A new bootstrap method improves hypothesis testing for roughness of time series data.
problem Improving hypothesis testing for roughness of time series data.
method Local fractional bootstrap method for high-frequency statistics of Brownian semistationary processes.
result The bootstrap method provides considerable finite-sample improvements over existing methods.
We introduce a simulation scheme for Brownian semistationary processes, which is based on discretizing the stochastic integral representation of the process in the time domain. We assume that the kernel function of the process is regularly varying at zero. The novel feature of the scheme is to approximate the kernel fu…
New models explain rough and persistent volatility patterns.
problem Understanding and modeling the rough and persistent nature of asset price volatility.
method Introduced a new class of continuous-time models based on the Brownian semistationary process.
result Models show evidence of roughness and long memory in volatility time series.
Method simulates volatility-modulated Volterra processes using HSPDEs.
problem Modeling turbulence, tumor growth, and financial prices.
method Finite difference scheme for HSPDEs.
result Scheme converges to VMV process solutions.
This paper introduces the class of volatility modulated Lévy-driven Volterra (VMLV) processes and their important subclass of Lévy semistationary (LSS) processes as a new framework for modelling energy spot prices. The main modelling idea consists of four principles: First, deseasonalised spot prices can be modelled di…
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.
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.
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.
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. The paper analyzes uncertainty quantification in sparse Gaussian process regression with a Brownian motion prior.
problem Analyzing uncertainty in sparse Gaussian process regression with a Brownian motion prior.
method Theoretical guarantees and limitations for pointwise credible sets are derived for a rescaled Brownian motion prior with a sparse variational Gaussian process method.
result Theoretical characterization of asymptotic frequentist coverage for credible sets, distinguishing conservative and overconfident cases.
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.
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.
Study predicts Gaussian Volterra processes with noisy Brownian motion.
problem Predicting Gaussian Volterra processes with hidden Brownian motion.
method Regular conditional law analysis under model disturbances.
result Developed method for variance reduction in measurement errors.
New Brownian motion defined in Minkowski normed spaces.
problem Constructing Brownian motion in non-Euclidean spaces.
method Singular McKean--Vlasov stochastic differential equation.
result Pathwise uniqueness of solutions to the stochastic differential equation.
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.
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.
The study examines hedging strategies in financial models with transaction costs.
problem Hedging strategies under transaction costs in financial models.
method Analysis of Gaussian Volterra processes and conditional-mean hedging.
result Derivation of a formula for prediction laws in Gaussian Volterra processes.
Unified treatment of eigenvalue processes using Riemannian geometry.
problem Eigenvalue processes in various settings.
method Riemannian submersion and gradient flow of isospectral orbits.
result Eigenvalue processes are projections of Brownian motion through Riemannian submersions.
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.
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.
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.
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.
The paper shows how particle movement on a manifold's grid approximates Brownian motion and heat diffusion.
problem Understanding particle movement on curved spaces.
method Analyzing symmetric exclusion process on random grids approximating a Riemannian manifold.
result Empirical density field converges to heat equation solution on the manifold.
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.
The paper explores anticipative binary information in financial markets using Brownian motion and Poisson processes.
problem Capturing anticipative information in financial markets with Brownian motion and Poisson processes.
method Using Malliavin calculus and filtration enlargement techniques, the paper computes the semimartingale decomposition of the processes.
result The paper provides the exact value of anticipative information in the pure jump case.
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.
Optimizes spending by adjusting a discount factor modelled as an exponential CIR process.
problem Maximizing discounted spendings/dividend payments given an exponential CIR discounting factor.
method Analytical and numerical methods for deterministic and stochastic surplus processes.
result Explicit expressions for optimal strategies in deterministic cases, and constant-barrier strategies for small volatility in stochastic cases.
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.
Dynamic Black-Litterman integrates expert views with portfolio optimization over varying time horizons.
problem Incorporating expert views with varying horizons in portfolio optimization.
method Exploiting graphical structure, deriving conditional distribution of asset returns, and using affine factor models.
result Explicit expression for optimal dynamic investment policy and hedging demand analysis.
The Epps effect helps distinguish between continuous and discrete financial tick data.
problem Determining whether financial tick data represents continuous or discrete events.
method Deriving and correcting the Epps effect, proposing experiments to discriminate between models.
result Tick data is better represented as discrete events rather than continuous Brownian diffusions.
