On a class of diverse market models

Abstract

A market model in Stochastic Portfolio Theory is a finite system of strictly positive stochastic processes. Each process represents the capitalization of a certain stock. If at any time no stock dominates almost the entire market, which means that its share of total market capitalization is not very close to one, then the market is called diverse. There are several ways to outperform diverse markets and get an arbitrage opportunity, and this makes these markets interesting. A feature of real-world markets is that stocks with smaller capitalizations have larger drift coefficients. Some models, like the Volatility-Stabilized Model, try to capture this property, but they are not diverse. In an attempt to combine this feature with diversity, we construct a new class of market models. We find simple, easy-to-test sufficient conditions for them to be diverse and other sufficient conditions for them not to be diverse.Keywords Stochastic Portfolio Theory · diverse markets · arbitrage opportunity · Feller's test JEL Classification Number G10 be the total capitalization of the market at time t and the market weights of stocks, respectively. Fix a threshold δ ∈ (0, 1). The market is called δ-diverse if for every t ≥ 0 and i = 1, . . . , n we have:This definition was introduced in [6]. Intuitively it means that, at any given moment, no stock dominates almost the entire market.

Multiple collisions in systems of competing Brownian particles

“…By [62,Lemma 2], this is a strictly increasing function on [0, τ ] with ∆(0) = 0. Denote τ 0 := ∆(τ ).…”

Diverse market models of competing Brownian particles with splits and mergers

“…It follows from the estimates in (28) and (29) that the probability of the event {Θ 2L ≤ T } which we would like to estimate, as in (26) and (27), is…”

“…This was shown in [8], Chapter 3; further examples of portfolios outperforming the market are given in [11,13], [12], Section 11. Some such models were constructed in [13,24,27,28] and [12], Chapter 9; see also the related articles [1,22].…”

“…(k) (t) = g 1 − min 2≤k≤N g k (1 − µ (1) (t)) ≤ 0.Let us make a random time change, using Lemma 2 from[28] (for σ = 1 in the notation of this lemma). The time change is as follows:t = T (s) = inf{t ≥ 0|∆(t) ≥ s}, s = ∆(t) := t 0 a(v) dv, t ∈ [0, τ ].Denoting by V (·) = {V (s), s ≥ 0} yet another standard Brownian motion, and Z(·) = {Z(s), s ≥ 0}, Z(s) = log µ (1) (T (s)), Λ(·) = {Λ(s), s ≥ 0}, Λ(s) = 1 2 Λ (1,2) (T (s)), γ(s) = β(T (s)) a(T (s)) SPLITS AND MERGERS 31 we get the following equation: dZ(s) = γ(s) ds + dV (s) + dΛ(s) for s ≤ ∆(τ ).…”

Capital distribution and portfolio performance in the mean-field Atlas model

“…Several models have been introduced to capture the growth rate and volatility size effects, see for instance the so-called Volatility-Stabilized Model by Fernholz and Karatzas [10,Section 12], which was later on discussed by Pal [19], Shkolnikov [29] and Sarantsev [27]. As we shall see below, both rebalancing and the volatility size effect play a key role in the analysis of portfolio performance.…”