Stochastic stability of monotone economies in regenerative environments

This paper presents an analysis of the stochastic recursion $$W_{i+1} = [V_iW_i+Y_i]^+$$ W i + 1 = [ V i W i + Y i ] + that can be interpreted as an autoregressive process of order 1, reflected at 0. We start our exposition by a discussion of the model’s stability condition. Writing $$Y_i=B_i-A_i$$ Y i = B i - A i , for independent sequences of nonnegative i.i.d. random variables $$\{A_i\}_{i\in {\mathbb N}_0}$$ { A i } i ∈ N 0 and $$\{B_i\}_{i\in {\mathbb N}_0}$$ { B i } i ∈ N 0 , and assuming $$\{V_i\}_{i\in {\mathbb N}_0}$$ { V i } i ∈ N 0 is an i.i.d. sequence as well (independent of $$\{A_i\}_{i\in {\mathbb N}_0}$$ { A i } i ∈ N 0 and $$\{B_i\}_{i\in {\mathbb N}_0}$$ { B i } i ∈ N 0 ), we then consider three special cases (i) $$V_i$$ V i equals a positive value a with certain probability $$p\in (0,1)$$ p ∈ ( 0 , 1 ) and is negative otherwise, and both $$A_i$$ A i and $$B_i$$ B i have a rational LST, (ii) $$V_i$$ V i attains negative values only and $$B_i$$ B i has a rational LST, (iii) $$V_i$$ V i is uniformly distributed on [0, 1], and $$A_i$$ A i is exponentially distributed. In all three cases, we derive transient and stationary results, where the transient results are in terms of the transform at a geometrically distributed epoch.

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“…. , a m+ (r ) can be determined from the linear systems (16) and (18) that will be given below. 2. if a = 1 then…”

Section: Model I: the Mixed Casementioning

confidence: 99%

“…Notable studies of stochastic recursions are [4,13,16]; see in addition [3]. For the non-reflected case, stochastic recursions of the form W i+1 = V i W i + Y i have been studied frequently, partly under the name 'Vervaat perpetuity'; we mention [9,12,14,17,19,22].…”

Section: Introductionmentioning

confidence: 99%

A multiplicative version of the Lindley recursion

et al. 2021

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“…It maybe of interest to generalize some of our results from Markov chains to stochastic recursive sequences driven by a stationary sequence. Such models have been investigated, for example, in [7] where they were applied to some economic models.…”

Section: Complements Examples and Open Problemsmentioning

confidence: 99%

Convergence of Markov chain transition probabilities

Scheutzow

Schindler

2021

Electron. Commun. Probab.

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Consider a discrete time Markov chain with rather general state space which has an invariant probability measure µ. There are several sufficient conditions in the literature which guarantee convergence of all or µ-almost all transition probabilities to µ in the total variation (TV) metric: irreducibility plus aperiodicity, equivalence properties of transition probabilities, or coupling properties. In this work, we review and improve some of these criteria in such a way that they become necessary and sufficient for TV convergence of all respectively µ-almost all transition probabilities.In addition, we discuss so-called generalized couplings.

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“…This is approach is also used byHall (2005), for example. What we want to capture is that there is an ex ante agreement on what actions should be taken, and how the resulting output should be split, and that failure to abide by it leads to the breakdown utilities.…”

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confidence: 99%

“…The Nash demand game is a simple way of implementing this idea -but we stress that our results are not sensitive to the way it is operationalized. For a fuller discussion, seeHall (2005).…”

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confidence: 99%

Dynamic relational contracts under complete information

Thomas

Worrall

2018

Journal of Economic Theory

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This paper considers a long-term relationship between two agents who both undertake an action or investment that produces a joint benefit. Agents have an opportunity to expropriate some of the joint benefit for their own use. Agents have quasi-linear preferences. Two cases are considered: where agents are risk averse but where limited liability constraints do not bind, and where agents are risk neutral and subject to limited liability constraints. We ask how to structure the investments and division of the surplus over time to avoid expropriation. In the risk-averse case, the dynamics of actions and surplus may or may not be monotonic depending on whether or not a first-best allocation can be sustained. Agents may underinvest but never overinvest. If the first-best allocation is not sustainable, there is a trade-off between risk sharing and surplus maximization; surplus may not be at its constrained maximum even in the long run and the "amnesia" property of pure risk-sharing models fails to hold. In contrast, in the risk-neutral case there may be an initial phase in which one agent overinvests and the other underinvests. Both actions and surplus converge monotonically to a stationary state, where surplus is maximized subject to the self-enforcing constraints.

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Stochastic stability of monotone economies in regenerative environments

Cited by 12 publications

References 31 publications

A multiplicative version of the Lindley recursion

A multiplicative version of the Lindley recursion

Convergence of Markov chain transition probabilities

Dynamic relational contracts under complete information

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