On collisions of Brownian particles

By Tomoyuki Ichiba, Ioannis Karatzas. 2008

67 citations recorded by the source corpus.

We examine the behavior of n Brownian particles diffusing on the real line, with bounded, measurable drift and bounded, piecewise continuous diffusion coefficients that depend on the currentconfigura- tion of particles. Sufficient conditions are established forthe absence of triple collisions, as well as for the presence of (infinitely-many) triple collisions among the particles. As an application to the Atlas model of equity markets, we study a special construction of such systems of diffusing particles using Brownian motions with reflection on polyhedral domains.

Subfields: Finance, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/b95034f703e5c3b9c6fbb1fc1aa2f867bed1bade

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Finance

1. Mykhaylo Shkolnikov
2. Soumik Pal
3. Tomoyuki Ichiba
4. Ioannis Karatzas
5. Constantinos Kardaras
6. Gordan Zitkovic
7. Marcel Nutz
8. Philip Protter

In Mathematical Physics

1. Mykhaylo Shkolnikov
2. Tomoyuki Ichiba
3. Ioannis Karatzas
4. Soumik Pal

In Molecular Biology

1. Mykhaylo Shkolnikov
2. Tomoyuki Ichiba
3. Ioannis Karatzas

Citations in the corpus

• Systems of Brownian particles with asymmetric collisions (2012, background): “2 of [21], or as in the proof of Lemmata 6 and 7 in [22], we need only consider the case b1 = b2 = · · · = bn = 0 in (1.”
• Convergence rates for rank-based models with applications to portfolio theory (2011, background): “This is a departure from the usual Black-Scholes models that cesses represented by continuous semimartingales (see, e.g., Duffie (1992) or Karatzas and Shreve (1998)).”
• HYBRID ATLAS MODELS (2009, unknown): “0) we obtain R(·)≥δ0 := min p∈Σn P3 ℓ=1 λ¯ ℓ,p max1≤ℓ≤3 ¯λ ℓ,p (A.12) >1 in Rn \Z, and so d(·) ≥δ0 >1 when Y(τ·)∈Rn \Z. By a comparison argument similar to that in the proof of Lemma 2.1 of [18], we may show that G(t)≥r(δ0)(t) for 0 ≤ t < ∞ a.s. Since √ 3η(t) ≥ g(Y (t)) = G(hΘi(t)) implie”
• Strong solutions of stochastic equations with rank-based coefficients (2011, methodology): “2 in [17] for the reflected Brownian motion Y (see the proof of Proposition 3 in [11] for more details).”