Utility maximization with addictive consumption habit formation in incomplete semimartingale markets

Yu, Xiang

doi:10.1214/14-aap1026

Cited by 32 publications

(40 citation statements)

References 30 publications

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“…Based on Mostovyi [2015], we derive a duality result for the two-parameter optimization problem, where the parameters are the initial wealth and the essential lower bound on consumption. In this regard, the formulation and derivation of our main optimization result (Theorem 27) is similar to the derivation of Hugonnier and Kramkov [2004] from the results of Schachermayer [1999, 2003], and to the work of Yu [2015].…”

Section: Introductionmentioning

confidence: 60%

“…This approach parallels the work of Kramkov and Schachermayer [2003] on maximization of the expected value of deterministic utility from terminal wealth. A different but common approach to establish duality for expected utility maximization with time-and/or scenariodependent utility is to require the utility to satisfy additional assumptions, including uniform reasonable asymptotic elasticity of Karatzas and Žitković [2003] (see also Žitković [2005], Bank and Kauppila [2017], Yu [2015]), a generalization of the reasonable asymptotic elasticity condition of Kramkov and Schachermayer [1999] for deterministic utility applied to terminal wealth.…”

Section: Introductionmentioning

confidence: 99%

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Utility maximization with ratchet and drawdown constraints on consumption in incomplete semimartingale markets

Tanana¹

2021

Preprint

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This paper studies expected utility maximization under ratchet and drawdown constraints on consumption in a general incomplete semimartingale market using duality methods. The optimization is considered with respect to two parameters: the initial wealth and the essential lower bound on consumption process. In order to state the problem and define the primal domains, we introduce a natural extension of the notion of running maximum to arbitrary non-negative optional processes and study its properties. The dual domains for optimization are characterized in terms of solidity with respect to an ordering that is introduced on the set of non-negative optional processes. The abstract duality result we obtain for the optimization problem is used in order to derive a more specific characterization of solution in case of ratchet constraint in a complete market. The method described in the paper might be useful for other control problems involving running maxima or increasing controls.

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Section: Introductionmentioning

confidence: 60%

Section: Introductionmentioning

confidence: 99%

Utility maximization with ratchet and drawdown constraints on consumption in incomplete semimartingale markets

Tanana¹

2021

Preprint

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“…In particular, to build the convex duality theory treating the problem with unhedgeable random endowment demands new techniques, especially if the endowment unbounded, see for example [4], [12], [19]. The references [18], [29], [23] and [27] also study this problem when the intermediate consumption is considered.…”

Section: Introductionmentioning

confidence: 99%

Optimal Investment with Random Endowments and Transaction Costs: Duality Theory and Shadow Prices

Bayraktar

2016

SSRN Journal

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Abstract. This paper studies the utility maximization problem on the terminal wealth with both random endowments and proportional transaction costs. To deal with unbounded random payoffs from some illiquid claims, we propose to work with the acceptable portfolios defined via the consistent price system (CPS) such that the liquidation value processes stay above some stochastic thresholds. In the market consisting of one riskless bond and one risky asset, we obtain a type of the super-hedging result. Based on this characterization of the primal space, the existence and uniqueness of the optimal solution for the utility maximization problem are established using the convex duality analysis. As an important application of the duality theory, we provide some sufficient conditions for the existence of a shadow price process with random endowments in a generalized form similar to [5] as well as in the usual sense using acceptable portfolios.

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“…It is worth noting that the Bipolar relationship plays an important role in the duality approach. In frictionless markets (see [32]), the Bipolar result relies on the consumption budget constraint which is taken as granted due to the optional decomposition theorem, see [22]. This inequality characterization of the consumption processes suggests us to define the correct auxiliary set and the auxiliary dual set in [32].…”

mentioning

confidence: 99%

“…To begin with, the consumption budget constraint provides an inequality characterization of the set of all financeable consumption strategies. By introducing the auxiliary primal process (c t ) t∈[0,T ] wherec t = c t − F (c) t and the auxiliary dual process (Γ t ) t∈[0,T ] defined via the SCPS, we can follow and modify the ideas and proofs in [32] to apply the convex duality approach for an abstract optimization problem. It is worth pointing out that the mathematical approach for habit formation may allow us to consider some related economic problems such as the consumption with durable goods under transaction 3 costs.…”

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confidence: 99%

Optimal consumption under habit formation in markets with transaction costs and random endowments

Yu¹

2017

Ann. Appl. Probab.

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This paper studies the optimal consumption via the habit formation preference in markets with transaction costs and unbounded random endowments. To model the proportional transaction costs, we adopt the Kabanov's multi-asset framework with a cash account. At the terminal time T , the investor can receive unbounded random endowments for which we propose a new definition of acceptable portfolios based on the strictly consistent price system (SCPS). We prove a type of super-hedging theorem using the acceptable portfolios which enables us to obtain the consumption budget constraint condition under market frictions. Following similar ideas in [32] with the path dependence reduction and the embedding approach, we obtain the existence and uniqueness of the optimal consumption using some auxiliary processes and the duality analysis. As an application of the duality theory, the market isomorphism with special discounting factors is also discussed in the sense that the original optimal consumption with habit formation is equivalent to the standard optimal consumption problem without the habits impact, however, in a modified isomorphic market model.1. Introduction. The study of consumption habit formation in financial economics dates back to [15] and [28]. It has been observed that the von Neumann-Morgenstern utilities can not reconcile the well-known magnitude of the equity premium (See [25] and [4]). Instead, the habit formation preference takes care of both the current consumption choice and its history pattern. Due to its time non-separable structure, a small change in the consumption can cause a large fluctuation in consumption net of the subsistence level, which may explain the sizable excess returns on risky assets in equilibrium models. Likewise this new preference can shed some light on the quantitative explanation of consumer's psychology in the context of the consumption behavior with investment opportunities. For instance, many empirical studies and psychological reports reveal that the consumer's satisfaction level and risk tolerance sometimes rely more on recent changes than the absolute levels. This preference has thereby been a surge in models for which the smooth consumption is more beneficial than the marked increase, such as the household consumption and other types of expenditures with commitment. In addition, as pointed out in [11], habit formation allows the model to match the response of real spending to monetary policy shocks and this new preference can also more accurately replicate the gradual decline of inflation during a disinflation. The diverse applications of this path-dependent preference motivate our research in general incomplete market models.

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Utility maximization with addictive consumption habit formation in incomplete semimartingale markets

Cited by 32 publications

References 30 publications

Utility maximization with ratchet and drawdown constraints on consumption in incomplete semimartingale markets

Utility maximization with ratchet and drawdown constraints on consumption in incomplete semimartingale markets

Optimal Investment with Random Endowments and Transaction Costs: Duality Theory and Shadow Prices

Optimal consumption under habit formation in markets with transaction costs and random endowments

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