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.
New clustering method for financial data with known cluster number.
problem Clustering financial data with known number of clusters.
method Introduced a covariance-based dissimilarity measure for multifractional Brownian motions.
result Asymptotically consistent clustering algorithms for multifractional Brownian motions.
When common factors strongly influence two power-law cross-correlated time series recorded in complex natural or social systems, using classic detrended cross-correlation analysis (DCCA) without considering these common factors will bias the results. We use detrended partial cross-correlation analysis (DPXA) to uncover…
Complex systems are composed of mutually interacting components and the output values of these components are usually long-range cross-correlated. We propose a method to characterize the joint multifractal nature of such long-range cross correlations based on wavelet analysis, termed multifractal cross wavelet analysis…
In this paper we study BSE Index financial time series for fractal and multifractal behaviour. We show that Bombay stock Exchange (BSE)Index time series is mono-fractal and can be represented by a fractional Brownian motion.
Many complex systems generate multifractal time series which are long-range cross-correlated. Numerous methods have been proposed to characterize the multifractal nature of these long-range cross correlations. However, several important issues about these methods are not well understood and most methods consider only o…
We consider the structure functions S^(q)(T), i.e. the moments of order q of the increments X(t+T)-X(t) of the Foreign Exchange rate X(t) which give clear evidence of scaling (S^(q)(T)~T^z(q)). We demonstrate that the nonlinearity of the observed scaling exponent z(q) is incompatible with monofractal additive stochasti…
A new method for direct determination of multifractal spectrum from moving average analysis.
problem Direct determination of multifractal spectrum from moving average analysis.
method Proposes a new approach based on detrending moving average method (MF-DMA) to directly determine multifractal spectrum.
result Direct determination of multifractal spectrum is possible with less computation cost compared to traditional approach.
A new model captures multifractal volatility in stock returns.
problem Capturing multifractal volatility in stock returns.
method Introduced mLog S-fBM model, defined mS-fBM, and developed calibration procedure.
result Model captures multifractal behavior in stock returns, validating on real data.
Mutually interacting components form complex systems and the outputs of these components are usually long-range cross-correlated. Using wavelet leaders, we propose a method of characterizing the joint multifractal nature of these long-range cross correlations, a method we call joint multifractal analysis based on wavel…
A new model captures multifractal volatility in stock returns.
problem Capturing multifractal volatility in stock returns.
method Introduced mLog S-fBM model, defined mS-fBM, and developed calibration procedure.
result Validated model on synthetic and real data, showing multifractal behavior.
In this paper, we use the generalized Hurst exponent approach to study the multi- scaling behavior of different financial time series. We show that this approach is robust and powerful in detecting different types of multiscaling. We observe a puzzling phenomenon where an apparent increase in multifractality is measure…
New multifractional processes model stock price roughness over time.
problem Modeling stock price roughness and informativeness over time.
method Introduced a new class of multifractional processes and estimated their PHEs using LGQV.
result Our estimator outperforms existing methods in estimating PHEs of multifractional processes.
This paper investigates multiscaling in the rough Bergomi model, finding it primarily due to fat-tailed returns.
problem Understanding multiscaling in the rough Bergomi model to improve financial modelling and risk management.
method Introducing a two-stage statistical testing procedure: first, testing for multiscaling against uniscaling; second, using shuffled surrogates to preserve return distributions.
result Multiscaling in the rough Bergomi model arises primarily from fat-tailed return distributions, not memory effects.
There are a number of situations in which several signals are simultaneously recorded in complex systems, which exhibit long-term power-law cross-correlations. The multifractal detrended cross-correlation analysis (MF-DCCA) approaches can be used to quantify such cross-correlations, such as the MF-DCCA based on detrend…
Introduces log S-fBM model to unify rough and multifractal volatility.
problem Modeling volatility with varying roughness and multifractality.
method Develops log S-fBM family of random measures and proposes estimation methods.
result Demonstrates the estimation of Hurst exponent H and intermittency coefficient λ².
Model predicts stock price volatility using stochastic differential equations.
problem Predicting stock price volatility in financial markets.
method Continuous cascade model using stochastic differential equations with two independent Brownian motions.
result The model accurately reproduces empirical volatility and multifractality.
The superfamily phenomenon of time series with different dynamics can be characterized by the motif rank patterns observed in the nearest-neighbor networks of the time series in phase space. However, the determinants of superfamily classification are unclear. We attack this problem by studying the influence of linear t…
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.
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.
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.
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.
Detrended fluctuation analysis (DFA) is a simple but very efficient method for investigating the power-law long-term correlations of non-stationary time series, in which a detrending step is necessary to obtain the local fluctuations at different timescales. We propose to determine the local trends through empirical mo…
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 …
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.
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.
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. 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.