arXiv research

The paper studies the question of whether the classical mirror and synchronous couplings of two Brownian motions minimise and maximise, respectively, the coupling time of the corresponding geometric Brownian motions. We establish a characterisation of the optimality of the two couplings over any finite time horizon and…

Develops non-Markovian couplings for sub-Riemannian Brownian motions.

Unified framework for Brownian motion distances on specific geometric manifolds.

Under a complete Ricci flow, we construct a coupling of two Brownian motion such that their $\mathcal{L}_0$-distance is a supermartingale. This recovers a result of Lott [J. Lott, Optimal transport and Perelman's reduced volume, Calc. Var. Partial Differential Equations 36 (2009), no. 1, 49--84.] on the monotonicity of…

We show that on a manifold whose Riemannian metric evolves under backwards Ricci flow two Brownian motions can be coupled in such a way that the expectation of their normalized L-distance is non-increasing. As an immediate corollary we obtain a new proof of a recent result of Topping (J. reine angew. Math. 636 (2009), …

We provide a probabilistic approach to studying minimal surfaces in three-dimensional Euclidean space. Following a discussion of the basic relationship between Brownian motion on a surface and minimality of the surface, we introduce a way of coupling Brownian motions on two minimal surfaces. This coupling is then used …

Maps on Sasakian manifolds limit to sub-Riemannian distance bounds.

The goal of this paper is twofold: we study metric measure spaces $(X,d,m)$ with variable lower bounds for the Ricci curvature and we study pathwise coupling of Brownian motions. Given any lower semicontinuous function $k:X\to \mathbb R$ we introduce the curvature-dimension condition $CD(k,\infty)$ which canonically ex…

We prove that minimal graphs (other than planes) are parabolic in the sense that any bounded harmonic function is determined by its boundary values. The proof relies on using the coupling introduced in the author's earlier paper "A martingale approach to minimal surfaces" to show that Brownian motion on such a minimal …

We extend martingale transport results to weak martingale transport.

Study of most probable paths for anisotropic Brownian motions on manifolds.

D2SRM solves complex PDEs using deep learning.

Constructs supermartingale couplings with full marginals constraints.

Stochastic representation for determinants derived from Brownian loop soups.

FDBM models use fractional Brownian motion to model complex stochastic processes.

The paper proves conditions for infinite lifetime of Brownian motion and regularity of heat flow.

Given a metric measure space $(X,\mathsf{d},\mathfrak{m})$ and a lower semicontinuous, lower bounded function $k\colon X\to\mathbb{R}$, we prove the equivalence of the synthetic approaches to Ricci curvature at $x\in X$ being bounded from below by $k(x)$ in terms of $\bullet$ the Bakry-Émery estimate $ΔΓ(f)/2 - Γ(f,Δf)…

In the setting of exponential investors and uncertainty governed by Brownian motions we first prove the existence of an incomplete equilibrium for a general class of models. We then introduce a tractable class of exponential-quadratic models and prove that the corresponding incomplete equilibrium is characterized by a …

In this paper we derive the optimal execution trajectory for a trader who wishes to buy or sell a large position of shares which evolve as a geometric Brownian process in contrast to the arithmetic model which prevails in the existing literature, and with a general temporary impact $h$. We provide a couple of examples …

This work extends stochastic localization to joint probability measures for data analysis.

Study of coupled Hawkes processes with rough-volatility limits.

The paper studies the geometry of multi-particle Schrödinger semigroups using $\mathrm{RCD}^*(K,N)$ spaces.

We prove the global existence of an incomplete, continuous-time finite-agent Radner equilibrium in which exponential agents optimize their expected utility over both running consumption and terminal wealth. The market consists of a traded annuity, and, along with unspanned income, the market is incomplete. Set in a Bro…

We use probabilistic methods to characterise time dependent optimal stopping boundaries in a problem of multiple optimal stopping on a finite time horizon. Motivated by financial applications we consider a payoff of immediate stopping of "put" type and the underlying dynamics follows a geometric Brownian motion. The op…

This paper provides guarantees for DFM models using KL divergence.

Model tracks structural changes in Brownian particle configurations on a sphere.

Corrects technical error in change of measure for HTB models.

We develop a second-order model for limit order books in a single scaling regime.

Continuous time random walks (CTRWs) are used in physics to model anomalous diffusion, by incorporating a random waiting time between particle jumps. In finance, the particle jumps are log-returns and the waiting times measure delay between transactions. These two random variables (log-return and waiting time) are typi…

The paper prices weather contracts using a complex temperature model.

We study the problem of stopping a Brownian motion at a given distribution $ν$ while optimizing a reward function that depends on the (possibly randomized) stopping time and the Brownian motion. Our first result establishes that the set $\mathcal{T}(ν)$ of stopping times embedding $ν$ is weakly dense in the set $\mathc…

Researchers created a continuous Markov martingale that mimics Brownian motion but lacks the strong Markov property.

Study on determinants of unitary Brownian motion and their asymptotic laws.

Study refracted skew Brownian motion, find densities and asymptotics.

New model uses generalized fractional Brownian motion for stock price prediction.

Study Brownian loops on hyperbolic surfaces, linking to Selberg zeta function.

Paper defines multi-dimensional fractional Brownian motion under volatility uncertainty.

Researchers calculate the Laplace transform of a geometric Brownian motion integral.

Proves CLT for Brownian paths on pinched negative curvature manifolds.

Geodesic walks converge to Brownian motion on Finsler manifolds.

Universal approximation for stochastic processes using Brownian motion.

New SDEs use $G$-Brownian motion, extending mean-field models.

A new model captures option price dynamics using sub-fractional Brownian motion.

The paper connects Riemann surface length spectra to Brownian loop measures.

This paper derives the non-analytic solution to the Fokker-Planck equation of fractional Brownian motion using the method of Laplace transform. Sequentially, by considering the fundamental solution of the non-analytic solution, this paper obtains the transition probability density function of the random variable that i…

Replacing Black-Scholes' driving process, Brownian motion, with fractional Brownian motion allows for incorporation of a past dependency of stock prices but faces a few major downfalls, including the occurrence of arbitrage when implemented in the financial market. We present the development, testing, and implementatio…

The discrete sum of geometric Brownian motions plays an important role in modeling stochastic annuities in insurance. It also plays a pivotal role in the pricing of Asian options in mathematical finance. In this paper, we study the probability distributions of the infinite sum of geometric Brownian motions, the sum of …

A nonlinear wave alternative for the standard Black-Scholes option-pricing model is presented. The adaptive-wave model, representing 'controlled Brownian behavior' of financial markets, is formally defined by adaptive nonlinear Schrödinger (NLS) equations, defining the option-pricing wave function in terms of the stock…