Study radial processes in sub-Riemannian Brownian motions, proving stochastic completeness and eigenvalue estimates.
problem Analyzing sub-Riemannian Brownian motions and their radial processes.
method Application of Itô's formula and sub-Laplacian comparison theorems to prove stochastic completeness and eigenvalue estimates.
result Proved Cheng's type estimates for Dirichlet eigenvalues of sub-Riemannian metric balls.
Constructs stochastic processes on sub-Riemannian manifolds using Cartan connections.
problem Developing stochastic processes on sub-Riemannian manifolds.
method Introduces stochastic development using Cartan connections, derives generator, and provides conditions for existence.
result Derives a general expression for the generator of the stochastic process and provides conditions for the existence of a Cartan connection.
Study random walks on sub-Riemannian manifolds using retractions.
problem Modeling random walks on sub-Riemannian manifolds.
method Use retractions to approximate normal geodesics and study convergence to Brownian motion.
result Convergence of geodesic random walks defined with different connections.
Maps on Sasakian manifolds limit to sub-Riemannian distance bounds.
problem Understanding sub-Riemannian distances on Sasakian manifolds.
method Parallel and mirror maps along geodesics of a taming Riemannian metric.
result Limits of transport maps outside sub-Riemannian cut-locus provide bounds on sub-Riemannian distance.
The book explores stochastic areas and heat kernels on manifolds.
problem Understanding stochastic area functionals and heat kernels on manifolds.
method Study of Brownian motions and heat kernels on Lie groups and Riemannian manifolds.
result Rich interactions between stochastic calculus, geometry, and random matrices.
This paper considers a classical question of approximation of Brownian motion by a random walk in the setting of a sub-Riemannian manifold M. To construct such a random walk we first address several issues related to the degeneracy of such a manifold. In particular, we define a family of sub-Laplacian operators natur…
Develops non-Markovian couplings for sub-Riemannian Brownian motions.
problem Constructing couplings for sub-Riemannian Brownian motions starting from points on the same vertical fiber.
method Uses global isometries to construct maximal couplings, satisfying a reflection principle.
result Estimates coupling time and applies to inequalities for the heat semigroup.
We discuss the geometric foundation behind the use of stochastic processes in the frame bundle of a smooth manifold to build stochastic models with applications in statistical analysis of non-linear data. The transition densities for the projection to the manifold of Brownian motions developed in the frame bundle lead …
We consider different sub-Laplacians on a sub-Riemannian manifold M. Namely, we compare different natural choices for such operators, and give conditions under which they coincide. One of these operators is a sub-Laplacian we constructed previously in \cite{GordinaLaetsch2014a}. This operator is canonical with respec…
Study on determinants of unitary Brownian motion and their asymptotic laws.
problem Understanding determinants of unitary Brownian motion and their behavior over time.
method Using Stiefel fibration and skew-product decomposition of the Stiefel Brownian motion.
result Prove asymptotic laws for determinants of block entries of unitary Brownian motion.
Researchers created a continuous Markov martingale that mimics Brownian motion but lacks the strong Markov property.
problem Constructing a continuous Markov martingale with Brownian marginals that misses the strong Markov property.
method Developed a new approach to create a continuous Markov martingale that differs from Brownian motion in terms of the strong Markov property.
result A continuous Markov martingale with Brownian marginals that lacks the strong Markov property was successfully constructed.
Study refracted skew Brownian motion, find densities and asymptotics.
problem Modeling and analyzing refracted skew Brownian motion.
method Perturbation approach to find potential densities, transition density, and asymptotic behaviors.
result Expressions and asymptotic behaviors of refracted skew Brownian motion.
Paper defines multi-dimensional fractional Brownian motion under volatility uncertainty.
problem Volatility uncertainty in fractional Brownian motion.
method Definition and study of multi-dimensional fractional Brownian motion (G-fBm) with Hurst index.
result First results on stochastic calculus for G-fBm with Hurst index > 0.5.
New model uses generalized fractional Brownian motion for stock price prediction.
problem Traditional models fail to accurately predict stock price fluctuations.
method Introduces generalized fractional Brownian motion as a new stochastic process for price modeling.
result Validates the new model for option pricing and risk assessment.
