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Ioannis Karatzas

Publications and source records attributed to Ioannis Karatzas.

At least 37 records · Page 2Linked to original sources
arXiv2017

Semimartingales on Rays, Walsh Diffusions, and Related Problems of Control and Stopping

We introduce a class of continuous planar processes, called "semimartingales on rays", and develop for them a change-of-variable formula involving quite general classes of test functions. Special cases of such planar processes are diffusions which choose, once they reach the origin, the rays for their subsequent voyage according to a fixed probability measure in the manner of Walsh (1978). We develop existence and uniqueness results up to an explosion time for these "Walsh diffusions", study their asymptotic behavior, and develop tests for explosions in finite time. We use these results to find an optimal strategy, in a problem of control with discretionary stopping involving Walsh diffusions.

math.PR↗
arXiv2016

Volatility and Arbitrage

The capitalization-weighted total relative variation $\sum_{i=1}^d \int_0^\cdot μ_i (t) \mathrm{d} \langle \log μ_i \rangle (t)$ in an equity market consisting of a fixed number $d$ of assets with capitalization weights $μ_i (\cdot)$ is an observable and nondecreasing function of time. If this observable of the market is not just nondecreasing, but actually grows at a rate which is 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.

q-fin.PM↗
arXiv2016

Diverse market models of competing Brownian particles with splits and mergers

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.

math.PR↗
arXiv2016

Trading Strategies Generated by Lyapunov Functions

Functional portfolio generation, initiated by E.R. Fernholz almost twenty years ago, is a methodology for constructing trading strategies with controlled behavior. It is based on very weak and descriptive assumptions on the covariation structure of the underlying market model, and needs no estimation of model parameters. In this paper, the corresponding generating functions $G$ are interpreted as Lyapunov functions for the vector process $μ(\cdot)$ of market weights; that is, via the property that $G(μ(\cdot))$ is a supermartingale under an appropriate change of measure. This point of view unifies, generalizes, and simplifies several existing results, and allows the formulation of conditions under which it is possible to outperform the market portfolio over appropriate time-horizons. From a probabilistic point of view, the present paper yields results concerning the interplay of stochastic discount factors and concave transformations of semimartingales on compact domains.

q-fin.MF↗
arXiv2016

Stochastic Integral Equations for Walsh Semimartingales

We construct planar semimartingales that include the Walsh Brownian motion as a special case, and derive Harrison-Shepp-type equations and a change-of-variable formula in the spirit of Freidlin-Sheu for these so-called "Walsh semimartingales". We examine the solvability of the resulting system of stochastic integral equations. In appropriate Markovian settings we study two types of connections to martingale problems, questions of uniqueness in distribution for such processes, and a few examples.

math.PR↗
arXiv2016

Pathwise solvability of stochastic integral equations with generalized drift and non-smooth dispersion functions

We study one-dimensional stochastic integral equations with non-smooth dispersion coefficients, and with drift components that are not restricted to be absolutely continuous with respect to Lebesgue measure. In the spirit of Lamperti, Doss and Sussmann, we relate solutions of such equations to solutions of certain ordinary integral equations, indexed by a generic element of the underlying probability space. This relation allows us to solve the stochastic integral equations in a pathwise sense.

math.PR↗
arXiv2015

Diversity-Weighted Portfolios with Negative Parameter

We analyze a negative-parameter variant of the diversity-weighted portfolio studied by Fernholz, Karatzas, and Kardaras (Finance Stoch 9(1):1-27, 2005), which invests in each company a fraction of wealth inversely proportional to the company's market weight (the ratio of its capitalization to that of the entire market). We show that this strategy outperforms the market with probability one, under a non-degeneracy assumption on the volatility structure and the assumption that the market weights admit a positive lower bound. Several modifications of this portfolio, which outperform the market under milder versions of this "no-failure" condition, are put forward, one of which is rank-based. An empirical study suggests that such strategies as studied here have indeed the potential to outperform the market and to be preferable investment opportunities, even under realistic proportional transaction costs.

q-fin.MF↗
arXiv2015

Distribution of the time to explosion for one-dimensional diffusions

We study the distribution of the time to explosion for one-dimensional diffusions. We relate this question to computing the expectations of suitable nonnegative local martingales, and to the distributions of related diffusions with unit variance. Moreover, we characterize the distribution function of the time to explosion as the minimal solution to a certain Cauchy problem for an appropriate parabolic differential equation; this leads to alternative characterizations of Feller's criterion for explosions. We discuss in detail several examples for which it is possible to obtain analytic expressions for the corresponding distribution of the time to explosion, using the methodologies developed in the paper.

math.PR↗
arXiv2014

Skew-Unfolding the Skorokhod Reflection of a Continuous Semimartingale

The Skorokhod reflection of a continuous semimartingale is unfolded, in a possibly skewed manner, into another continuous semimartingale on an enlarged probability space according to the excursion-theoretic methodology of Prokaj (2009). This is done in terms of a skew version of the Tanaka equation, whose properties are studied in some detail. The result is used to construct a system of two diffusive particles with rank-based characteristics and skew-elastic collisions. Unfoldings of conventional reflections are also discussed, as are examples involving skew Brownian Motions and skew Bessel processes.

math.PR↗
arXiv2014

Impulse Control of a Diffusion with a Change Point

This paper solves a Bayes sequential impulse control problem for a diffusion, whose drift has an unobservable parameter with a change point. The partially-observed problem is reformulated into one with full observations, via a change of probability measure which removes the drift. The optimal impulse controls can be expressed in terms of the solutions and the current values of a Markov process adapted to the observation filtration. We shall illustrate the application of our results using the Longstaff-Schwartz algorithm for multiple optimal stopping times in a geometric Brownian motion stock price model with drift uncertainty.

