arXiv research

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 …

Solves optimal liquidation problem for stock price following geometric Brownian motion.

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…

Reflected geometric Brownian motion models are not arbitrage-free.

Paper compares stock price prediction models using Heston and Geometric Brownian Motion.

In this paper we revisit the integral functional of geometric Brownian motion $I_t= \int_0^t e^{-(μs +σW_s)}ds$, where $μ\in\mathbb{R}$, $σ> 0$, and $(W_s )_s>0$ is a standard Brownian motion. Specifically, we calculate the Laplace transform in $t$ of the cumulative distribution function and of the probability density …

The evolution of prices on ideal market is given by geometrical Brownian motion, where Gaussian white noise describes fluctuations. We study the effect of correlations introduced by a color noise.

New method calculates geometric Brownian motion with affine drift and its integral.

An innovative extension of Geometric Brownian Motion model is developed by incorporating a weighting factor and a stochastic function modelled as a mixture of power and trigonometric functions. Simulations based on this Modified Brownian Motion Model with optimal weighting factors selected by goodness of fit tests, sub…

Unified geometric framework for Brownian motion on various manifolds.

Study bounds for Brownian motion on manifolds with sticky boundary conditions.

The paper proposes estimators for bid-ask spreads with and without serial dependence.

Geometric Bass martingales linked to Brownian motion and geometric Brownian motion.

Paper offers a fast method to assess DeFi liquidation risk.

Modeling price dynamics in AMMs with fees using geometric Brownian motion.

We construct a binomial tree model fitting all moments to the approximated geometric Brownian motion. Our construction generalizes the classical Cox-Ross-Rubinstein, the Jarrow-Rudd, and the Tian binomial tree models. The new binomial model is used to resolve a discontinuity problem in option pricing.

The time average of geometric Brownian motion plays a crucial role in the pricing of Asian options in mathematical finance. In this paper we consider the asymptotics of the discrete-time average of a geometric Brownian motion sampled on uniformly spaced times in the limit of a very large number of averaging time steps.…

Asymptotic Laplace transform for geometric Brownian motion applied to bond pricing.

To convert standard Brownian motion $Z$ into a positive process, Geometric Brownian motion (GBM) $e^{βZ_t}, β>0$ is widely used. We generalize this positive process by introducing an asymmetry parameter $α\geq 0$ which describes the instantaneous volatility whenever the process reaches a new low. For our new process, …

Exponential functionals of Brownian motion have been extensively studied in financial and insurance mathematics due to their broad applications, for example, in the pricing of Asian options. The Black-Scholes model is appealing because of mathematical tractability, yet empirical evidence shows that geometric Brownian m…

GMMNs model cross-sectional dependence for better option pricing and simulation.

Entropy corrections improve GBM's predictive accuracy for non-log-normal distributions.

We study how resetting affects geometric Brownian motion, showing it becomes stationary but remains non-ergodic.

Researchers develop a generalised geometric Brownian motion for better asset pricing.

The aim of this paper is to evaluate geometric Asian option by a mixed fractional subdiffusive Black-Scholes model. We derive a pricing formula for geometric Asian option when the underlying stock follows a time changed mixed fractional Brownian motion. We then apply the results to price Asian power options on the stoc…

The long-term dependence of Bitcoin (BTC), manifesting itself through a Hurst exponent $H>0.5$, is exploited in order to predict future BTC/USD price. A Monte Carlo simulation with $10^4$ geometric fractional Brownian motion realisations is performed as extensions of historical data. The accuracy of statistical inferen…

Geometric Brownian motion (GBM) is a model for systems as varied as financial instruments and populations. The statistical properties of GBM are complicated by non-ergodicity, which can lead to ensemble averages exhibiting exponential growth while any individual trajectory collapses according to its time-average. A com…

Estimates returns for dollar cost averaging using geometric Brownian motion.

Geometric Brownian motion simulates stock prices for Brazilian small caps index.

Study investigates ruin probability with random premiums and risky investments.

We extend martingale transport results to weak martingale transport.

Study finds GBM model accurately predicts stock prices on Ghana Stock Exchange.

Study of a generalized geometric Brownian motion with varying entry and exit rates.

Study on the geometric Dyson Brownian motion of non-square matrix products.

Innovative extensions to option pricing models using asymmetric Brownian motion and random walk approaches.

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

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

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

We study the radial part of sub-Riemannian Brownian motion in the context of totally geodesic foliations. Itô's formula is proved for the radial processes associated to Riemannian distances approximating the Riemannian one. We deduce very general stochastic completeness criteria for the sub-Riemannian Brownian motion. …

Study refracted skew Brownian motion, find densities and asymptotics.

In the paper "On Truncated Variation of Brownian Motion with Drift" (Bull. Pol. Acad. Sci. Math. 56 (2008), no.4, 267 - 281) we defined truncated variation of Brownian motion with drift, $W_t = B_t + μt, t\geq 0,$ where $(B_t)$ is a standard Brownian motion. Truncated variation differs from regular variation by neglect…

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

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

Invariance principle proved for lifted geodesic walks on Riemannian submersions.

In this paper, we consider the asset-liability management under the mean-variance criterion. The financial market consists of a risk-free bond and a stock whose price process is modeled by a geometric Brownian motion. The liability of the investor is uncontrollable and is modeled by another geometric Brownian motion. W…

Geodesic walks converge to Brownian motion on Finsler manifolds.

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

We consider a limit order book, where buyers and sellers register to trade a security at specific prices. The largest price buyers on the book are willing to offer is called the market bid price, and the smallest price sellers on the book are willing to accept is called the market ask price. Market ask price is always …