescribed goal. which can be regarded as the security probability for player 1 as well as the risk probability for player 2. Firstly, we give the characterization of the probability, and establish the Shapley equation and the existence of a saddle point under a suitable condition. Then, we develop a value iterative algorithm to compute an epsilon-saddle point and to approach the value of the game by solving a series of matrix games. The construction of the epsilon-saddle point and the convergence of the algorithm are also shown. Finally, we demonstrate the application of our main results by an example on an inventory system.

 

Optimal management of DC pension fund under the relative performance ratio and VaR constraint

关国卉  中国人民大学

This paper investigates the optimal management of defined contribution pension plan under the Omega ratio and Value-at-Risk (VaR) constraint. Interest and inflation risks are considered, and the financial market consists of cash, a zero-coupon bond, an inflation-indexed zero-coupon bond, and a stock. The goal of the pension manager is to maximize the performance ratio of the real terminal wealth under the VaR constraint. An auxiliary process is introduced to transform the original problem into a self-financing problem. We obtain the optimal terminal wealth under different cases by combining the linearization method, the Lagrange dual method, the martingale method, and the concavification method. There are fourteen cases for the convex penalty function, and there are six cases for the concave penalty function. Besides, when the penalty and reward functions are both power functions, the explicit forms of the optimal investment strategies are obtained. Numerical examples are shown to illustrate the impacts of the performance ratio and VaR constraint.

 

Risk measurement of joint risk of portfolios: a liquidity shortfall aspect

胡亦钧  武汉大学

In this talk, we present a novel axiomatic framework of measuring the joint risk of a portfolio consisting of several financial positions. From the liquidity shortfall aspect, we construct a distortion-type risk measure to measure the joint risk of portfolios, which we referred to as multivariate distortion joint risk measure, representing the liquidity shortfall caused by the joint risk of portfolios. After its fundamental properties have been studied, we axiomatically characterize it by proposing a novel set of axioms. Furthermore, based on the representations for multivariate distortion joint risk measures, we also propose a new class of vector-valued multivariate distortion joint risk measures, as well as with sensible financial interpretation. Their fundamental properties are also investigated. It turns out that this new class is large enough, as it can not only induce new vector-valued multivariate risk measures, but also recover some popular vector-valued multivariate risk measures known in the literature with alternative financial interpretation. Examples are given to illustrate the proposed multivariate distortion joint risk measures. This paper mainly gives some theoretical results, helping one to have an insight look at the measurement of joint risk of portfolios. This talk is based on a joint work with Suo Gong and Linxiao Wei.

On the dual risk model with Parisian implementation delays under a mixed dividend strategy

李婧超  深圳大学

In this paper, we consider a mixed dividend strategy in a dual risk model. The mixed dividend strategy is the combination of a threshold dividend and a Parisian implementation delays dividend under periodic observation. Given a series of discrete observation points, when the surplus level is larger than the predetermined bonus barrier at observation point, the Parisian implementation delays in dividend is immediately carried out, and the threshold dividend is per-formed continuously during the delayed period. We study the Gerber-Shiu expected discounted penalty function and the expected discounted dividend payments before ruin in such a dual risk model. Numerical illustrations are given to study the influence of relevant parameters on the ruin-related quantities and the selection of the optimal dividend barrier for a given initial surplus level.

 

A Stackelberg - Nash equilibria with investment and reinsurance in the mixed leadership game

梁志彬  南京师范大学

In this paper, we investigate the optimal reinsurance and investment problem from joint interests of the insurer and reinsurer under the framework of the mixed leadership game —the Stochastic Stackelberg-Nash game. More specifically, the reinsurer is the leader to decide on an optimal reinsurance premium, while the insurer acts as the leader to determine the amount he/she invests into the risky assets to hedge the liabilities. A correlation between insurer’s liabilities and the risky assets is introduced, and both the insurer and the reinsurer aim to maximize the expected utility on the terminal wealth. By solving the Hamilton-Jacobi-Bellman equation, the explicit expressions of the optimal results are derived, and some interesting analysis and comparisons are given. Compare to the traditional Stackelberg differential game, we find that both the insurer and the reinsurer make better choice in the mixed leadership game. In the end, some numerical examples are provided to investigate the impacts of some important parameters on the optimal results.

