Large Deviations Asymptotics and the Spectral Theory of Multiplicatively Regular Markov Processes

Kontoyiannis, Ioannis; Meyn, Sean

doi:10.1214/ejp.v10-231

Cited by 122 publications

(268 citation statements)

References 45 publications

(143 reference statements)

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“…Some of these results were subsequently proved for continuous-time models in [4]. In the present paper we prove results similar to Theorem 5 of [10] using different arguments and technical tools (the large deviation results of [9] instead of those in [8]). In this way we manage to cover some well-known models for asset prices which were untractable in the setting of [10], see Examples 3.14, 3.15 below.…”

Section: Introductionsupporting

confidence: 57%

“…Under a new set of conditions on µ, σ and (ε t ) t∈N (see (A 1 ), (A 2 ), (A 3 ), (A 4 ) below), which are neither stronger nor weaker than the corresponding conditions in [10] recalled above, we show again the existence of AEA with GDP F (see Theorem 2.3 below), using classical large deviations techniques from [2], Markov chains tools from [11] and ergodicity results on Markov chains from [9]. Moreover, the trading strategies generating those arbitrage opportunities will be explicitly constructed; a contribution we already obtained in [10] under different conditions, but it was absent from the inspiring continuous-time work [6].…”

Section: Introductionmentioning

confidence: 54%

“…The proof of Theorem 2.3 will be based on results from [9]. In order to apply the results of that paper we will need to verify that the Markov chain Φ t satisfies condition (DV 3+) below.…”

Section: Large Deviation Estimatesmentioning

confidence: 99%

See 2 more Smart Citations

Asymptotic Exponential Arbitrage and Utility-Based Asymptotic Arbitrage in Markovian Models of Financial Markets

Bidima

Rásonyi

2014

Acta Appl Math

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Consider a discrete-time infinite horizon financial market model in which the logarithm of the stock price is a time discretization of a stochastic differential equation. Under conditions different from those given in [10], we prove the existence of investment opportunities producing an exponentially growing profit with probability tending to 1 geometrically fast. This is achieved using ergodic results on Markov chains and tools of large deviations theory.Furthermore, we discuss asymptotic arbitrage in the expected utility sense and its relationship to the first part of the paper.

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Section: Introductionsupporting

confidence: 57%

Section: Introductionmentioning

confidence: 54%

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Asymptotic Exponential Arbitrage and Utility-Based Asymptotic Arbitrage in Markovian Models of Financial Markets

Bidima

Rásonyi

2014

Acta Appl Math

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“…3 for a recent thorough analysis of existence and uniqueness of continuation value processes, but the sufficient conditions given there impose restrictions that preclude some of the parametric models used in practice.) In this paper we establish a connection between the solution to this equation and to an arguably simpler eigenvalue equation of the type that occurs in the study of large deviations for Markov processes (4)(5)(6).…”

mentioning

confidence: 99%

Recursive utility in a Markov environment with stochastic growth

Hansen

Scheinkman

2012

Proc. Natl. Acad. Sci. U.S.A.

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Recursive utility models that feature investor concerns about the intertemporal composition of risk are used extensively in applied research in macroeconomics and asset pricing. These models represent preferences as the solution to a nonlinear forward-looking difference equation with a terminal condition. In this paper we study infinite-horizon specifications of this difference equation in the context of a Markov environment. We establish a connection between the solution to this equation and to an arguably simpler Perron-Frobenius eigenvalue equation of the type that occurs in the study of large deviations for Markov processes. By exploiting this connection, we establish existence and uniqueness results. Moreover, we explore a substantive link between large deviation bounds for tail events for stochastic consumption growth and preferences induced by recursive utility.R ecursive utility models of the type suggested by ref. 1 and featured in the asset-pricing literature by ref. 2 and others represent preferences as the solution to a nonlinear forwardlooking difference equation with a terminal condition. Such preferences are used in economic dynamics because seemingly simple parametric versions provide a convenient device to change risk aversion while maintaining the same elasticity of intertemporal substitution. In this paper we explore infinite-horizon specifications in the context of a Markov environment. Even under the Markov specification, establishing the existence of a solution to this forward-looking recursion used to depict preferences can be challenging. (See ref. 3 for a recent thorough analysis of existence and uniqueness of continuation value processes, but the sufficient conditions given there impose restrictions that preclude some of the parametric models used in practice.) In this paper we establish a connection between the solution to this equation and to an arguably simpler eigenvalue equation of the type that occurs in the study of large deviations for Markov processes (4-6).The remainder of the paper is organized as follows. First, we state formally the recursive utility problem and a related PerronFrobenius eigenvalue problem. We use the latter problem to construct a change in probability that plays a central role in our analysis. Under this change of measure, we establish several inequalites leading up to our main analytical result. We conclude the paper by expanding on some of the ramifications of our analysis and linking our results to the study of large deviations applied to a Markov process. Two Related ProblemsConsider a discrete-time specification of recursive preferences of the type suggested by refs. 1 and 2. We use the homogeneous-ofdegree-one aggregator specified in terms of current period consumption C t and the continuation value V t ,whereadjusts the continuation value V tþ1 for risk. With these preferences, 1 ρ is the elasticity of intertemporal substitution and δ is a subjective discount rate. The parameter ζ does not alter preferences, but gives some additional flexibility, an...

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“…For example, see [15,Theorem 1], where [5, Theorem 2] provides general mixing and uniform tail exponent conditions under which the prerequisites of this theorem hold. This encompasses increment processes that are Harris recurrent Markov chains, subject to a Foster-Lyapunov drift condition [17]. The existence of such an LDP will be the primary assumption in our proof of eq.…”

Section: A Heuristic Argumentmentioning

confidence: 99%

Large Deviation Asymptotics for Busy Periods

Duffy

Meyn

2014

Stochastic Systems

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The busy period for a queue is cast as the area swept under the random walk until it first returns to zero. Encompassing non-i.i.d. increments, the large-deviations asymptotics of the busy period B is addressed, under the assumption that the increments satisfy standard conditions, including a negative drift. The main conclusions provide insight on the probability of a large busy period, and the manner in which this occurs. The scaled probability of a large busy period has the asymptote, for any b > 0, [Formula: see text], where [Formula: see text], with [Formula: see text], and with Λ denoting the scaled cumulant generating function of the increments process. The most likely path to a large swept area is found to be a simple rescaling of the path on [0, 1] given by [Formula: see text]. In contrast to the piecewise linear most likely path leading the random walk to hit a high level, this is strictly concave in general. While these two most likely paths have distinctly different forms, their derivatives coincide at the start of their trajectories, and at their first return to zero. These results partially answer an open problem of Kulick and Palmowski [18] regarding the tail of the work done during a busy period at a single server queue. The paper concludes with applications of these results to the estimation of the busy period statistics (λ*, K) based on observations of the increments, offering the possibility of estimating the likelihood of a large busy period in advance of observing one.

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Large Deviations Asymptotics and the Spectral Theory of Multiplicatively Regular Markov Processes

Cited by 122 publications

References 45 publications

Asymptotic Exponential Arbitrage and Utility-Based Asymptotic Arbitrage in Markovian Models of Financial Markets

Asymptotic Exponential Arbitrage and Utility-Based Asymptotic Arbitrage in Markovian Models of Financial Markets

Recursive utility in a Markov environment with stochastic growth

Large Deviation Asymptotics for Busy Periods

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