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BACKWARD STOCHASTIC RICCATI EQUATION WITH JUMPS ASSOCIATED WITH STOCHASTIC LINEAR QUADRATIC OPTIMAL CONTROL WITH JUMPS AND RANDOM COEFFICIENTS

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2020年第1期58卷 393-424页

作者： Zhang, Fu Dong, Yuchao Meng, Qingxin Univ Shanghai Sci & Technol Coll Sci Shanghai 200093 Peoples R China Univ Angers Dept Math 2 Bd Lavoisier F-49045 Angers 01 France Fudan Univ Dept Math Shanghai 200433 Peoples R China Huzhou Univ Dept Math Huzhou 313000 Zhejiang Peoples R China

In this paper, we investigate the solvability of matrix valued backward stochastic Riccati equations with jumps (BSREJ), which are associated with a stochastic linear quadratic (SLQ) optimal control problem with random coefficients and driven by both Brownian motion and a Poisson process. By the dynamic programming principle, Doob-Meyer decomposition, and inverse flow technique, the existence and uniqueness of the solution for the BSREJ are established. The difficulties addressed in this issue not only are brought from the high nonlinearity of the generator of the BSREJ like the case driven only by Brownian motion, but also from that (i) the inverse flow of the controlled linear stochastic differential equation driven by Poisson process may not exist without additional technical conditions, and (ii) showing that the inverse matrix term involving a jump process in the generator is well-defined. Utilizing the structure of the optimal control problem, we overcome these difficulties and establish the existence of the solution. In addition, we show the construction of the optimal feedback control with the help of the Riccati equation and the relation between the solution of the Riccati equation and the value function of the SLQ problem, which also implies the uniqueness of BSREJ.

关键词： dynamic programming principle Doob-Meyer decomposition stochastic differential equation Poisson process backward stochastic Riccati equation with jumps

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Stochastic recursive optimal control problem of reflected stochastic differential systems

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INTERNATIONAL JOURNAL OF CONTROL 2020年第9期93卷 2187-2198页

作者： Feng, Xinwei Chinese Univ Hong Kong Dept Stat Shatin Hong Kong Peoples R China

In this paper, we study one kind of stochastic recursive optimal control problem in which the control system is stochastic differential equations reflected in a domain and the cost functional is defined by generalised backward stochastic differential equations with reflection. We establish the dynamic programming principle for the value function and show that it is a viscosity solution of the associated obstacle problem for the corresponding Hamilton-Jacobi-Bellman equation with a nonlinear Neumann boundary condition.

关键词： Stochastic recursive optimal control generalised reflected backward stochastic differential equations dynamic programming principle viscosity solution nonlinear Neumann boundary condition

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VISCOSITY SOLUTIONS TO PARABOLIC MASTER EQUATIONS AND MCKEAN-VLASOV SDES WITH CLOSED-LOOP CONTROLS

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ANNALS OF APPLIED PROBABILITY 2020年第2期30卷 936-986页

作者： Wu, Cong Zhang, Jianfeng Wells Fargo Secur San Francisco CA 94108 USA Univ Southern Calif Dept Math Los Angeles CA 90007 USA

The master equation is a type of PDE whose state variable involves the distribution of certain underlying state process. It is a powerful tool for studying the limit behavior of large interacting systems, including mean field games and systemic risk. It also appears naturally in stochastic control problems with partial information and in time inconsistent problems. In this paper we propose a novel notion of viscosity solution for parabolic master equations, arising mainly from control problems, and establish its wellposedness. Our main innovation is to restrict the involved measures to a certain set of semimartingale measures which satisfy the desired compactness. As an important example, we study the HJB master equation associated with the control problems for McKean-Vlasov SDEs. Due to practical considerations, we consider closed-loop controls. It turns out that the regularity of the value function becomes much more involved in this framework than the counterpart in the standard control problems. Finally, we build the whole theory in the path dependent setting, which is often seen in applications. The main result in this part is an extension of Dupire's (2009) functional Ito formula. This Ito formula requires a special structure of the derivatives with respect to the measures, which was originally due to Lions in the state dependent case. We provided an elementary proof for this well known result in the short note (2017), and the same arguments work in the path dependent setting here.

