arXiv subjects

Mykhaylo Shkolnikov

Publications and source records attributed to Mykhaylo Shkolnikov.

At least 37 records · Page 2Linked to original sources

arXiv2018

Construction of Forward Performance Processes in Stochastic Factor Models and an Extension of Widder's Theorem

We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. Given multiple traded assets, the prices of which depend on multiple observable stochastic factors, we construct a large class of forward performance processes with power-utility initial data, as well as the corresponding optimal portfolios. This is done by solving the associated non-linear parabolic partial differential equations (PDEs) posed in the "wrong" time direction, for stock-factor correlation matrices with eigenvalue equality (EVE) structure, which we introduce here. Along the way we establish on domains an explicit form of the generalized Widder's theorem of Nadtochiy and Tehranchi [NT15, Theorem 3.12] and rely hereby on the Laplace inversion in time of the solutions to suitable linear parabolic PDEs posed in the "right" time direction.

q-fin.MF↗

arXiv2018

Dynamics of observables in rank-based models and performance of functionally generated portfolios

In the seminal work [9], several macroscopic market observables have been introduced, in an attempt to find characteristics capturing the diversity of a financial market. Despite the crucial importance of such observables for investment decisions, a concise mathematical description of their dynamics has been missing. We fill this gap in the setting of rank-based models and expect our ideas to extend to other models of large financial markets as well. The results are then used to study the performance of multiplicatively and additively functionally generated portfolios, in particular, over short-term and medium-term horizons.

math.PR↗

arXiv2017

Edge of spiked beta ensembles, stochastic Airy semigroups and reflected Brownian motions

We access the edge of Gaussian beta ensembles with one spike by analyzing high powers of the associated tridiagonal matrix models. In the classical cases beta=1, 2, 4, this corresponds to studying the fluctuations of the largest eigenvalues of additive rank one perturbations of the GOE/GUE/GSE random matrices. In the infinite-dimensional limit, we arrive at a one-parameter family of random Feynman-Kac type semigroups, which features the stochastic Airy semigroup of Gorin and Shkolnikov [13] as an extreme case. Our analysis also provides Feynman-Kac formulas for the spiked stochastic Airy operators, introduced by Bloemendal and Virag [6]. The Feynman-Kac formulas involve functionals of a reflected Brownian motion and its local times, thus, allowing to study the limiting operators by tools of stochastic analysis. We derive a first result in this direction by obtaining a new distributional identity for a reflected Brownian bridge conditioned on its local time at zero.

math.PR↗

arXiv2017

Particle systems with singular interaction through hitting times: application in systemic risk modeling

We propose an interacting particle system to model the evolution of a system of banks with mutual exposures. In this model, a bank defaults when its normalized asset value hits a lower threshold, and its default causes instantaneous losses to other banks, possibly triggering a cascade of defaults. The strength of this interaction is determined by the level of the so-called non-core exposure. We show that, when the size of the system becomes large, the cumulative loss process of a bank resulting from the defaults of other banks exhibits discontinuities. These discontinuities are naturally interpreted as systemic events, and we characterize them explicitly in terms of the level of non-core exposure and the fraction of banks that are "about to default". The main mathematical challenges of our work stem from the very singular nature of the interaction between the particles, which is inherited by the limiting system. A similar particle system is analyzed in [DIRT15a] and [DIRT15b], and we build on and extend their results. In particular, we characterize the large-population limit of the system and analyze the jump times, the regularity between jumps, and the local uniqueness of the limiting process.

math.PR↗

arXiv2017

Large deviations for diffusions interacting through their ranks

We prove a Large Deviations Principle (LDP) for systems of diffusions (particles) interacting through their ranks, when the number of particles tends to infinity. We show that the limiting particle density is given by the unique solution of the approriate McKean-Vlasov equation and that the corresponding cumulative distribution function evolves according to the porous medium equation with convection. The large deviations rate function is provided in explicit form. This is the first instance of a LDP for interacting diffusions, where the interaction occurs both through the drift and the diffusion coefficients and where the rate function can be given explicitly. In the course of the proof, we obtain new regularity results for a certain tilted version of the porous medium equation.

