Mckean vlasov limit in portfolio optimization pdf

Mckean vlasov processes may be regarded as a limit approximation of systems composed by a large number of interacting particles. Every optimal portfolio invests in a combination of the riskfree asset and the market portfolio. A second kind of local methods is based on some local optimization problems solved at each. Pdf we propose several algorithms to solve mckeanvlasov forward backward. In the particular case where the price process is a martingale, i. Mckeanvlasov limit for interacting random processes in. We get a new type of controlled backward stochastic differential equations bsdes, namely, the bsdes, coupled with value function. Portfolio optimization, meanvariance criterion, optimal control, time inconsistency, dynamic programming, mckeanvlasov limit. Portfolio optimization constraints estimating return expectations and covariance alternative risk measures. Mckeanvlasov limit in portfolio optimization request pdf. The optimal execution problem is then the second optimization problem, which. Limit theory for controlled mckeanvlasov dynamics siam. In this approach, the original optimal control problem, which is time inconsistent, is viewed as the mckean vlasov limit of a family of controlled manycomponent weakly interacting systems. Portfolio optimization in a semimarkov modulated market.

Financial risk modelling and portfolio optimization with r. Meanvariance optimization and the capm these lecture notes provide an introduction to meanvariance analysis and the capital asset pricing model capm. Mckean vlasov limit in portfolio optimization, stochastic analysis and applications, vol. Semi closedloop strategies are introduced, and following the dynamic programming approach in pham and wei. This paper is concerned with linear quadratic optimal control problems for meanfield backward stochastic differential equations mfbsdes, for short with deterministic coefficients. A simple meanvariance portfolio optimization problem in continuous time is solved using the mean field approach. Consider the controlled stochastic mckeanvlasov dynamics in \mathbb. We provide a version of this maximum principle based on the differential. Ams transactions of the american mathematical society. Generally speaking, the purpose is to understand the continuum limit of. Mean field games and interacting particle systems columbia. Vassili kolokoltsov department of statistics university of warwick, uk young women in probability bonn, may 2014 d. Control of mckean vlasov systems and applications postdoc uc berkeley from june 2019 man ngo 2019, jvn institute hochiminh city.

Multilevel monte carlo for stochastic mckeanvlasov. Limit theory for controlled mckean vlasov dynamics. Semianalytical solution of a mckean vlasov equation with feedback through hitting a boundary alexander lipton, vadim kaushansky, christoph reisinger. These systems, which we call the stochastic mckean vlasov limits for the approximating finite systems, are described as stochastic evolutions in a space of probability measures onr d and are obtained as weak limits of the sequence of empirical measures for the finite systems, which are highly correlated and driven by dependent brownian motions. Professor of operations research and financial engineering, princeton university. Due to the randomness in the interaction, the mckean vlasov equation is a collection of coupled pdes indexed by the state space of the single components in the medium. The coefficients of the system and the weighting matrices in the cost functional are allowed to be adapted processes with respect to the common noise filtration. The dynamics of each particle depends not only on the empirical measure of the whole population but also on those of different populations. Pdf we propose several algorithms to solve mckeanvlasov forward. The stochastic processes could be an independent system, a weakly interacting system mckean vlasov limit, a strongly interacting system hydrodynamic limit, measurevalued process, or a random perturbation of a deterministic dynamical system. Some contributions in portfolio optimization and risk management. Linearquadratic optimal control problems for meanfield ams. Linear quadratic optimal control of conditional mckean. Request pdf mckeanvlasov limit in portfolio optimization this article considers a sectorwise allocation in a portfolio consisting of a very large number of stocks.

Murgoci, a general theory of markovian time inconsistent stochasitc control problem, working paper. Linear quadratic optimal control problems for meanfield. Continuous time meanvariance portfolio optimization through the mean field approach. Linear quadratic optimal control of conditional mckeanvlasov. We also mention the recent paper, where the meanvariance problem is viewed as the mckean vlasov limit of a family of controlled manycomponent weakly interacting systems. Mckean vlasov equation, nonlinear parabolic equation, portfolio optimization mathematics subject classification.

Investment decisions rules are made according to the objective of maximizing the expected return for. We shall argue that the dynamics of the total liquidity follows a purejump, mckean vlasov type sde with reflection, and is the optimal trajectory of a representing seller in the limiting bertrand game. Mckeanvlasov equation with random coefficients and. Multilevel monte carlo for stochastic mckeanvlasov equations lukasz szpruch joint work with shuren tan and alvin tse edinburgh school of mathemtics university of edinburgh lmsepsrc durham symposium 2017 lukasz szpruch university of edinburgh mlmc for mvsdes march 2017 1 30. A cubature based algorithm to solve decoupled mckean. These two methods di er in the order in which optimization and passage to the limit are performed. We apply largedeviation theory to particle systems with a random meanfield interaction in the mckean vlasov limit. To learn about our use of cookies and how you can manage your cookie settings, please see our cookie policy.

Suresh kumar, mckeanvlasov limit in portfolio optimization, stoch. In particular, we describe large deviations and normal fluctuations around the mckean vlasov equation. The limits of such systems as the number of particles tends to infinity are investigated. Otherwise, it reads as an optimization problem over controlled dynamics of mckean vlasov type. Chapter 1 introduction to portfolio theory updated. Analytical approximations of nonlinear sdes of mckean. This work focuses on stochastic systems of weakly interacting particles containing different populations represented by multiclasses. Mertons model views equity as a call option on the asset of the firm. Cambridge core probability theory and stochastic processes diffusions, markov processes, and martingales by l.