关键词： Master equation McKean-Vlasov SDEs viscosity solution functional Ito formula path dependent PDEs Wasserstein spaces dynamic programming principle

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ASYMPTOTICS OF VALUES IN dynamic GAMES ON LARGE INTERVALS

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ST PETERSBURG MATHEMATICAL JOURNAL 2020年第1期31卷 157-179页

作者： Khlopin, D., V IMM UB RAS NN Krasovskii Inst Math & Mech Ul S Kovalevskoi 16 Ekaterinburg 620999 Russia

dynamic antagonistic games are treated. The dependence of the game value on the payoff function is explored for games with one and the same dynamics, running cost, and possibilities of the players. Each payoff function is viewed as a running cost, averaged in terms of a certain probability distribution on the semiaxis. It is shown that if the dynamic programming principle is fulfilled, then a Tauberian-type theorem holds, specifically, the existence of a uniform limit of the values (as the scaling parameter tends to zero) for games with exponential and/or uniform distribution implies that the same limit exists (as the scaling parameter tends to zero) with arbitrary piecewise-continuous density. Also, a version of this theorem for games with discrete time is proved. The general results are applied to various game settings, both in the deterministic and in the stochastic case. Also, asymptotics are studied as the time horizon tends to infinity.

关键词： dynamic programming principle antagonistic dynamic games differential games Abel mean Cesaro mean

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Asymptotic Lipschitz Regularity for Tug-of-War Games with Varying Probabilities

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POTENTIAL ANALYSIS 2020年第2期53卷 565-589页

作者： Arroyo, Angel Luiro, Hannes Parviainen, Mikko Ruosteenoja, Eero Univ Jyvaskyla Dept Math & Stat POB 35 FI-40014 Jyvaskyla Finland Norwegian Univ Sci & Technol Dept Math Sci NTNU NO-7491 Trondheim Norway

We prove an asymptotic Lipschitz estimate for value functions of tug-of-war games with varying probabilities defined in omega subset of Double-struck capital R-n. The method of the proof is based on a game-theoretic i... 详细信息

关键词： dynamic programming principle Local Lipschitz estimates Stochastic games Normalizedp(x)-Laplacian

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Hedging longevity risk in defined contribution pension schemes

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COMPUTATIONAL MANAGEMENT SCIENCE 2023年第1期20卷 11-11页

作者： Agarwal, Ankush Ewald, Christian-Oliver Wang, Yongjie Univ Glasgow Adam Smith Business Sch Glasgow City G12 8QQ Scotland Inland Norway Univ Appl Sci Business Sch Lillehammer Norway

Pension schemes all over the world are under increasing pressure to efficiently hedge longevity risk imposed by ageing populations. In this work, we study an optimal investment problem for a defined contribution pension scheme that decides to hedge longevity risk using a mortality-linked security, typically a longevity bond. The pension scheme promises a minimum guarantee which allows the members to purchase lifetime annuities upon retirement. The scheme manager invests in the risky and riskless assets available on the market, including the longevity bond. We transform the corresponding constrained optimal investment problem into a single investment portfolio optimization problem by replicating future contributions from members and the minimum guarantee provided by the scheme. We solve the resulting optimization problem using the dynamic programming principle. Through a series of numerical studies, we show that the longevity risk has an important impact on the investment strategy performance. Our results add to the growing evidence supporting the use of mortality-linked securities for efficient hedging of longevity risk.

关键词： Defined contribution pension scheme Longevity bond Stochastic control dynamic programming principle

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Stochastic optimal control with random coefficients and associated stochastic Hamilton-Jacobi-Bellman equations

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ADVANCES IN CONTINUOUS AND DISCRETE MODELS 2022年第1期2022卷 3-3页

作者： Moon, Jun Hanyang Univ Dept Elect Engn Seoul 04763 South Korea

We consider the optimal control problem for stochastic differential equations (SDEs) with random coefficients under the recursive-type objective functional captured by the backward SDE (BSDE). Due to the random coefficients, the associated Hamilton-Jacobi-Bellman (HJB) equation is a class of second-order stochastic PDEs (SPDEs) driven by Brownian motion, which we call the stochastic HJB (SHJB) equation. In addition, as we adopt the recursive-type objective functional, the drift term of the SHJB equation depends on the second component of its solution. These two generalizations cause several technical intricacies, which do not appear in the existing literature. We prove the dynamic programming principle (DPP) for the value function, for which unlike the existing literature we have to use the backward semigroup associated with the recursive-type objective functional. By the DPP, we are able to show the continuity of the value function. Using the Ito-Kunita's formula, we prove the verification theorem, which constitutes a sufficient condition for optimality and characterizes the value function, provided that the smooth (classical) solution of the SHJB equation exists. In general, the smooth solution of the SHJB equation may not exist. Hence, we study the existence and uniqueness of the solution to the SHJB equation under two different weak solution concepts. First, we show, under appropriate assumptions, the existence and uniqueness of the weak solution via the Sobolev space technique, which requires converting the SHJB equation to a class of backward stochastic evolution equations. The second result is obtained under the notion of viscosity solutions, which is an extension of the classical one to the case for SPDEs. Using the DPP and the estimates of BSDEs, we prove that the value function is the viscosity solution to the SHJB equation. For applications, we consider the linear-quadratic problem, the utility maximization problem, and the European option pricing problem

