41 research outputs found
Ergodic BSDEs driven by Markov Chains
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
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
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
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
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
In this paper we study an Ergodic Markovian BSDE involving a forward process 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