Related papers: Fractional smoothness and applications in finance
Based on empirical market data, a stochastic volatility model is proposed with volatility driven by fractional noise. The model is used to obtain a risk-neutrality option pricing formula and an option pricing equation.
Based on a criterium of mathematical simplicity and consistency with empirical market data, a stochastic volatility model has been obtained with the volatility process driven by fractional noise. Depending on whether the stochasticity…
We present a conformal theory for intermittent scalar fields. As an example, we consider the energy flux from large to small scales in the developed turbulent flow. The conformal correlation functions are found in the inertial range of…
The paper deals with the fast-slow motions setups in the continuous time $\frac {dX^(t)}{dt}=\frac 1\varepsilon B(X^\varepsilon(t),\xi(t/\varepsilon^2))+b(X^\varepsilon(t),\,\xi(t/\varepsilon^2)),\, t\in [0,T]$ and the discrete time…
In this article, we consider the reconstruction of $\rho(t)$ in the (time-fractional) diffusion equation $(\partial_t^\alpha-\triangle)u(x,t)=\rho(t)g(x)$ ($0<\alpha \le 1$) by the observation at a single point $x_0$. We are mainly…
In this article, we consider the following stochastic fractional diffusion equation \begin{equation*} \left(\partial^{\beta}+\dfrac{\nu}{2}\left(-\Delta\right)^{\alpha / 2}\right) u(t, x)= \lambda\: I_{0_+}^{\gamma}\left[u(t, x) \dot{W}(t,…
The sub-Gaussian stable distribution is a heavy-tailed elliptically contoured law which has interesting applications in signal processing and financial mathematics. This work addresses the problem of feasible estimation of distributions. We…
It is well known that the probability distribution of high-frequency financial returns is characterized by a leptokurtic, heavy-tailed shape. This behavior undermines the typical assumption of Gaussian log-returns behind the standard…
Increasingly larger data sets of processes in space and time ask for statistical models and methods that can cope with such data. We show that the solution of a stochastic advection-diffusion partial differential equation provides a…
A weak invariant of a stochastic system is defined in such a way that its expectation value with respect to the distribution function as a solution of the associated Fokker-Planck equation is constant in time. A general formula is given for…
The fractional stable motion is a prototypical stochastic process exhibiting both heavy tails and long-range dependence, parameterized via a stability index $\alpha$ and a Hurst exponent $H$. We consider a nonstationary extension where the…
In the present work, the notion of Super Fractal Interpolation Function (SFIF) is introduced for finer simulation of the objects of the nature or outcomes of scientific experiments that reveal one or more structures embedded in to another.…
Fractional diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogues. They are used in physics to model anomalous diffusion. This paper develops strong solutions of space-time fractional…
We present analytical investigations of a multiplicative stochastic process that models a simple investor dynamics in a random environment. The dynamics of the investor's budget, $x(t)$, depends on the stochasticity of the return on…
We give an infinitesimal meaning to the symbol $dX_t$ for a continuous semimartingale $X$ at an instant in time $t$. We define a vector space structure on the space of differentials at time $t$ and deduce key properties consistent with the…
Assuming frictionless trading, classical stochastic portfolio theory (SPT) provides relative arbitrage strategies. However, the costs associated with real-world execution are state-dependent, volatile, and under increasing stress during…
We derive explicit solutions for time-fractional anomalous diffusion equations with diffusivity coefficients that depend on both space and time variables. These solutions are expressed in Fox-H and generalized Wright functions, which are…
The aim of this paper is to develop a sequence of discrete approximations to a one-dimensional It\^o diffusion that almost surely converges to a weak solution of the given stochastic differential equation. Under suitable conditions, the…
This paper concerns the instantaneous frequency (IF) of continuous-time, zero-mean, complex-valued, proper, mean-square differentiable nonstationary Gaussian stochastic processes. We compute the probability density function for the IF for…
This dissertation discusses the intermitency phenomenon in three models of turbulence, employing analytical and numerical techniques in the analysis of stochastic processes and the probability distributions which they induce. The initial…