The Eigenvalue Problem of Singular Ergodic Control

This paper provides a rigorous asymptotic analysis of long‐term growth rates under both proportional and Morton–Pliska transaction costs. We consider a general incomplete financial market with an unspanned Markov factor process that includes the Heston stochastic volatility model and the Kim–Omberg stochastic excess return model as special cases. Using a dynamic programming approach, we determine the leading‐order expansions of long‐term growth rates and explicitly construct strategies that are optimal at the leading order. We further analyze the asymptotic performance of Morton–Pliska strategies in settings with proportional transaction costs. We find that the performance of the optimal Morton–Pliska strategy is the same as that of the optimal one with costs increased by a factor of 2. Finally, we demonstrate that our strategies are in fact pathwise optimal, in the sense that they maximize the long‐run growth rate path by path.

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“…Remark Note that for each fixed value v of the factor process the leading‐order VIs and can be regarded as eigenvalue problems, with eigenfunction

ψ (v, \cdot)

and eigenvalue

Q_{l o c} false(v false)

in the terminology of ergodic theory; see, for example, Possamaï et al. () and Hynd ().…”

Section: Heuristics For Leading‐order Vismentioning

confidence: 99%

Small‐cost asymptotics for long‐term growth rates in incomplete markets

Melnyk

Seifried

2017

Mathematical Finance

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“…The method used in (1.7) is usually called penalty method and was introduced by L. C. Evans to establish existence and regularity of solutions to second order elliptic equations with gradient constraints [12]. This method has also been used in other works like [20,37,18,19].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Finally, Hypothesis (H3) is a classical assumption for differential operators called ellipticity property, see, e.g. [12,20,24,17,16,14,11,18,6].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Solution to HJB Equations with an Elliptic Integro-Differential Operator and Gradient Constraint

Moreno-Franco

2016

Appl Math Optim

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“…Finally, since V g j is differentiable almost everywhere, we can define the optimal contract through the maximisers in the Hamiltonian (5.5). Then, using the classical result (see for instance [32] for related arguments) that the domain in which the diffusion equation is not saturated is bounded, it follows that the optimal controls (h 1,g, , h 2,g, , h 1,b,c, , h 2,b,c, ) are bounded and the corresponding SDEs admit weak solutions…”

Section: Value Function Of the Investormentioning

confidence: 99%

Bank Monitoring Incentives Under Moral Hazard and Adverse Selection

Santibáñez

Possamaï

Zhou

2019

J Optim Theory Appl

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In this paper, we extend the optimal securitisation model of Pagès [50] and Possamaï and Pagès [51] between an investor and a bank to a setting allowing both moral hazard and adverse selection. Following the recent approach to these problems of Cvitanić, Wan and Yang [14], we characterise explicitly and rigorously the so-called credible set of the continuation and temptation values of the bank, and obtain the value function of the investor as well as the optimal contracts through a recursive system of first-order variational inequalities with gradient constraints. We provide a detailed discussion of the properties of the optimal menu of contracts.Principal-Agent problems with moral hazard have an extremely rich history, dating back to the early static models of the 70s, see among many others Zeckhauser [68], Spence and Zeckhauser [63], or Mirrlees [42,43,44,45], as well as the seminal papers by Grossman and Hart [26], Jewitt, [33], Holmström [30] or Rogerson [57]. If moral hazard results from the inability of the Principal to monitor, or to contract upon, the actions of the Agent, there is a second fundamental feature of the Principal-Agent relationship which has been very frequently studied in the literature, namely that of adverse selection, corresponding to the inability to observe private information of the Agent, which is often referred to as his type. In this case, the Principal offers to the Agent a menu of contracts, each having been designed for a specific type. The so-called revelation principle, states then that it is always optimal for the Principal to propose menus for which it is optimal for the Agent to truthfully reveal his type. Pioneering research in the latter direction were due to Mirrlees [46], Mussa and Rosen [47], Roberts [55], Spence [62], Baron and Myerson [7], Maskin and Riley [38], Guesnerie and Laffont [27], and later by Salanié [59], Wilson [67], or Rochet and Choné [56]. However, despite the early realisation of the importance of considering models involving both these features at the same time, the literature on Principal-Agent problems involving both moral hazard and adverse selection has remained, in comparison, rather scarce. As far as we know, they were considered for the first time by Antle [2], in the context of auditor contracts, and then, under the name of generalised Principal-Agent problems, by Myerson [48] 1 . These generalised agency problems were then studied in a wide variety of economic settings, notably by Dionne and Lasserre [17], Laffont and Tirole [35], McAfee and McMillan [39], Picard [54], Baron and Besanko [4, 5], Melumad and Reichelstein [40, 41], Guesnerie, Picard and Rey [28], Page [49], Zou [69], Caillaud, Guesnerie and Rey [10], Lewis and Sappington [36], or Bhattacharyya [8] 2 .All the previously mentioned models are either in static or discrete-time settings. The first study of the continuous time problem with moral hazard and adverse selection was made by Sung [64], in which the author extends the seminal finite horizon and continuous-time model of Hol...

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The Eigenvalue Problem of Singular Ergodic Control

Cited by 22 publications

References 13 publications

Small‐cost asymptotics for long‐term growth rates in incomplete markets

Small‐cost asymptotics for long‐term growth rates in incomplete markets

Solution to HJB Equations with an Elliptic Integro-Differential Operator and Gradient Constraint

Bank Monitoring Incentives Under Moral Hazard and Adverse Selection

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