Diffusions, markov processes, and martingales by l. When optimizing rst, the asymptotic problem is usually referred to as a mean eld game. Mckean vlasov fbsdes with a common noise motivated by the theory of mean eld games, i will address the solvability and the smoothness. This gives a more compact description using a timevarying drift characterized in terms of a measurevalued process that satisfies a nonlinear parabolic equation. Limit theory for controlled mckean vlasov dynamics siam journal on control and optimization. This density evolves according to a stochastic partial differential equation, and we establish existence and uniqueness for the solution taking values in a suitable function space. A brief introduction to mckeanvlasov processes and nonlinear diffusions in general dialid santiago prof.

Conversely, any distribution on the set of optimal controlstate pairs for the mckean vlasov problem can be realized as a limit in this manner. Mckean vlasov limit for interacting systems with simultaneous jumps luisa andreis prof. Optimal execution problem under mckeanvlasov liquidity. The classical portfolio optimization problem is then addressed in this framework. Thus the asset is partially observed through the equity. Their interdependence is captured by the dependence of the drift coefficient of each stock on an averaged effect of the sectors. Robust markowitz meanvariance portfolio selection under. The optimality system, which is a linear meanfield forwardbackward stochastic differential equation with constraint, is obtained by a variational method. A class of finitedimensional numerically solvable mckeanvlasov. Arguments are based on controlled martingale problems, which lend themselves naturally to existence proofs. Amamef is the acronym standing for advanced mathematical methods in fi.

Continuous time meanvariance portfolio optimization. We provide analytical approximations for the law of the solutions to a certain class of scalar mckean vlasov stochastic differential equations mkvsdes with random initial datum. Then using nonlinear filtering an explicit expression for likelihood ratio for underlying parameters in terms of the nonlinear filter is obtained. We begin with the meanvariance analysis of markowitz 1952 when there is no riskfree asset and then move on to the case where there is a riskfree asset available. In this paper, we study stochastic optimal control problem for general mckean. Their interdependence is captured by the dependence of. Vlasov limit of a family of controlled manycomponent weakly. In this approach, the original optimal control problem, which is time inconsistent, is viewed as the mckean vlasov limit of a family of controlled. Continuous time meanvariance portfolio optimization through the.

Conversely, any distribution on the set of optimal controlstate pairs for the mckeanvlasov problem can be realized as a limit in this manner. In the context of stochastic portfolio theory, shkol nikov 109. Robust markowitz meanvariance portfolio selection under ambiguous covariance matrix. By considering the large portfolio limit of this system we show the existence of a density process for the asset values. The optimal expected terminal utility is obtained by constructing sub and supersolutions of the marginal hamiltonjacobibellman equation associ. This gives a more compact description using a timevarying drift characterized in terms of a measurevalued process that satisfies a. Semianalytical solution of a mckeanvlasov equation with. Control of mckeanvlasov dynamics versus mean field games. Zhang, viscosity solutions to parabolic master equations and mckean vlasov sdes with closedloop controls, annals of applied probability, accepted, arxiv.

We consider the optimal control problem for a linear conditional mckean vlasov equation with quadratic cost functional. Stochastic analysis and applications 28 5, 884906, 2010. Pathdependent pdes, siam journal on control and optimization, 58 2020, 277302. We prove the existence and the uniqueness theorem as well as a comparison theorem for such bsdes coupled with value function by using the approximation method. Financial risk modelling and portfolio optimization with r,secondedition. Chapter 24 large deviation techniques and financial. Propagation of chaos results 15 connect this class of sdes with the macroscopic limiting behavior of a particle, evolving within a meanfield.

An international journal of probability and stochastic processes, preprint 2019 arxiv 21 pagliarani s. Continuous time meanvariance portfolio optimization through the mean field approach article in esaim probability and statistics 20 january 2016 with 37 reads how we measure reads. This leads to a decoupled dynamics in the limit of large numbers, akin to the mean field limit leading to the mckean vlasov equation in statistical. The optimal control of mckeanvlasov also called meanfield.

A cubature based algorithm to solve decoupled mckeanvlasov forwardbackward stochastic differential equations. Request pdf optimal control of sdes of mckeanvlasov type the. Chabakauri, dynamic meanvariance asset allocation, rev. Optimal control of sdes of mckeanvlasov type request pdf. Mckeanvlasov limit in portfolio optimization, stoch. Workshop on bsdes and spdes edinburgh, 37 july 2017 organisers. This chapter introduces modern portfolio theory in a simpli. Inthefinalchapterofpartichapter5,themarkowitzportfolioframe. This article considers a sectorwise allocation in a portfolio consisting of a very large number of stocks. I will discuss the optimal portfolio management of a population of fund managers who trade in. This leads to a decoupled dynamics in the limit of large numbers, akin to the omean fieldo limit leading to the mckean vlasov equation in statistical physics. Let p be the optimal portfolio for target expected return 0. Excel modeling and estimation in investments third. Pdf numerical resolution of mckeanvlasov fbsdes using neural.