Researchers calculate the Laplace transform of a geometric Brownian motion integral.
problem Calculating the Laplace transform of a specific integral functional of geometric Brownian motion.
method Analytical calculation of the Laplace transform of the cumulative distribution and probability density functions.
result The Laplace transform of the integral functional of geometric Brownian motion is derived.
Geodesic walks converge to Brownian motion on Finsler manifolds.
problem Understanding random walks on Finsler manifolds.
method Analyzing convergence of geodesic random walks to diffusion processes.
result The Brownian motion on a Riemannian metric is a key result.
New SDEs use G-Brownian motion, extending mean-field models.
problem Extending mean-field models to new types of stochastic processes.
method Introduced G-SDEs with coefficients dependent on current state and solution as random variable. result Validated new SDE framework for complex stochastic systems.
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…
This paper classifies Legendre singularities of sub-Riemannian geodesics on surfaces.
problem Classifying singularities of sub-Riemannian geodesics.
method Complete local classification using Legendre fibrations.
result Legendre singularities are completely classified for sub-Riemannian geodesics.
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 …
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…
Explains planetary motion in a sub-Riemannian setting.
problem Kepler-Heisenberg problem
method Classical Kepler problem adapted to sub-Riemannian geometry
result Rich and mysterious dynamical system with tractable questions
Modeling financial markets with memory using fractional calculus and Brownian motion.
problem Capturing memory effects in financial markets using stochastic models.
method Fractional Langevin equation with colored noise generated by fractional Brownian motion.
result Anomalous marginal glass phase observed in some regions of the system.
Estimates spectral gap for Brownian motion on sticky-reflecting domains.
problem Estimating spectral gap for Brownian motion on sticky-reflecting domains.
method Interpolation method and novel applications of Reilly formula.
result Lower bounds for spectral gap derived for general domains.
Two insurance companies collaborate to maximize the probability of none going bankrupt.
problem Maximizing the probability of no company bankruptcy in a correlated Brownian motion model.
method Analyzing optimal strategies and deriving explicit formulas for minimal ruin probability.
result Maximizing collaboration benefits when Brownian motions are positively correlated.
The paper extends Merton model to price equity warrants under subdiffusive fractional Brownian motion of the short rate.
problem Equity warrant pricing under subdiffusive fractional Brownian motion of the short rate.
method The paper applies subdiffusive mechanism to analyze equity warrant in a fractional Brownian motion environment, deriving a pricing formula for equity warrant.
result The paper provides a pricing formula for equity warrants under subdiffusive fractional Brownian motion model of the short rate.
Solves optimal liquidation problem for stock price following geometric Brownian motion.
problem Optimal liquidation problem for stock price process following geometric Brownian motion.
method Functional analysis tools; working in terms of cash.
result Explicit solution to the problem, extending to stochastic drift.
Study Brownian motions and heat kernel bounds on Kähler and quaternion Kähler manifolds.
problem Understanding Brownian motions and heat kernel bounds on specific geometric manifolds.
method Sharp Laplacian comparison theorems and Cheeger-Yau type lower bounds for heat kernels.
result Sharp Cheeger-Yau type lower bounds for heat kernels and Dirichlet eigenvalues of metric balls.
Universal approximation for stochastic processes using Brownian motion.
problem Approximating stochastic processes with linear functionals.
method Establishing Lp-type universal approximation theorems for rough path spaces. result Linear functionals on the signature of time-extended Brownian motion can approximate any p-integrable stochastic process. Quaternionic Brownian motion on flag manifold linked to sphere diffusion.
problem Modeling quaternionic stochastic areas on quaternionic flag manifolds.
method Relating quaternionic Brownian motion to symplectic Brownian motion and using radial dynamics.
result Quaternionic stochastic areas follow a multivariate normal distribution.
Upper bounds on constants for Brownian motion with sticky boundary.
problem Bounding constants for Brownian motion with sticky boundary.
method Interpolation approach based on energy interactions and Reilly formula.
result Upper bounds on Poincaré and Logarithmic Sobolev constants.
