Fundamental theorems of asset pricing for piecewise semimartingales of stochastic dimension

Strong, Winslow

doi:10.1007/s00780-014-0230-2

Cited by 9 publications

(30 citation statements)

References 34 publications

(48 reference statements)

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“…This is in contrast with the models from [12], where splits/mergers are not allowed. Indeed, it was observed in [33] that in the presence of splits/mergers, diversity might not lead to arbitrage; in [34], Strong and Fouque established this for their models with a fixed number of companies.…”

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confidence: 94%

“…This comes at a price, which is both "technical" and substantive: the number of companies in the model is now fluctuating randomly, in ways that need to be understood before any reasonable analysis can go through. The foundational theory for generic market models with a randomly varying number of stocks was developed by Strong in the important and very useful article [33].…”

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confidence: 99%

“…The authors would like to thank E. Robert Fernholz, Jean-Pierre Fouque, Soumik Pal, Vasileios Papathanakos, Walter Schachermayer, Johannes Ruf and Philip Whitman for very helpful discussions, and Johannes Ruf for his detailed comments. They are obliged to Winslow Strong for an updated version of the manuscript [33]. They are deeply indebted to the referee for reading of the manuscript very carefully, and for making many incisive comments and suggestions.…”

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confidence: 99%

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Diverse market models of competing Brownian particles with splits and mergers

Karatzas¹,

Sarantsev²

2016

Ann. Appl. Probab.

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We study models of regulatory breakup, in the spirit of Strong and Fouque [Ann. Finance 7 (2011) 349-374] but with a fluctuating number of companies. An important class of market models is based on systems of competing Brownian particles: each company has a capitalization whose logarithm behaves as a Brownian motion with drift and diffusion coefficients depending on its current rank. We study such models with a fluctuating number of companies: If at some moment the share of the total market capitalization of a company reaches a fixed level, then the company is split into two parts of random size. Companies are also allowed to merge, when an exponential clock rings. We find conditions under which this system is nonexplosive (i.e., the number of companies remains finite at all times) and diverse, yet does not admit arbitrage opportunities.

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confidence: 94%

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confidence: 99%

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confidence: 99%

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Diverse market models of competing Brownian particles with splits and mergers

Karatzas¹,

Sarantsev²

2016

Ann. Appl. Probab.

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“…In contrast to Kabanov and Kramkov (, ), Björk and Näslund () assumed that a large market consists of one probability space, but the number of traded assets is countable, and among other contributions developed the arbitrage pricing theory results in such settings. Note that the models with countably many assets embrace the ones with the stochastic dimension of the stock price process (considered, e.g., in Strong ). De Donno et al.…”

Section: Introductionmentioning

confidence: 99%

“…Note that the models with countably many assets embrace the ones with the stochastic dimension of the stock price process (considered e.g. in [26]). [9] extended the formulation in [1] to a model driven by a sequence of semimartingales and established the standard conclusions of the theory for the utility maximization from terminal wealth problem as well as obtained the dual characterization of the superreplicable claims.…”

Section: Introductionmentioning

confidence: 99%

Utility Maximization in a Large Market

Mostovyi¹

2016

Mathematical Finance

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We study the problem of expected utility maximization in a large market, i.e., a market with countably many traded assets. Assuming that agents have von NeumannMorgenstern preferences with stochastic utility function and that consumption occurs according to a stochastic clock, we obtain the "usual" conclusions of the utility maximization theory. We also give a characterization of the value function in a large market in terms of a sequence of value functions in finite-dimensional models.

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Trading strategies generated pathwise by functions of market weights

Karatzas

Kim

2019

Finance Stoch

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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 developed recently 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, possibly less smooth than twicedifferentiable, involve the current market weights, as well as additional bounded-variation functions related to the market weights. This generalization leads to a wider class of functionally-generated portfolios than was heretofore possible, to novel methods for dealing with the "size" and "momentum" effects, and to improved conditions for outperforming the market portfolio over suitable timehorizons.Definition 2.1. A continuous function X = (X 1 , X 2 , · · · , X d ) is said to have a pathwise quadratic covariation along a given nested sequence of partitions (T n ) n∈N of [0, T ], if the limit of the sequenceexists for any t ∈ [0, T ] as n → ∞ and the resulting mapping, denoted by t → X i , X k (t), is realvalued and continuous for every 1 ≤ i, k ≤ d. We call X i , X k the pathwise quadratic covariation of X i and X k , and define the pathwise quadratic variation of X i by X i := X i , X i as usual.We stress that the existence of pathwise covariations and quadratic variations for the components of X depends heavily on the choice of the nested, or "refining" sequence (T n ) n∈N of partitions. Example 5.3.2 in Cont (2016), and the arguments following, illustrate this fact. We note also that the existence of pathwise covariations and quadratic variations is required for Itô's formula to hold in a pathwise sense.Next, we state the original one-dimensional pathwise Itô formula, introduced by Föllmer (1981).Theorem 2.2 (Pathwise Itô formula for paths with quadratic variation, Föllmer (1981)). Fix a continuous function X which admits the quadratic variation along the given nested sequence of partitions T = (T n ) n≥1 of [0, T ]. Then for every C 2 (R, R) function f , the pathwise change of variable formulaHere, the Föllmer-Itô integral is defined as the pointwise limit t 0 f X(s) dX(s) := lim n→∞ [t j ,t j+1 ]∈Tn t j ≤tf (X(t j ))(X(t j+1 ) − X(t j )), (2.3) and the last integral of the right-hand side of (2.2) is Lebesgue-Stieltjes integral.

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Fundamental theorems of asset pricing for piecewise semimartingales of stochastic dimension

Cited by 9 publications

References 34 publications

Diverse market models of competing Brownian particles with splits and mergers

Diverse market models of competing Brownian particles with splits and mergers

Utility Maximization in a Large Market

Trading strategies generated pathwise by functions of market weights

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