41 research outputs found

    Ergodic BSDEs driven by Markov Chains

    Full text link
    We consider ergodic backward stochastic differential equations, in a setting where noise is generated by a countable state uniformly ergodic Markov chain. We show that for Lipschitz drivers such that a comparison theorem holds, these equations admit unique solutions. To obtain this result, we show by coupling and splitting techniques that uniform ergodicity estimates of Markov chains are robust to perturbations of the rate matrix, and that these perturbations correspond in a natural way to EBSDEs. We then consider applications of this theory to Markov decision problems with a risk-averse average reward criterion

    Ergodic backward stochastic difference equations

    Get PDF
    We consider ergodic backward stochastic differential equations in a discrete time setting, where noise is generated by a finite state Markov chain. We show existence and uniqueness of solutions, along with a comparison theorem. To obtain this result, we use a Nummelin splitting argument to obtain ergodicity estimates for a discrete time Markov chain which hold uniformly under suitable perturbations of its transition matrix. We conclude with an application of this theory to a treatment of an ergodic control problem

    Systems of ergodic BSDEs arising in regime switching forward performance processes

    Get PDF
    We introduce and solve a new type of quadratic backward stochastic differential equation (BSDE) systems defined in an infinite time horizon, called ergodic BSDE systems. Such systems arise naturally as candidate solutions to characterize forward performance processes and their associated optimal trading strategies in a regime switching market. In addition, we develop a connection between the solution of the ergodic BSDE system and the long-term growth rate of classical utility maximization problems, and use the ergodic BSDE system to study the large time behavior of PDE systems with quadratic growth Hamiltonians

    Ergodic BSDEs with jumps and time dependence

    Full text link
    In this paper we look at ergodic BSDEs in the case where the forward dynamics are given by the solution to a non-autonomous (time-periodic coefficients) Ornstein-Uhlenbeck SDE with L\'evy noise, taking values in a separable Hilbert space. We establish the existence of a unique bounded solution to an infinite horizon discounted BSDE. We then use the vanishing discount approach, together with coupling techniques, to obtain a Markovian solution to the EBSDE. We also prove uniqueness under certain growth conditions. Applications are then given, in particular to risk-averse ergodic optimal control and power plant evaluation under uncertainty

    Representation of homothetic forward performance processes in stochastic factor models via ergodic and infinite horizon BSDE

    Get PDF
    In an incomplete market, with incompleteness stemming from stochastic factors imperfectly correlated with the underlying stocks, we derive representations of homothetic (power, exponential and logarithmic) forward performance processes in factor-form using ergodic BSDE. We also develop a connection between the forward processes and infinite horizon BSDE, and, moreover, with risk-sensitive optimization. In addition, we develop a connection, for large time horizons, with a family of classical homothetic value function processes with random endowments.Comment: 34 page

    Ergodic BSDEs with Multiplicative and Degenerate Noise

    Get PDF
    In this paper we study an Ergodic Markovian BSDE involving a forward process XX that solves an infinite dimensional forward stochastic evolution equation with multiplicative and possibly degenerate diffusion coefficient. A concavity assumption on the driver allows us to avoid the typical quantitative conditions relating the dissipativity of the forward equation and the Lipschitz constant of the driver. Although the degeneracy of the noise has to be of a suitable type we can give a stochastic representation of a large class of Ergodic HJB equations; morever our general results can be applied to get the synthesis of the optimal feedback law in relevant examples of ergodic control problems for SPDEs
    corecore