Proves CLT for Brownian paths on pinched negative curvature manifolds.
problem Distribution of Brownian paths on pinched negative curvature manifolds.
method Proof of central limit theorem for distances and Green functions.
result Central limit theorem holds for Brownian paths in pinched negative curvature.
The sub-fractional Brownian motion (sfBm) is a stochastic process, characterized by non-stationarity in their increments and long-range dependency, considered as an intermediate step between the standard Brownian motion (Bm) and the fractional Brownian motion (fBm). The mixed process, a linear combination between a Bm …
Rough volatility models are becoming increasingly popular in quantitative finance. In this framework, one considers that the behavior of the log-volatility process of a financial asset is close to that of a fractional Brownian motion with Hurst parameter around 0.1. Motivated by this, we wish to define a natural and re…
Study bounds for Brownian motion on manifolds with sticky boundary conditions.
problem Proving geometric bounds for Brownian motion on manifolds with sticky boundary conditions.
method Interpolation involving energy interactions between boundary and interior of the manifold.
result Explicit geometric bounds on Steklov eigenvalues, boundary trace operators, and boundary trace logarithmic Sobolev constants.
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…
Time-subordinated Brownian motion models improve financial market stochastic distribution.
problem Improving stochastic distribution modeling in financial markets.
method Fourier theory and methodology for time-subordinated Brownian motion models, extending real domain to complex plane.
result Characterization and direct study of stochastic time-change from full process.
A new option pricing model uses a time-varying Hurst exponent for more accurate financial predictions.
problem Inaccurate modeling of financial time series due to constant memory parameter limitations.
method Modeling price fluctuations with multifractional Brownian motion and deriving option pricing formula.
result Empirical performance shows the multifractional model fits market quotes better than standard models.
We provide an explicit formula giving the optimal number of paths needed to simulate two correlated Brownian motions.
This paper constructs Brownian motion on complex flag manifolds and finds joint distribution of stochastic areas.
problem Modeling stochastic areas on complex partial flag manifolds.
method Constructs Brownian motion on complex partial flag manifolds and uses it to find joint distribution of stochastic areas.
result Limit law of stochastic areas is a multivariate Cauchy distribution.
In this paper, we develop a theory of common decomposition for two correlated Brownian motions, in which, by using change of time method, the correlated Brownian motions are represented by a triplet of processes, (X,Y,T), where X and Y are independent Brownian motions. We show the equivalent conditions for the tr…
Reflected geometric Brownian motion models are not arbitrage-free.
problem No-arbitrage condition violation in financial markets.
method Analysis of reflected geometric Brownian motion models.
result Models violate even the weakest no-arbitrage condition.
We consider so-called regular invertible Gaussian Volterra processes and derive a formula for their prediction laws. Examples of such processes include the fractional Brownian motions and the mixed fractional Brownian motions. As an application, we consider conditional-mean hedging under transaction costs in Black-Scho…
Paper compares stock price prediction models using Heston and Geometric Brownian Motion.
problem Predicting stock prices accurately.
method Developed Heston and Geometric Brownian Motion models using Ito's lemma and Euler-Maruyama methods.
result Models outperform statistical indicators in predicting stock prices.
Study Brownian motion on Grassmann manifold using matrix stochastic calculus.
problem Understanding Brownian motion on non-compact Grassmann manifold.
method Realize Brownian motion as matrix diffusion process, use matrix stochastic calculus, and hyperbolic Stiefel fibration.
result Connection to generalized Maass Laplacian of complex hyperbolic space.
Modeling stock price fluctuations using Brownian motion and stochastic differential equations.
problem Capturing the stochastic behavior of stock prices.
method Developed a stochastic differential equation to model stock price fluctuations, incorporating Itô integration.
result Backtesting showed a strong correlation coefficient between the model and actual stock price movements.
Researchers prove long-time existence for two landmark Brownian motion.
problem Proving long-time existence of Brownian motion on configurations of two landmarks.
method Classification and analysis of long-time existence for configurations of exactly two landmarks, using a radial kernel.
result For configurations of exactly two landmarks, long-time existence is possible for certain kernels, but not for others.