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We develop a class of non-Gaussian translation processes that extend classical stochastic differential equations (SDEs) by prescribing arbitrary absolutely continuous marginal distributions. Our approach uses a copula-based transformation…
We develop a novel approach for the construction of quantile processes governing the stochastic dynamics of quantiles in continuous time. Two classes of quantile diffusions are identified: the first, which we largely focus on, features a…
In finance, the price of a volatile asset can be modeled using fractional Brownian motion (fBm) with Hurst parameter $H>1/2.$ The Black-Scholes model for the values of returns of an asset using fBm is given as, [Y_t=Y_0…
Continuous time stochastic processes are useful models especially for financial and insurance purposes. The numerical simulation of such models is dependant of the time discrete discretization, of the parametric estimation and of the choice…
The volatility of financial instruments is rarely constant, and usually varies over time. This creates a phenomenon called volatility clustering, where large price movements on one day are followed by similarly large movements on successive…
Given a positive random variable $X$, $X\ge0$ a.s., a null hypothesis $H_0:E(X)\le\mu$ and a random sample of infinite size of $X$, we construct test supermartingales for $H_0$, i.e. positive processes that are supermartingale if the null…
We study a quasilinear parabolic Cauchy problem with a cumulative distribution function on the real line as an initial condition. We call 'probabilistic solution' a weak solution which remains a cumulative distribution function at all…
It is shown that superpositions of path integrals with arbitrary Hamiltonians and different scaling parameters v ("variances") obey the Chapman-Kolmogorov relation for Markovian processes if and only if the corresponding smearing…
Modern regression applications can involve hundreds or thousands of variables which motivates the use of variable selection methods. Bayesian variable selection defines a posterior distribution on the possible subsets of the variables…
We study the diffusion phenomena on the negatively curved surface made up of congruent heptagons. Unlike the usual two-dimensional plane, this structure makes the boundary increase exponentially with the distance from the center, and hence…
A model for diffusion on a cubic lattice with a random distribution of traps is developed. The traps are redistributed at certain time intervals. Such models are useful for describing systems showing dynamic disorder, such as ion-conducting…
In this paper we explain that the natural filtration of a continuous Hunt process is continuous, and show that martingales over such a filtration are continuous. We further establish a martingale representation theorem for a class of…
Quantum theory is used to model secondary financial markets. Contrary to stochastic descriptions, the formalism emphasizes the importance of trading in determining the value of a security. All possible realizations of investors holding…
The proposed model modifies option pricing formulas for the basic case of log-normal probability distribution providing correspondence to formulated criteria of efficiency and completeness. The model is self-calibrating by historic…
Classical diffusion in a random medium involves an exponential functional of Brownian motion. This functional also appears in the study of Brownian diffusion on a Riemann surface of constant negative curvature. We analyse in detail this…
The class of $\alpha$-stable distributions received much interest for modelling impulsive phenomena occur in engineering, economics, insurance, and physics. The lack of non-analytical form for probability density function is considered as…
We show that our generalization of the Black-Scholes partial differential equation (pde) for nontrivial diffusion coefficients is equivalent to a Martingale in the risk neutral discounted stock price. Previously, this was proven for the…
Accurate modeling of the temporal evolution of asset prices is crucial for understanding financial markets. We explore the potential of discrete-time quantum walks to model the evolution of asset prices. Return distributions obtained from a…
This paper provides the mathematical foundation for polynomial diffusions. They play an important role in a growing range of applications in finance, including financial market models for interest rates, credit risk, stochastic volatility,…
Various phenomenological models of particle multiplicity distributions are discussed using a general form of the grand canonical partition function. These phenomenological models include a wide range of varied processes such as coherent…