math.PR↗

arXiv2016

Intertwinings of beta-Dyson Brownian motions of different dimensions

We show that for all positive beta the semigroups of beta-Dyson Brownian motions of different dimensions are intertwined. The proof relates beta-Dyson Brownian motions directly to Jack symmetric polynomials and omits an approximation of the former by discrete space Markov chains, thereby disposing of the technical assumption beta>1 in [GS]. The corresponding results for beta-Dyson Ornstein-Uhlenbeck processes are also presented.

math.PR↗

arXiv2016

SPDE limit of the global fluctuations in rank-based models

We consider systems of diffusion processes ("particles") interacting through their ranks (also referred to as "rank-based models" in the mathematical finance literature). We show that, as the number of particles becomes large, the process of fluctuations of the empirical cumulative distribution functions converges to the solution of a linear parabolic SPDE with additive noise. The coefficients in the limiting SPDE are determined by the hydrodynamic limit of the particle system which, in turn, can be described by the porous medium PDE. The result opens the door to a thorough investigation of large equity markets and investment therein. In the course of the proof we also derive quantitative propagation of chaos estimates for the particle system.

math.PR↗

arXiv2016

Stochastic Airy semigroup through tridiagonal matrices

We determine the operator limit for large powers of random tridiagonal matrices as the size of the matrix grows. The result provides a novel expression in terms of functionals of Brownian motions for the Laplace transform of the Airy$_β$ process, which describes the largest eigenvalues in the $β$ ensembles of random matrix theory. Another consequence is a Feynman-Kac formula for the stochastic Airy operator of Ramírez, Rider, and Virág. As a side result, we find that the difference between the area underneath a standard Brownian excursion and one half of the integral of its squared local times is a Gaussian random variable.

math.PR↗

arXiv2015

Asymptotic analysis of forward performance processes in incomplete markets and their ill-posed HJB equations

We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. The dynamics of the prices of the traded assets depend on a pair of stochastic factors, namely, a slow factor (e.g. a macroeconomic indicator) and a fast factor (e.g. stochastic volatility). We analyze the associated forward performance SPDE and provide explicit formulae for the leading order and first order correction terms for the forward investment process and the optimal feedback portfolios. They both depend on the investor's initial preferences and the dynamically changing investment opportunities. The leading order terms resemble their time-monotone counterparts, but with the appropriate stochastic time changes resulting from averaging phenomena. The first-order terms compile the reaction of the investor to both the changes in the market input and his recent performance. Our analysis is based on an expansion of the underlying ill-posed HJB equation, and it is justified by means of an appropriate remainder estimate.

q-fin.MF↗

arXiv2015

Multidimensional sticky Brownian motions as limits of exclusion processes

We study exclusion processes on the integer lattice in which particles change their velocities due to stickiness. Specifically, whenever two or more particles occupy adjacent sites, they stick together for an extended period of time, and the entire particle system is slowed down until the ``collision'' is resolved. We show that under diffusive scaling of space and time such processes converge to what one might refer to as a sticky reflected Brownian motion in the wedge. The latter behaves as a Brownian motion with constant drift vector and diffusion matrix in the interior of the wedge, and reflects at the boundary of the wedge after spending an instant of time there. In particular, this leads to a natural multidimensional generalization of sticky Brownian motion on the half-line, which is of interest in both queuing theory and stochastic portfolio theory. For instance, this can model a market, which experiences a slowdown due to a major event (such as a court trial between some of the largest firms in the market) deciding about the new market leader.