关键词： Stochastic Hamilton-Jacobi-Bellman equation Random coefficients Viscosity solution dynamic programming principle Verification theorem Moon

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On the dynamic representation of some time-inconsistent risk measures in a Brownian filtration

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MATHEMATICS AND FINANCIAL ECONOMICS 2020年第3期14卷 433-460页

作者： Backhoff-Veraguas, Julio Tangpi, Ludovic Univ Wien Fak Math Oskar Morgenstern Pl 1 A-1090 Vienna Austria Princeton Univ Dept Operat Res & Financial Engn Sherrerd Hall Princeton NJ 08540 USA

It is well-known from the work of Kupper and Schachermayer that most law-invariant risk measures are not time-consistent, and thus do not admit dynamic representations as backward stochastic differential equations. In this work we show that in a Brownian filtration the "Optimized Certainty Equivalent" risk measures of Ben-Tal and Teboulle can be computed through PDE techniques, i.e. dynamically. This can be seen as a substitute of sorts whenever they lack time consistency, and covers the cases of conditional value-at-risk and monotone mean-variance. Our method consists of focusing on the convex dual representation, which suggests an expression of the risk measure as the value of a stochastic control problem on an extended the state space. With this we can obtain a dynamic programming principle and use stochastic control techniques, along with the theory of viscosity solutions, which we must adapt to cover the present singular situation.

关键词： Time-inconsistency Risk measures Optimized certainty equivalent HJB equation Viscosity solution Unbounded stochastic control dynamic programming principle Singular Hamiltonian

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Robust Framework for Quantifying the Value of Information in Pricing and Hedging

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SIAM JOURNAL ON FINANCIAL MATHEMATICS 2020年第1期11卷 27-59页

作者： Aksamit, Anna Hou, Zhaoxu Obloj, Jan Univ Sydney Sch Math & Stat Sydney NSW 2006 Australia Univ Oxford Math Inst Oxford OX2 6GG England Univ Oxford Math Inst Oxford OX1 3JP England Univ Oxford St Johns Coll Oxford OX1 3JP England

We investigate asymmetry of information in the context of the robust approach to pricing and hedging of financial derivatives. We consider two agents, one who only observes the stock prices and another with some additional information, and investigate when the pricing-hedging duality for the former extends to the latter. We introduce a general framework to express the superhedging and market model prices for an informed agent. Our key insight is that an informed agent can be seen as a regular agent who can restrict her attention to a certain subset of possible paths. We use results of Hou and Obloj [Finance Stoch., 22 (2018), pp. 511-567], on the robust approach with beliefs to establish the pricing-hedging duality for an informed agent. Our results cover a number of scenarios, including information arriving before trading starts, arriving after the static position in European options is formed but before dynamic trading starts, or arriving at some point before maturity. For the latter we show that the superhedging value satisfies a suitable dynamic programming principle, which is of independent interest. Finally, we explore how our results allow us to develop robust valuation of information.

关键词： robustness superhedging pricing-hedging duality informed investor asymmetry of information filtration enlargement path restrictions dynamic programming principle modeling with beliefs

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Representation formula for viscosity solution to a PDE problem involving Pucci's extremal operator

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NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS 2021年 57卷 103199-103199页

作者： Pozza, Marco Sapienza Univ Roma Dipartimento Matemat G Castelnuovo Piazzale Aldo Moro 5 I-00185 Rome Italy

We provide a representation formula for viscosity solutions to an elliptic Dirichlet problem involving Pucci's extremal operators. This is done through a dynamic programming principle derived from Denis et al. (2010). The formula can be seen as a nonlinear extension of the Feynman-Kac formula. (C) 2020 Elsevier Ltd. All rights reserved.

关键词： Nonlinear Feynman-Kac formula Pucci's extremal operators Viscosity solutions dynamic programming principle

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