Conjugate duality in problems of constrained utility maximization

Labbé, Chantal; Heunis, Andrew J.

doi:10.1080/17442500802525401

Cited by 6 publications

(12 citation statements)

References 14 publications

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“…In the rest of this section, we formulate the dual problem following the approach in [15]. Given any continuous {F t } semimartingale process X, we write…”

Section: Market Model and Primal And Dual Problemsmentioning

confidence: 99%

“…The duality relation follows from [15,Corollary 4.12]. In this paper, instead of applying the convex duality method of [1], we use the machinery of stochastic maximum principle and BSDEs to derive the necessary and sufficient conditions of the primal and dual problems separately.…”

Section: )mentioning

confidence: 99%

“…According to [15,Proposition 4.15], there exists some (ŷ,v) ∈ (0, ∞) × D such that < ∞, the associated adjoint equation for the dual problem is the following linear BSDE in the unknown processes p 2 ∈ H 2 (0, T ; R) and q 2 ∈ H 2 (0, T ; R N ):…”

Section: )mentioning

confidence: 99%

“…However, despite the evident power of this approach, it is nevertheless true that obtaining the corresponding dual problem remains a challenge as it often involves clever experimentation and subsequently show to work as desired. To bring some transparency to the dual problem, Labbé and Heunis [15] established a simple synthetic method of arriving at a dual functional, bypassing the need to formulate a fictitious market. It often happens that the dual problem is much nicer than the primal problem in the sense that it is easier to show the existence of a solution and in some cases explicitly obtain a solution to the dual problem than it is to do likewise for the primal problem.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we follow the approach as in Labbé and Heunis [15] by first converting the original problem into a static problem in an abstract space. Then we apply convex analysis to derive its dual problem and get the specific dual stochastic control problem.…”

Section: Introductionmentioning

confidence: 99%

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Dynamic convex duality in constrained utility maximization

Zheng

2018

Stochastics

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In this paper, we study a constrained utility maximization problem following the convex duality approach. After formulating the primal and dual problems, we construct the necessary and sufficient conditions for both the primal and dual problems in terms of FBSDEs plus additional conditions. Such formulation then allows us to explicitly characterize the primal optimal control as a function of the adjoint process coming from the dual FBSDEs in a dynamic fashion and vice versa. Moreover, we also find that the optimal primal wealth process coincides with the adjoint process of the dual problem and vice versa. Finally we solve three constrained utility maximization problems, which contrasts the simplicity of the duality approach we propose and the technical complexity of solving the primal problems directly.

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“…In the rest of this section, we formulate the dual problem following the approach in [15]. Given any continuous {F t } semimartingale process X, we write…”

Section: Market Model and Primal And Dual Problemsmentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Dynamic convex duality in constrained utility maximization

Zheng

2018

Stochastics

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Utility maximization in a multidimensional semimartingale model with nonlinear wealth dynamics

Junca

Serrano

2021

Math Finan Econ

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Utility Maximization in a Regime Switching Model with Convex Portfolio Constraints and Margin Requirements: Optimality Relations and Explicit Solutions

Heunis¹

2015

SIAM J. Control Optim.

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We study a problem of stochastic control in mathematical finance, with the goal of maximizing expected utility of investment and consumption over a finite trading horizon. The asset prices are modeled by Itô processes, for which the market parameters are subject to regime switching in the sense of being adapted to the joint filtration of the driving Brownian motion and a finite-state Markov chain which models "regime states" of the market. The vector of portfolios is constrained to a specified closed and convex set, and margin payments are levied on the investor, resulting in a wealth equation which is nonlinear in the portfolio. We proceed by the method of conjugate duality to construct a dual optimization problem together with optimality relations between putative solutions of the given (i.e., "primal") optimization problem and the dual optimization problem. These optimality relations are then used to address the specific cases of power-type and logarithmic utility functions, with convex cone portfolio constraints, and a higher rate of interest for borrowing than for lending. We get completely explicit optimal portfolios and characterize the optimal consumption rate as the solution of a backward stochastic differential equation (BSDE) "driven" by the canonical martingales of the regime-state Markov chain. For the power utility function this is a rather unconventional BSDE, to which standard existence results do not apply, and accordingly we establish existence and uniqueness of solutions for this BSDE. Introduction.We address a problem of stochastic optimal control with the goal of trading over a finite time horizon to maximize expected utility from wealth at close of trade together with utility from intertemporal consumption. Our market model is quite classical, comprising a single risk-free asset and a finite number of risky assets, the prices of which are modeled by continuous-time Itô processes driven by a standard multidimensional Brownian motion. The problem includes (i) portfolio constraints, namely, the portfolio vector of holdings in the risky assets must take values in a specified closed convex set; (ii) margin payments levied on the investor, resulting in a wealth equation which is generally nonlinear in the portfolio; (iii) regime switching in the market model in the following sense: besides the standard Brownian motion driving the stock price models a further source of randomness in the model is a finite-state continuous-time Markov chain which is independent of the Brownian motion, it being stipulated that the market parameters (i.e., the risk-free interest rate, mean rate of return, and volatility of stocks) are adapted to the joint filtration of the Brownian motion and the Markov chain. Each state of the Markov chain models a market "mode" or "regime state"; a simple but apt example is that of a two-state

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Conjugate duality in problems of constrained utility maximization

Cited by 6 publications

References 14 publications

Dynamic convex duality in constrained utility maximization

Dynamic convex duality in constrained utility maximization

Utility maximization in a multidimensional semimartingale model with nonlinear wealth dynamics

Utility Maximization in a Regime Switching Model with Convex Portfolio Constraints and Margin Requirements: Optimality Relations and Explicit Solutions

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