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.
New fake Brownian motions derived from calibrated LSV models.
problem Calibrating LSV models to market data with stochastic volatility.
method Particle methods and McKean SDEs for calibration; Fokker-Planck PDEs for existence proof.
result Existence of new fake Brownian motions derived from calibrated LSV models.
Paper explores Skorokhod embedding with local time for robust hedging in finance.
problem Robust hedging of options with local time.
method Stochastic control approach, relaxation of monotonicity assumption.
result New solution to two-marginal Skorokhod embedding, relaxation of monotonicity.
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.
Formula calculates optimal number of paths for correlated Brownian motions.
problem Determining the optimal number of paths for simulating correlated Brownian motions.
method Provides an explicit formula for the optimal number of paths.
result Optimal number of paths for simulating correlated Brownian motions is calculated.
Paper solves fractional Brownian motion using Laplace transforms.
problem Fractional Brownian motion and its applications.
method Non-analytic solution via Laplace transform.
result Transition probability density function derived for fractional Brownian motion.
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.
The paper studies discrete sums of geometric Brownian motions in finance.
problem Modeling stochastic annuities and pricing Asian options.
method Analyzes probability distributions and asymptotic behavior of discrete sums of geometric Brownian motions.
result Derives tail asymptotics and computes asymptotic distribution functions for discrete sums.
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.
Researchers define a limit for fractional Brownian motion as Hurst parameter approaches zero.
problem Defining a limit for fractional Brownian motion with zero Hurst parameter.
method Developed a Gaussian random distribution and log-correlated random field as limits.
result Fractional Brownian motion converges to a Gaussian random distribution when Hurst parameter approaches zero.
Alternative model for financial derivatives pricing using Gaussian Markov process.
problem Inaccurate pricing of financial derivatives due to past dependency of stock prices.
method Developed a simplified Gaussian Markov process alternative to fractional Brownian motion.
result Improved accuracy in pricing derivatives by allowing past dependency.
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.
A new model captures option price dynamics using sub-fractional Brownian motion.
problem Capturing the complex price dynamics of financial options.
method Developed a CEV model driven by a mixed sub-fractional Brownian motion.
result Empirical tests show the model effectively captures option price dynamics.
Paper simplifies proving transition density for a specific type of Brownian motion.
problem Proving transition density for Hyperbolic Brownian motion with drift.
method Expansion of transition density with a simplified proof condition.
result Simplified proof for transition density simplifies option pricing.
New binomial model fits all moments to geometric Brownian motion.
problem Discontinuity problem in option pricing.
method Constructs a generalized binomial tree model.
result Resolves discontinuity problem in option pricing.
Develops a theory of common decomposition for correlated Brownian motions.
problem Tackles the modeling of correlated Brownian motions in financial applications.
method Uses change of time method to represent correlated Brownian motions as a triplet of processes.
result Shows equivalent conditions for the triplet being independent and proposes a new method for constructing correlated Brownian motions.
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…
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.
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.
Study describes recurrence/transience for Brownian motion in warped product manifolds.
problem Understanding recurrence and transience in Brownian motion on warped product manifolds.
method Integral test for cut-off between recurrence and transience in unbounded domains of warped product manifolds.
result Recovery and extension of classical tests for standard Brownian motion in model spaces and generalized tube domains.
Improved model better fits financial data.
problem Inaccurate fitting of financial data distributions.
method Modified Brownian Motion with weighting and stochastic functions.
result Model outperforms basic Geometric Brownian Motion.
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 high-frequency trading with fractional Brownian motion, finding optimal strategies and convergence.
problem Maximizing utility in high-frequency trading with fractional Brownian motion.
method Spectral methods for stationary Gaussian sequences, asymptotic growth rate analysis, finite-dimensional distribution convergence.
result Suitably rescaled optimal positions converge to a Gaussian white-noise-type field.
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.
Study prices compound and extendible options using mixed fractional Brownian motion with jumps.
problem Pricing compound and extendible options under mixed fractional Brownian motion with jumps.
method Analytic formula derived under risk-neutral measure, applied to extendible options, discussed special cases, provided numerical results.
result An analytic formula for pricing compound options derived.
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. Calculates first exit times for Brownian motion in Euclidean and hyperbolic planes.
problem Computing expected first exit times for Brownian motion.
method Analytical computation for Brownian motion in Euclidean and hyperbolic planes.
result Results in expected first exit times for specified domains.
Model predicts Bitcoin prices using fractional Brownian motion.
problem Predicting Bitcoin prices with long-term dependence.
method Monte Carlo simulation with geometric fractional Brownian motion.
result Most probable Bitcoin price at the start of 2018 was 6358 USD.
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.
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.
A new process generalizes geometric Brownian motion with asymmetry.
problem Creating a positive process with asymmetry parameter.
method Introducing asymmetry parameter α to describe volatility at new lows.
result Preserves GBM properties while expressing volatility as weighted mean.
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.
Kinetic theory explains financial Brownian motion from trader dynamics.
problem Understanding financial Brownian motion from high-frequency trading dynamics.
method Deriving time-evolution equations, Bogoliubov-Born-Green-Kirkwood-Yvon hierarchies, Boltzmann-like and Langevin-like equations.
result Mathematical foundation for financial Brownian motion parallels physical Brownian motion.
Optimizes Brownian motion with minimal evaluations.
problem Optimizing a random Brownian motion realization.
method Adaptive algorithm with sample complexity of log^2(1/ε).
result Improves sample complexity over previous methods.
The paper studies the discrete-time average of geometric Brownian motion and its application to Asian options pricing.
problem Understanding the pricing of Asian options with discrete-time averaging.
method Deriving asymptotics for the discrete-time average of geometric Brownian motion and analyzing its impact on Asian options pricing.
result Derives the asymptotics for the price of Asian options with discrete-time averaging in the Black-Scholes model.
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.
Eigenfunctions constructed via Brownian motion exit times on curved spaces.
problem Constructing eigenfunctions of Laplacian on curved spaces.
method Using exit times of Brownian motion on a Riemannian manifold.
result Eigenfunctions can be constructed for a wide range of Laplacian eigenvalues.
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.
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.