 

Convergence rates and CLT for stochastic inviscid Leray- model with transport noise

罗德军  中科院数学与系统科学研究院

The stochastic inviscid Leray- model perturbed by multiplicative transport noise is considered on the torus. Under a suitable scaling of the noise, it is shown that the weak solutions converge, in some negative Sobolev spaces, to the unique solution of the deterministic viscous Leray- model. This implies that transport noise regularizes the inviscid Leray- model so that it enjoys “approximate weak uniqueness”. Interpreting such limit result as a law of large numbers, we also study the underlying central limit theorem and provide an explicit convergence rate.

 

The continuous-time pre-commitment KMM problem in incomplete markets

梁宗霞  清华大学

We study the continuous-time pre-commitment KMM problem proposed by Klibanoff, Marinacci and Mukerji (2005) in incomplete financial markets, which concerns with the portfolio selection under smooth ambiguity. The decision maker (DM) is uncertain about the dominated priors of the financial market, which are characterized by a second-order distribution (SOD). The KMM model separates risk attitudes and ambiguity attitudes apart and the aim of the DM is to maximize the two-fold utility of terminal wealth, which does not belong to the classical subjective utility maximization problem. By constructing the efficient frontier, the original KMM problem is first simplified as an one-fold expected utility problem on the second-order space. To solve the equivalent simplified problem, we introduce a new distorted Legendre transformation to establish the bipolar relation and the distorted duality theorem. Then, under some assumptions, we obtain uniqueness and existence of the solution the KMM problem, and the semi-explicit forms of the optimal terminal wealth and the optimal strategy. Explicit forms of optimal strategies are presented for CRRA, CARA and HARA utilities in the case of Gaussian SOD in a Black-Scholes financial market, which show that DM with higher ambiguity aversion tends to be more concerned about extreme market conditions with larger bias.

 

The policy iteration algorithm for dividend problem under Cramér-Lundberg model: the case of bounded dividend rates

刘国欣  河北工业大学

In this talk, we focus on the policy iteration algorithm (PIA) for the optimal dividend problem with restriction of dividend rate under the Cramér- Lundberg risk model. We propose a new approach to study the optimal strategy and the optimal value function by an auxiliary optimization prob- lem, which is maximizing dividend payment with terminal reward up to the arrival-time of the first claim. We discover the different geometric relation- ship between the optimal value function and an accompanied (or auxiliary) function in case of that the ceiling rate is less, equal or greater than the premium rate. These help us to solve the auxiliary optimization problem completely and get the explicit formulae of the optimal strategy and the optimal value function for the different cases. We also characterize the optimal value function of the original problem as the minimal nonnegative solution of an optimization equation. So, in any case, it can be iteratively approximated by starting with the initial zero-valued policy, the policy of no dividend payment at all. This characterization, together with the results on the auxiliary optimization problem, established a rigorous analysis for PIA. Though PIAs here are different in different cases of the ceiling rate to the premium rate, they are independent of the model parameters, especially for the distribution of the claim size. At last, the policy iteration algorithms are presented and explained by some numerical examples.

 

Optimal entry decision of unemployment insurance under partial information

马敬堂  西南财经大学

The aim of this paper is to study the optimal time for the individual to join an unemployment insurance scheme which is intended to protect workers against the consequences of job loss and to encourage the unemployed workers to find a new job as early as possible. The wage dynamic is described by a geometric Brownian motion model under drift uncertainty and the problem is a kind of two-dimensional degenerate optimal stopping problems which is hard to analyze. The optimal time of decision for the workers is given by the first time at which the wage process hits the free boundary which therefore plays a key role in solving the problem. This paper analyzes the monotonicity and continuity of the free boundary and derives a nonlinear integral equation for it. For a particular case the closed-form formula of free boundary is obtained and for the general case the free boundary is solved by the numerical solution of the nonlinear integral equation. The key in the analysis is to convert the degenerate problem into the non-degenerate one using the probability approach.  (This is joint work with Jie Xing).

 

Model Aggregation for Risk Evaluation and Robust Optimization

毛甜甜  中国科学技术大学

We introduce a new approach for prudent risk evaluation based on stochastic dominance, which will be called the model aggregation (MA) approach. In contrast to the classic worst-case risk (WR) approach, the MA approach produces not only  a robust value of  risk evaluation but also  a robust distributional model which is useful for modeling, analysis and simulation, independent of any specific risk measure. The MA approach is easy to implement even if the uncertainty set is non-convex or the risk measure is computationally complicated, and it provides great tractability in distributionally robust optimization. Via an equivalence property between the MA and the WR approaches, new axiomatic characterizations are obtained for a few classes of popular risk measures. In particular, the Expected Shortfall (ES, also known as CVaR) is the unique risk measure satisfying the equivalence property for convex uncertainty sets  among a very large class. The MA  approach for  Wasserstein and  mean-variance uncertainty sets admits  explicit formulas for the obtained robust models, and the new approach is illustrated with various risk measures and  examples from portfolio optimization.