math.OC↗
arXiv2014

Diffusions with rank-based characteristics and values in the nonnegative quadrant

We construct diffusions with values in the nonnegative orthant, normal reflection along each of the axes, and two pairs of local drift/variance characteristics assigned according to rank; one of the variances is allowed to vanish, but not both. The construction involves solving a system of coupled Skorokhod reflection equations, then ``unfolding'' the Skorokhod reflection of a suitable semimartingale in the manner of Prokaj (Statist. Probab. Lett. 79 (2009) 534-536). Questions of pathwise uniqueness and strength are also addressed, for systems of stochastic differential equations with reflection that realize these diffusions. When the variance of the laggard is at least as large as that of the leader, it is shown that the corner of the quadrant is never visited.

math.PR↗
arXiv2013

Time-reversal of reflected Brownian motions in the orthant

We determine the processes obtained from a large class of reflected Brownian motions (RBMs) in the nonnegative orthant by means of time reversal. The class of RBMs we deal with includes, but is not limited to, RBMs in the so-called Harrison-Reiman class [4] having diagonal covariance matrices. For such RBMs our main result resolves the long-standing open problem of determining the time reversal of RBMs beyond the skew-symmetric case treated by R.J. Williams in [16]. In general, the time-reversed process itself is no longer a RBM, but its distribution is absolutely continuous with respect to a certain auxiliary RBM. In the course of the proof we introduce a novel discrete approximation scheme for the class of RBMs described above, and use it to determine the semigroups dual to the semigroups of such RBMs.

math.PR↗
arXiv2013

A second-order stock market model

A first-order model for a stock market assigns to each stock a return parameter and a variance parameter that depend only on the rank of the stock. A second-order model assigns these parameters based on both the rank and the name of the stock. First- and second-order models exhibit stability properties that make them appropriate as a backdrop for the analysis of the idiosyncratic behavior of individual stocks. Methods for the estimation of the parameters of second-order models are developed in this paper.

q-fin.ST↗
arXiv2012

Systems of Brownian particles with asymmetric collisions

We study systems of Brownian particles on the real line, which interact by splitting the local times of collisions among themselves in an asymmetric manner. We prove the strong existence and uniqueness of such processes and identify them with the collections of ordered processes in a Brownian particle system, in which the drift coefficients, the diffusion coefficients, and the collision local times for the individual particles are assigned according to their ranks. These Brownian systems can be viewed as generalizations of those arising in first-order models for equity markets in the context of stochastic portfolio theory, and are able to correct for several shortcomings of such models while being equally amenable to computations. We also show that, in addition to being of interest in their own right, such systems of Brownian particles arise as universal scaling limits of systems of jump processes on the integer lattice with local interactions. A key step in the proof is the analysis of a generalization of Skorokhod maps which include `local times' at the intersection of faces of the nonnegative orthant. The result extends the convergence of TASEP to its continuous analogue. Finally, we identify those among the Brownian particle systems which have a probabilistic structure of determinantal type.

math.PR↗
arXiv2012

Two Brownian Particles with Rank-Based Characteristics and Skew-Elastic Collisions

We construct a two-dimensional diffusion process with rank-dependent local drift and dispersion coefficients, and with a full range of patterns of behavior upon collision that range from totally frictionless interaction, to elastic collision, to perfect reflection of one particle on the other. These interactions are governed by the left- and right-local times at the origin for the distance between the two particles. We realize this diffusion in terms of appropriate, apparently novel systems of stochastic differential equations involving local times, which we show are well posed. Questions of pathwise uniqueness and strength are also discussed for these systems. The analysis depends crucially on properties of a skew Brownian motion with two-valued drift of the bang-bang type, which we also study in some detail. These properties allow us to compute the transition probabilities of the original planar diffusion, and to study its behavior under time reversal.

math.PR↗
arXiv2012

Planar Diffusions with Rank-Based Characteristics: Transition Probabilities, Time Reversal, Maximality and Perturbed Tanaka equations

We construct a planar diffusion process whose infinitesimal generator depends only on the order of the components of the process. Speaking informally and a bit imprecisely for the moment, imagine you run two Brownian-like particles on the real line. At any given time, you assign positive drift g and diffusion σ to the laggard; and you assign negative drift -h and diffusion ρ to the leader. We compute the transition probabilities of this process, discuss its realization in terms of appropriate systems of stochastic differential equations, study its dynamics under a time reversal, and note that these involve singularly continuous components governed by local time. Crucial in our analysis are properties of Brownian and semimartingale local time; properties of the generalized perturbed Tanaka equation which we study here in detail; and those of a one-dimensional diffusion with bang-bang drift. We also show that our planar diffusion can be represented in terms of a process with bang-bang drift, its local time at the origin, and an independent standard Brownian motion, in a form which can be construed as a two-dimensional analogue of the stochastic equation satisfied by the so-called skew Brownian motion.

math.PR↗
arXiv2012

Optimal arbitrage under model uncertainty

In an equity market model with "Knightian" uncertainty regarding the relative risk and covariance structure of its assets, we characterize in several ways the highest return relative to the market that can be achieved using nonanticipative investment rules over a given time horizon, and under any admissible configuration of model parameters that might materialize. One characterization is in terms of the smallest positive supersolution to a fully nonlinear parabolic partial differential equation of the Hamilton--Jacobi--Bellman type. Under appropriate conditions, this smallest supersolution is the value function of an associated stochastic control problem, namely, the maximal probability with which an auxiliary multidimensional diffusion process, controlled in a manner which affects both its drift and covariance structures, stays in the interior of the positive orthant through the end of the time-horizon. This value function is also characterized in terms of a stochastic game, and can be used to generate an investment rule that realizes such best possible outperformance of the market.

math.PR↗