Showing papers by "Ioannis Karatzas published in 2018"
TL;DR: In this article, the authors show that strong arbitrage is not possible in general, and also show that, under appropriate additional conditions, outperformance over any time horizon indeed becomes possible, and exhibit investment strategies that effect it.
Abstract: The capitalization-weighted cumulative variation d i=1 0 µi(t)d(log µi)(t) in an equity market consisting of a fixed number d of assets with capitalization weights µi(·) ; is an observable and a nondecreasing function of time. If this observable of the market is not just nondecreasing but actually grows at a rate bounded away from zero, then strong arbitrage can be constructed relative to the market over sufficiently long time horizons. It has been an open issue for more than ten years, whether such strong outperformance of the market is possible also over arbitrary time horizons under the stated condition. We show that this is not possible in general, thus settling this long-open question. We also show that, under appropriate additional conditions, outperformance over any time horizon indeed becomes possible, and exhibit investment strategies that effect it.
25 citations
TL;DR: In this article, the semimartingales planaires de Walsh were constructed, including le mouvement brownien de Walsh comme cas particulier, and a formule de changement de variable dans l'esprit de Freidlin-Sheu.
Abstract: Nous construisons des semimartingales planaires qui incluent le mouvement brownien de Walsh comme cas particulier, et nous etablissons pour ces « semimartingales de Walsh » des equations de type Harrison–Shepp, et une formule de changement de variable dans l’esprit de Freidlin–Sheu. Dans des cadres markoviens appropries, nous etudions deux types de relations aux problemes de martingale, des questions d’unicite en loi pour de tels processus, et quelques exemples.
18 citations
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TL;DR: In this paper, the authors revisited the variational characterization of diffusion as entropic gradient flux and provided for it a probabilistic interpretation based on stochastic calculus.
Abstract: We revisit the variational characterization of diffusion as entropic gradient flux and provide for it a probabilistic interpretation based on stochastic calculus. It was shown by Jordan, Kinderlehrer, and Otto that, for diffusions of Langevin-Smoluchowski type, the Fokker-Planck probability density flow minimizes the rate of relative entropy dissipation, as measured by the distance traveled in the ambient space of probability measures with finite second moments, in terms of the quadratic Wasserstein metric. We obtain novel, stochastic-process versions of these features, valid along almost every trajectory of the diffusive motion in both the forward and, most transparently, the backward, directions of time, using a very direct perturbation analysis. By averaging our trajectorial results with respect to the underlying measure on path space, we establish the minimum rate of entropy dissipation along the Fokker-Planck flow and measure exactly the deviation from this minimum that corresponds to any given perturbation. As a bonus of our perturbation analysis we derive the so-called HWI inequality relating relative entropy (H), Wasserstein distance (W) and relative Fisher information (I).
14 citations
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TL;DR: In this paper, the authors generalized Fernholz and Karatzas' functional portfolio generation to a pathwise, probability-free setting, where the current market weights are replaced by path-dependent functionals, which involve the market weights, as well as additional bounded-variation functions of past and present market weights.
Abstract: Almost twenty years ago, E.R. Fernholz introduced portfolio generating functions which can be used to construct a variety of portfolios, solely in the terms of the individual companies' market weights. I. Karatzas and J. Ruf recently developed another methodology for the functional construction of portfolios, which leads to very simple conditions for strong relative arbitrage with respect to the market. In this paper, both of these notions of functional portfolio generation are generalized in a pathwise, probability-free setting; portfolio generating functions are substituted by path-dependent functionals, which involve the current market weights, as well as additional bounded-variation functions of past and present market weights. This generalization leads to a wider class of functionally-generated portfolios than was heretofore possible, and yields improved conditions for outperforming the market portfolio over suitable time-horizons.
4 citations
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26 Sep 2018
Abstract: Almost twenty years ago, E.R. Fernholz introduced portfolio generating functions which can be used to construct a variety of portfolios, solely in the terms of the individual companies' market weights. I. Karatzas and J. Ruf recently developed another methodology for the functional construction of portfolios, which leads to very simple conditions for strong relative arbitrage with respect to the market. In this paper, both of these notions of functional portfolio generation are generalized in a pathwise, probability-free setting; portfolio generating functions are substituted by path-dependent functionals, which involve the current market weights, as well as additional bounded-variation functions of past and present market weights. This generalization leads to a wider class of functionally-generated portfolios than was heretofore possible, and yields improved conditions for outperforming the market portfolio over suitable time-horizons.
4 citations
Posted Content•
21 Nov 2018
TL;DR: In this article, the authors revisited the variational characterization of diffusion as entropic gradient flux and provided for it a probabilistic interpretation based on stochastic calculus.
Abstract: We revisit the variational characterization of diffusion as entropic gradient flux and provide for it a probabilistic interpretation based on stochastic calculus. It was shown by Jordan, Kinderlehrer, and Otto that, for diffusions of Langevin-Smoluchowski type, the Fokker-Planck probability density flow minimizes the rate of relative entropy dissipation, as measured by the distance traveled in the ambient space of probability measures with finite second moments, in terms of the quadratic Wasserstein metric. We obtain novel, stochastic-process versions of these features, valid along almost every trajectory of the diffusive motion in both the forward and, most transparently, the backward, directions of time, using a very direct perturbation analysis. By averaging our trajectorial results with respect to the underlying measure on path space, we establish the minimum rate of entropy dissipation along the Fokker-Planck flow and measure exactly the deviation from this minimum that corresponds to any given perturbation. As a bonus of our perturbation analysis we derive the so-called HWI inequality relating relative entropy (H), Wasserstein distance (W) and relative Fisher information (I).
3 citations
Posted Content•
TL;DR: In this article, a probabilistic interpretation of diffusion as entropic gradient flux is provided for it based on stochastic calculus, and the Cordero-Erausquin version of the so-called HWI inequality relating relative entropy, Fisher information and Wasserstein distance.
Abstract: We revisit the [JKO98] variational characterization of diffusion as entropic gradient flux, and provide for it a probabilistic interpretation based on stochastic calculus. It was shown by Jordan, Kinderlehrer, and Otto in [JKO98] that, for diffusions of Langevin-Smoluchowski type, the Fokker-Planck probability density flow minimizes the rate of relative entropy dissipation, as measured by the distance traveled in terms of the quadratic Wasserstein metric. We obtain novel, stochastic-process versions of these features, valid along almost every trajectory of the diffusive motion in both the forward and, most transparently, the backward, directions of time, using a very direct perturbation analysis; the original results follow then simply by taking expectations. As a bonus, we derive the Cordero-Erausquin version of the so-called HWI inequality relating relative entropy, Fisher information and Wasserstein distance.
2 citations
10 Jul 2018
TL;DR: In this article, a stochastic differential equation with rank-based characteristics on the plane is studied and its flow solutions and coalescence is characterized, and the authors find its flow solution and characterize coalescence.
Abstract: We study a stochastic differential equation with rank-based characteristics on the plane. We find its flow solutions and characterize coalescence.