 

Optimal investment problem for a hybrid pension with intergenerational risk-sharing and longevity trend under model uncertainty

荣喜民  天津大学

This paper studies the optimal investment problem for a hybrid pension plan under model uncertainty, where both the contribution and the benefit are adjusted depending on the performance of the plan. Furthermore, an age and time-dependent force of mortality and a linear maximum age are considered to capture the longevity trend. Suppose that the plan manager is ambiguity averse and is allowed to invest in a risk-free asset and a stock. The plan manager aims to find optimal investment strategies and optimal intergenerational risk-sharing arrangements by minimizing the cost of unstable contribution risk, the cost of unstable benefit risk and the cost of discontinuity risk under the worst-case scenario. By applying the stochastic optimal control approach, closed-form solutions are derived under a penalized quadratic cost function. Through numerical analysis and three special cases, we find that the intergeneration risk-sharing is achieved in our collective hybrid pension plan effectively. And it also shows that when people live longer, postponing the retirement seems a feasible way to alleviate the stress of the aging problem.

 

From the optimal singular stochastic control to the optimal stopping for regime-switching processes

邵井海  天津大学

We introduce a result generalizing the connection between optimal singular stochastic control problem and optimal stopping problem for regime-switching processes. Via the optimal singular stochastic control, the optimal stopping time and the continuation region are characterized. Moreover, we prove the existence of optimal singular stochastic control for a finite horizon singular control problem with the cost function containing the terminal cost. We prove it directly by the compactification method, which is based on an elaborate application of the properties of probability measures over the cadlag space. Such a problem was left open in Haussmann and Suo (SICON, 1995). In addition, our compactification method can remove the convexity condition on the coefficients used in Dufour and Miller (SICON, 2004).

 

Mean Field Games with Mean-Field-Dependent Volatility, and Associated Coupled Nonlocal Quasilinear Forward-Backward Parabolic Equations

汤善健  复旦大学

We consider mean field games with mean-field-dependent volatility, and associated fully coupled nonlocal quasilinear forward-backward PDEs (FBPDEs). We give the global in time existence of classical solutions of the FBPDEs, and  the uniqueness under an additional monotonicity condition. A verification theorem is also obtained and the solution of the FBPDEs is used to construct an optimal strategy of the mean field game. Finally, we discuss the linear-quadratic case. (This is a joint work with Ziyu Huang, Fudan University.)

 

马氏过程经验分布Wasserstein距离的估计方法

王凤雨  天津大学

介绍估计马氏过程经验分布Wasserstein距离的几种有效方法，以紧流形上扩散过程为例验证这些方法的精确性。

 

Pricing fair premium for the loss from MBS in a CDO under a reduced form framework with regime switching

王过京  苏州大学

We introduce a homogeneous portfolio reduced form credit risk model with regime switching to describe the default behaviors for the mortgage holders. We use the occupation times of Markov chain to express the default loss and the fair premium for the loss in the different tranche investors in the CDO. We derive some explicit formulas for the distributions of those occupation times. Basing on these results, we obtain some explicit expressions for the default loss and its fair premium. We also present some numerical results to illustrate the influence of the model parameters on the default loss and the fair premium. (This talk is based on a joint work with Professor S.N. Chiu and Dr. G. Wang).

 

Two-sided heat kernel estimates for Schrodinger operators with unbounded potential

王健  福建师范大学

Consider the Schr\"odinger operator $L^V=-\Delta+V$ on $\R^d$,  where  $V:\R^d\to [0,\infty)$ is a nonnegative and locally bounded potential on $\R^d$ so that for all $x\in \R^d$ with $|x|\ge 1$, $c_1g(|x|)\le V(x)\le c_2g(|x|)$ with some constants $c_1,c_2>0$ and a nondecreasing and strictly positive function $g:[0,\infty)\to [1,+\infty)$ that satisfies $g(2r)\le c_0 g(r)$ for all $r>0$ and $\lim_{r\to \infty} g(r)=\infty.$ Two-sided heat kernel (i.e., density function) estimates for the associated Schr\"{o}dinger semigroup are established.