Paper extends Brownian bridge with random length and pinning point for financial modeling.
problem Modeling financial information flow with uncertainty in pinning point.
method Introduced an extension of Brownian bridge with random length and pinning point, derived formulae for conditional expectations.
result The extended Brownian bridge fails to be Markovian if pinning point distribution is absolutely continuous.
In this paper we consider a new mathematical extension of the Black-Scholes model in which the stochastic time and stock share price evolution is described by two independent random processes. The parent process is Brownian, and the directing process is inverse to the totally skewed, strictly α-stable process. The subo…
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.
We study systems of Brownian particles on the real line, which interact by splitting the local times of collisions among themselves in an asymmetric manner. We prove the strong existence and uniqueness of such processes and identify them with the collections of ordered processes in a Brownian particle system, in which …
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.
The paper analyzes optimal execution strategies for traders with inventory processes influenced by Brownian motion.
problem Optimal execution strategies for traders with inventory processes influenced by Brownian motion.
method Statistical tests and empirical analysis of intra-day data from the Toronto Stock Exchange.
result Empirical evidence supports the presence of a non-zero Brownian motion component in inventories and wealth processes.
Optimizes dividend policies in a Brownian model with controlled rates.
problem Realistic optimal dividend policies in a stochastic control problem.
method Delayed linear control strategies for refracted diffusion processes.
result Optimality of delayed linear control strategies for dividend payments.
The paper studies variable annuity benefits using exponential functionals of Levy processes.
problem Modeling equity returns with a Levy process to better fit market features.
method Uses exponential functionals of a Levy process to compute the distribution of variable annuity guaranteed benefits.
result Explicitly computes the distribution of certain exponential functionals.
New process from fractional BM and OU process yields simpler variance.
problem Simpler model for autocovariance structure.
method Construct new process using fractional BM and OU process, analyze increments.
result Variance of new process easier to compute than FARIMA.
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.
Non-Markovian point process shows power-law scaling, similar to nonlinear Markovian process.
problem Understanding the scaling behavior of non-Markovian point processes.
method Analyzed a confined fractional Brownian motion-driven point process and compared it to a nonlinear Markovian process.
result A nonlinear Markovian process can reproduce the power-law scaling behavior of a non-Markovian point process.
Modeling stock order book dynamics with bouncing GBMs.
problem Capturing the dynamics of order book prices in financial markets.
method Modeling order book bid and ask prices as bouncing geometric Brownian motions.
result The logarithmic trading price process converges to a standard Brownian motion as δ approaches 0.
Proves Brownian bridges on manifolds are semimartingales.
problem Semimartingale property of Brownian bridges on Riemannian manifolds.
method Localized Hamilton-type gradient estimate by Arnaudon/Thalmaier.
result Every adapted Brownian bridge on a geodesically complete Riemannian manifold is a semimartingale.
We study how resetting affects geometric Brownian motion, showing it becomes stationary but remains non-ergodic.
problem Effects of stochastic resetting on geometric Brownian motion.
method Analysis of geometric Brownian motion under stochastic resetting.
result Resetting makes geometric Brownian motion stationary but non-ergodic.
Defines a bridge process to predict financial market risks.
problem Predicting financial market risks before a default occurs.
method Defines a Brownian bridge process conditioned to zero at default.
result Leaks information about default before it happens.
GMMNs model cross-sectional dependence for better option pricing and simulation.
problem Modeling cross-sectional dependence between stochastic processes.
method Generative moment matching networks (GMMNs) for geometric Brownian motions and ARMA-GARCH models.
result GMMNs produce dependent quasi-random samples with variance reduction.
New algorithm for non-Markovian optimal stopping problems using Brownian motion.
problem Optimal stopping time problems for non-Markovian state processes.
method Longstaff-Schwartz-type algorithm based on statistical learning theory.
result Error estimates for approximation architecture spaces with finite Vapnik-Chervonenkis dimension.
Study of most probable paths for anisotropic Brownian motions on manifolds.
problem Characterizing paths of Brownian motions with anisotropic diffusion on manifolds.
method Using stochastic development and fiber bundle of linear frames, the study provides a comprehensive characterization of most probable paths.
result Explicit equations and integration methods for most probable paths on different geometries, including constant curvature surfaces.