Convergence results for the indifference value based on the stability of BSDEs

Frei, Christoph

doi:10.1080/17442508.2011.653568

Cited by 9 publications

(11 citation statements)

References 13 publications

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“…B M O 1 (P) ≤ Xπ ,1 B M O 1 (P), we obtain from the variant of the John-Nirenberg inequality stated in Theorem 2.1 of Kazamaki[13] that EP exp qpη 2 λ 2 Xπ that there exists p > 1 such that E( (η 2 Z (k) + θ) dŴ ) satisfies the reverse Hölder inequality R p (P) uniformly in k; compare (3.2). This implies by Theorem 3.3 of Kazamaki[13] that the B M O(P)-norm of Z (k) dŴ is bounded uniformly in k. One can now show similarly to the proof of Theorem 2.1 of Frei[9] that one has lim k→∞ Y (Y (∞) , Z (∞) ) is the solution of the BSDE related to Vπ 2 . Therefore, we obtain lim k→∞ Vπ k 2 = − lim k→∞ exp(η 2 Y = Vπ 2 .…”

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confidence: 68%

A financial market with interacting investors: does an equilibrium exist?

2011

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While trading on a financial market, the agents we consider take the performance of their peers into account. By maximizing individual utility subject to investment constraints, the agents may ruin each other even unintentionally so that no equilibrium can exist. However, when the agents are willing to waive little expected utility, an approximated equilibrium can be established. The study of the associated backward stochastic differential equation (BSDE) reveals the mathematical reason for the absence of an equilibrium. Presenting an illustrative counterexample, we explain why such multidimensional quadratic BSDEs may not have solutions despite bounded terminal conditions and in contrast to the one-dimensional case.

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confidence: 68%

A financial market with interacting investors: does an equilibrium exist?

2011

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“…An approach similar to ours was used in [12] for the exponential indifference value when the mean-variance tradeoff process is bounded. There one has λ n ≡ λ and β n · M ≡ L n ≡ 0 so that the quadratic growth and locally Lipschitz assumptions on the respective BSDEs are uniform in n so that a corresponding stability result can be used.…”

Section: Model Formulationmentioning

confidence: 99%

The Stability of the Constrained Utility Maximization Problem: A BSDE Approach

Mocha

Westray²

2013

SIAM J. Finan. Math.

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This article studies the sensitivity of the power utility maximization problem with respect to the investor's relative risk aversion, the statistical probability measure, the investment constraints and the market price of risk. We extend previous descriptions of the dual domain then exploit the link between the constrained utility maximization problem and continuous semimartingale quadratic BSDEs to reduce questions on sensitivity to results on stability for such equations. This then allows us to prove appropriate convergence of the primal and dual optimizers in the semimartingale topology. * Corresponding author: n.westray@imperial.ac.uk AMS 2000 subject classifications: 60H30, 93D99, 91B28

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“…Therein stability of the optimal payoff with respect to market variations is studied while a utility defined on

R_{+}

is fixed. This type of stability problem has recently been investigated in Frei () and Bayraktar and Kravitz () for the exponential utility maximization problem.…”

Section: Introductionmentioning

confidence: 99%

Stability of the Exponential Utility Maximization Problem With Respect to Preferences

Xing

2014

Mathematical Finance

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This paper studies stability of the exponential utility maximization when there are small variations on agent's utility function. Two settings are considered. First, in a general semimartingale model where random endowments are present, a sequence of utilities defined on R converges to the exponential utility. Under a uniform condition on their marginal utilities, convergence of value functions, optimal payoffs, and optimal investment strategies are obtained, their rate of convergence is also determined. Stability of utility-based pricing is studied as an application. Second, a sequence of utilities defined on R + converges to the exponential utility after shifting and scaling. Their associated optimal strategies, after appropriate scaling, converge to the optimal strategy for the exponential hedging problem. This complements Theorem 3.2 in [Nutz, M. (2012): Risk aversion asymptotics for power utility maximization. Probab. Theory & Relat. Fields 152,, which establishes the convergence for a sequence of power utilities.

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Convergence results for the indifference value based on the stability of BSDEs

Cited by 9 publications

References 13 publications

A financial market with interacting investors: does an equilibrium exist?

A financial market with interacting investors: does an equilibrium exist?

The Stability of the Constrained Utility Maximization Problem: A BSDE Approach

Stability of the Exponential Utility Maximization Problem With Respect to Preferences

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