Related papers: Modified Brownian Motion Approach to Modelling Ret…
Following a Geometrical Brownian Motion extension into an Irrational Fractional Brownian Motion model, we re-examine agent behaviour reacting to time dependent news on the log-returns thereby modifying a financial market evolution. We…
In this paper, we combine the techniques of enlargement of filtrations and stochastic control theory to establish an extension of the verification theorem, where the coefficients of the stochastic controlled equation are adapted to the…
This study presents a long-term alternative formula for stock price variation described by a geometric Brownian motion on the basis of median instead of mean or expected values. The proposed method is motivated by the observation made in…
We identify an issue in recent approaches to learning-based control that reformulate systems with uncertain dynamics using a stochastic differential equation. Specifically, we discuss the approximation that replaces a model with fixed but…
The diversity of diffusive systems exhibiting long-range correlations characterized by a stochastically varying Hurst exponent calls for a generic multifractional model. We present a simple, analytically tractable model which fills the gap…
We study some functional inequalities satisfied by the distribution of the solution of a stochastic differential equation driven by fractional Brownian motions. Such functional inequalities are obtained through new integration by parts…
A new extension of the sub-fractional Brownian motion, and thus of the Brownian motion, is introduced. It is a linear combination of a finite number of sub-fractional Brownian motions, that we have chosen to call the mixed sub-fractional…
We study the numerical evaluation of several functions appearing in the small time expansion of the distribution of the time-integral of the geometric Brownian motion as well as its joint distribution with the terminal value of the…
In this paper we derive novel change of variable formulas for stochastic integrals w.r.t. a time-changed Brownian motion where we assume that the time-change is a general increasing stochastic process with finitely many jumps in a bounded…
We investigated the quality of forecasting of fractional Brownian motion, and new method for estimating of Hurst exponent is validated. Stochastic model of the time series in the form of converted fractional Brownian motion is proposed. The…
We study integral representations of random variables with respect to general H\"older continuous processes and with respect to two particular cases; fractional Brownian motion and mixed fractional Brownian motion. We prove that arbitrary…
A multivariate fractional Brownian motion (mfBm) with component-wise Hurst exponents is used to model and forecast realized volatility. We investigate the interplay between correlation coefficients and Hurst exponents and propose a novel…
The invariance properties of Brownian motion are investigated and revisited within a recent Lie symmetry approach to stochastic differential equations. Some notable properties of the process can be recovered by a related integration by…
In this paper, we investigate the optimal control problem for systems driven by mixed fractional Brownian motion (including a fractional Brownian motion with Hurst parameter $H>1/2$ and the standard Brownian motion). By using Malliavin…
Modeling financial data often relies on assumptions that may prove insufficient or unrealistic in practice. The Geometric Brownian Motion (GBM) model is frequently employed to represent stock price processes. This study investigates whether…
The aim of this paper is to present the new results concerning some functionals of Brownian motion with drift and present their applications in financial mathematics. We find a probabilistic representation of the Laplace transform of…
In this paper we present a dynamical system to generate Brownian motion based on the Langevin equation without stochastic term and using fractional derivatives, i.e., a deterministic Brownian motion model is proposed. The stochastic process…
Many studies assume stock prices follow a random process known as geometric Brownian motion. Although approximately correct, this model fails to explain the frequent occurrence of extreme price movements, such as stock market crashes. Using…
We consider fractional Brownian motion with the Hurst parameters from (1/2,1). We found that the increment of a fractional Brownian motion can be represented as the sum of a two independent Gaussian processes one of which is smooth in the…
Exact generalized stochastic representation of deterministic interaction between two dynamical (quantum or classical) systems is derived which helps when considering one of them to replace another by equivalent commutative ($c$-number…