 

Stochastic distortion and its transformed copula

杨静平  北京大学

Motivated by wide applications of distortion functions and copulas in insurance and finance, we generalize the notion of a deterministic distortion function to a stochastic distortion, i.e., a random process, and employ the defined stochastic distortion to construct a so-called transformed copula by stochastic distortions. One method for constructing stochastic distortions is provided with a focus on using time-changed processes. After giving some families of the transformed copulas by stochastic distortions, a particular class of transformed copulas is applied to a portfolio credit risk model, where a numeric study shows the advantage of using the transformed copulas over the conventional Gaussian copula and the double t copula in terms of the fitting accuracy and the ability of catching tail dependence. It is a joint work with Feng Lin, Liang Peng and Jiehua Xie.

 

Infinite horizon FBSDEs and LQ optimal control problems

于志勇  山东大学

In this talk, we introduce a new infinite horizon domination-monotonicity framework. In this framework, by the method of continuation and some subtle techniques, we obtain an existence and uniqueness result and a pair of estimates for the solutions to a kind of infinite horizon coupled forward-backward stochastic differential equations (FBSDEs, for short). Then, the theoretical result of FBSDEs is applied to solve a stochastic linear-quadratic (LQ, for short) optimal control problem with random time-varying coefficients on infinite horizon. The unique open-loop optimal control is characterized by the solution of an infinite horizon FBSDE. Moreover, we find and illustrate a different phenomenon between the LQ problems on infinite horizon and finite horizon.

 

A Mean Field Game Approach to Equilibrium Consumption under External Habit Formation

余翔  香港理工大学

This paper studies the equilibrium consumption under external habit formation in a large population of agents. We first formulate problems under two types of conventional habit formation preferences, namely linear and multiplicative external habit formation, in a mean field game framework. In a log-normal market model with the asset specialization, we characterize one mean field equilibrium in analytical form in each problem, allowing us to understand some quantitative properties of the equilibrium strategy and conclude distinct financial implications caused by different consumption habits from a mean field perspective. In each problem with n agents, we then construct an approximate Nash equilibrium for the n-player game using the obtained mean field equilibrium when n is sufficiently large. The explicit convergence order in each problem can also be obtained. Joint work with Lijun Bo and Shihua Wang.

 

Impact of Insurers’ Digital Technology Accessibility and Information Asymmetry on Insurance Market Structure

曾燕  中山大学

This paper studies the impact of the technology accessibility divide among insurers on insurers’ technology adoption decision and market share and reveals the role of the asymmetric information about technology accessibility between individuals and insurers. Specifically, this paper studies the competitive equilibrium in a theoretical framework featuring heterogenous individuals with different search costs and two insurers with their operation costs as their private information: One of them has access to a new technology while the other does not. Our result show that when individuals overestimate insurers’ technology accessibility, the insurer without access to the technology is less likely to be crowded out of the market, and the market share divide between the two insurers is narrowed and may even be reversed when the technology is immature (i.e., when it is not cost-saving enough or brings a high additional cost). However, when the technology becomes mature, individuals’ overestimation of insurers’ technology accessibility accelerates the appearance of a “winner-tales-all” market. Our results help explain the digital divide and different market share dynamic of insurance companies in practice, and highlight the importance of releasing precise information to individuals and eliminating the technology accessibility divide among insurance companies.

 

Mckean-Vlasov stochastic differential equations with oblique reflection on non-smooth time dependent domains

翟建梁  中国科学技术大学

We consider a class of Mckean-Vlasov stochastic differential equation with oblique reflection over an non-smooth time dependent domain. We establish the existence and uniqueness results of this class, address the propagation of chaos and prove a Fredlin-Wentzell type large deviations principle (LDP).

 

The Parameter Estimations on Forward-Backward Stochastic Differential Equations

张奇  复旦大学

We provide a method for the parameter estimations of forward-backward stochastic differential equations, where only the backward equations can be observed. To get the parameter estimations in this case, the nonlinear Feynman-Kac formula and some statistical methods are used. This is a joint work with Minxuan Li.

 

Stochastic 2D Navier-Stokes equations on time-dependent domains

张土生  中国科学技术大学

We establish the existence and uniqueness of solutions to stochastic 2D Navier-Stokes equations in a time-dependent domain driven by Brownian motion. A martingale solution is first constructed through a careful domain transformation and appropriate time-dependent Galerkin approximations. The probabilistic strong solution follows from the pathwise uniqueness and the Yamada-Watanable theorem.