math.PR↗

arXiv2015

A construction of infinite Brownian particle systems

The paper identifies families of quasi-stationary initial conditions for infinite Brownian particle systems within a large class and provides a construction of the particle systems themselves started from such initial conditions. Examples of particle systems falling into our framework include Brownian versions of TASEP-like processes such as the diffusive scaling limit of the q-TASEP process. In this context the spacings between consecutive particles form infinite-dimensional versions of the softly reflected Brownian motions recently introduced in the finite-dimensional setting by O'Connell and Ortmann and are of independent interest. The proof of the main result is based on intertwining relations satisfied by the particle systems involved which can be regarded as infinite-dimensional analogues of the suitably generalized Burke's Theorem.

math.PR↗

arXiv2014

Interacting particle systems at the edge of multilevel Dyson Brownian motions

We study the joint asymptotic behavior of spacings between particles at the edge of multilevel Dyson Brownian motions, when the number of levels tends to infinity. Despite the global interactions between particles in multilevel Dyson Brownian motions, we observe a decoupling phenomenon in the limit: the global interactions become negligible and only the local interactions remain. The resulting limiting objects are interacting particle systems which can be described as Brownian versions of certain totally asymmetric exclusion processes. This is the first appearance of a particle system with local interactions in the context of general $β$ random matrix models.

math.PR↗

arXiv2014

Multilevel Dyson Brownian motions via Jack polynomials

We introduce multilevel versions of Dyson Brownian motions of arbitrary parameter $β>0$, generalizing the interlacing reflected Brownian motions of Warren for $β=2$. Such processes unify $β$ corners processes and Dyson Brownian motions in a single object. Our approach is based on the approximation by certain multilevel discrete Markov chains of independent interest, which are defined by means of Jack symmetric polynomials. In particular, this approach allows to show that the levels in a multilevel Dyson Brownian motion are intertwined (at least for $β\ge 1$) and to give the corresponding link explicitly.

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

Large deviations for interacting Bessel-like processes and applications to systemic risk

We establish a process level large deviation principle for systems of interacting Bessel-like diffusion processes. By establishing weak uniqueness for the limiting non-local SDE of McKean-Vlasov type, we conclude that the latter describes the process level hydrodynamic limit of such systems and obtain a propagation of chaos result. This is the first instance of results of this type in the context of interacting diffusion processes under explicit assumptions on the coefficients, where the diffusion coefficients are allowed to be both non-Lipschitz and degenerate. In the second part of the paper, we explain how systems of this type naturally arise in the study of stability of the interbank lending system and describe some financial implications of our results.

math.PR↗

arXiv2012

Fluctuations of Martingales and Winning Probabilities of Game Contestants

Within a contest there is some probability M_i(t) that contestant i will be the winner, given information available at time t, and M_i(t) must be a martingale in t. Assume continuous paths, to capture the idea that relevant information is acquired slowly. Provided each contestant's initial winning probability is at most b, one can easily calculate, without needing further model specification, the expectations of the random variables N_b = number of contestants whose winning probability ever exceeds b, and D_{ab} = total number of downcrossings of the martingales over an interval [a,b]. The distributions of N_b and D_{ab} do depend on further model details, and we study how concentrated or spread out the distributions can be. The extremal models for N_b correspond to two contrasting intuitively natural methods for determining a winner: progressively shorten a list of remaining candidates, or sequentially examine candidates to be declared winner or eliminated. We give less precise bounds on the variability of D_{ab}. We formalize the setting of infinitely many contestants each with infinitesimally small chance of winning, in which the explicit results are more elegant. A canonical process in this setting is the Wright-Fisher diffusion associated with an infinite population of initially distinct alleles; we show how this process fits our setting and raise the problem of finding the distributions of N_b and D_{ab} for this process.

math.PR↗

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

Small time central limit theorems for semimartingales with applications

We give conditions under which the normalized marginal distribution of a semimartingale converges to a Gaussian limit law as time tends to zero. In particular, our result is applicable to solutions of stochastic differential equations with locally bounded and continuous coefficients. The limit theorems are subsequently extended to functional central limit theorems on the process level. We present two applications of the results in the field of mathematical finance: to the pricing of at-the-money digital options with short maturities and short time implied volatility skews.

math.PR↗