 

Strong convergence of McKean-Vlasov SDEs with singular interactions

张希承  北京理工大学

In this work we show the strong convergence of propagation of chaos for the particle approximation of McKean-Vlasov SDEs with singular $L^p$-interactions as well as for the moderate interaction particle systems on the level of particle trajectories. One of the main obstacles is to establish the strong well-posedness of the SDEs for particle systems with singular interaction. To this end, we extend the results on strong well-posedness of Krylov and Rockner to the case of mixed $L^p$-drifts, where the heat kernel estimates play a crucial role. Moreover, when the interaction kernel is bounded measurable, we also obtain the optimal rate of strong convergence, which is partially based on Jabin and Wang's entropy method and Zvonkin's transformation. (This is a joint work with Zimo Hao and Michael Rockner.)

 

Mean-variance asset-liability management with affine diffusion factor process and a reinsurance option

张鑫  东南大学

This talk considers an optimal asset-liability management (ALM) problem for an insurer under the mean-variance criterion. It is assumed that the value of liabilities is described by a geometric Brownian motion (GBM). The insurer’s surplus process is modeled by a general jump process generated by a marked point process. The financial market consists of one risk-free asset and n risky assets with the risk premium relying on an affine diffusion factor process. By transferring a proportion of insurance risk to a reinsurer and investing the surplus into the financial market, the insurer aims to maximize the expected terminal net wealth and, at the same time, minimize the corresponding variance of the terminal net wealth. By using a backward stochastic differential equation (BSDE) approach, closed-form expressions for both the efficient strategy and efficient frontier are derived. To illustrate the main results, we study an example with the Heston stochastic volatility (SV) model and numerically analyze the economic behavior of the efficient frontier. Finally, a generalization of the Mutual Fund Theorem is obtained.

 

Credibility theory for mean-variance premium principles

张艺赢  南方科技大学

The method of credibility theory plays a critical role in various research areas of actuarial science. Among others, the hypothetical mean and process variance are two quantities that convey crucial information to insurance companies when determining premiums for the insureds. The classical credibility model charges premiums by applying a linear combination of the mean of past claims and the population mean, and the corresponding estimator is proved to be the best estimator of hypothetical mean under mean squared loss criterion. Enlightened by the prestigious mean-variance premium principle, we propose a credibility approach to estimate the linear combination of hypothetical mean and process variance under the quadratic loss function. Our proposed estimator consists of the linear form of observations and their quadratic terms, as well as some quantities representing population information.  Meanwhile, a spin-off result is found and utilized to compare with the classical credibility model and the $q$-credibility model. The non-parametric estimators of structural quantities are also provided for ease of its practical usage, yielding the empirical credibility estimator. Several numerical illustrations are carried out to demonstrate the performance of the estimator. A real dataset in Switzerland insurance company is also analyzed for the practical application of our results.

 

Efficient Valuation of Guaranteed Minimum Maturity Benefits in Regime Switching Jump Diffusion Models with Surrender Risk

张志民  重庆大学

We present an efficient valuation approach for guaranteed minimum maturity benefits (GMMBs) embedded in variable annuity (VA) contracts in a regime-switching jump diffusion model. We allow early surrender of the VA contract and impose surrender charges, which are important in practice to discourage early termination/lapse of the contract. We consider both continuously-monitored and discretely-monitored surrender behaviors before maturity, and utilize an intensity-based framework. Based on the continuous-time Markov chain (CTMC) approximation combined with the Fourier cosine series expansion method, we find that the valuation problem can be solved under a regime-switching jump diffusion framework. Both error analysis and numerical experiments demonstrate the accuracy and efficiency of the proposed method.

 

Large ranking games with diffusion control

周超  新加坡国立大学

We consider a symmetric stochastic differential game where each player can control the diffusion intensity of an individual dynamic state process, and the players whose states at a deterministic finite time horizon are among the best of all states receive a fixed prize. Within the mean field limit version of the game we compute an explicit equilibrium, a threshold strategy that consists in choosing the maximal fluctuation intensity when the state is below a given threshold, and the minimal intensity otherwise. We show that for large n the symmetric n-tuple of the threshold strategy provides an approximate Nash equilibrium of the n-player game. We also derive the rate at which the approximate equilibrium reward and the best response reward converge to each other, as the number of players n tends to infinity. Finally, we compare the approximate equilibrium for large games with the equilibrium of the two-player case. This talk is based on the joint work with Stefan Ankirchner, Nabil Kazi-Tani and Julian Wendt.