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This paper gives an accessible (but still technical) self-contained proof to the fact that the intersection probabilities for planar Brownian motion are given in terms of the intersection exponents, up to a bounded multiplicative error, and…
We apply the formalism of the continuous time random walk to the study of financial data. The entire distribution of prices can be obtained once two auxiliary densities are known. These are the probability densities for the pausing time…
We compute the joint distribution of the site and the time at which a $d$-dimensional standard Brownian motion $B_t$ hits the surface of the ball $ U(a) =\{|{\bf x}|<a\}$ for the first time. The asymptotic form of its density is obtained…
We consider a Brownian particle moving on a ring. We study the probability distributions of the total number of turns and the net number of counter-clockwise turns the particle makes till time t. Using a method based on the renewal…
Aging in spin glasses is analyzed via the Probability Density Function (PDF) of the heat transfer between system and bath over a small time $\delta t$. The PDF contains a Gaussian part, describing reversible fluctuations, and an exponential…
We consider a general piecewise deterministic Markov process (PDMP) $X=\{X_t\}_{t\geqslant 0}$ with measure-valued generator $\mathcal{A}$, for which the conditional distribution function of the inter-occurrence time is not necessarily…
We study the survival probability and the first-passage time distribution for a Brownian motion in a planar wedge with infinite absorbing edges. We generalize existing results obtained for wedge angles of the form $\pi/n$ with $n$ a…
We consider the paths of a Gaussian random process $x(t)$, $x(0)=0$ not exceeding a fixed positive level over a large time interval $(0,T)$, $T\gg 1$. The probability $p(T)$ of such event is frequently a regularly varying function at…
The binary information collects all those events that may or may not occur. With this kind of variables, a large amount of information can be captured, in particular, about financial assets and their future trends. In our paper, we assume…
Circular Dyson Brownian motion describes the Brownian dynamics of particles on a circle (periodic boundary conditions), interacting through a logarithmic, long-range two-body potential. Within the log-gas picture of random matrix theory, it…
Diffusion processes driven by Fractional Brownian motion (FBM) have often been considered in modeling stock price dynamics in order to capture the long range dependence of stock price observed in reality. Option prices for such models had…
The Bessel process with parameter $D>1$ and the Dyson model of interacting Brownian motions with coupling constant $\beta >0$ are extended to the processes in which the drift term and the interaction terms are given by the logarithmic…
This paper addresses the question of how Brownian-like motion can arise from the solution of a deterministic differential delay equation. To study this we analytically study the bifurcation properties of an apparently simple differential…
This paper derives a complete analytical solution for the probability distribution of the configuration of a non-holonomic vehicle that moves in two spatial dimensions by satisfying the unicycle kinematic constraints and in presence of…
In this paper we discuss a closed-form approximation of the likelihood functions of an arbitrary diffusion process. The approximation is based on an exponential ansatz of the transition probability for a finite time step $\Delta t$, and a…
We derive closed-form solutions to the optimal stopping problems related to the pricing of perpetual American standard and lookback put and call options in the extensions of the Black-Merton-Scholes model with progressively enlarged…
It is shown that time reversibility of Hamiltonian microscopic dynamics and Gibbs canonical statistical ensemble of initial conditions for it together produce an exact virial expansion for probability distribution of path of molecular…
We present an empirical study of the subordination hypothesis for a stochastic time series of a stock price. The fluctuating rate of trading is identified with the stochastic variance of the stock price, as in the continuous-time random…
In this paper we show how to approximate the transition density of a CARMA(p, q) model driven by means of a time changed Brownian Motion based on the Gauss-Laguerre quadrature. We then provide an analytical formula for option prices when…
We consider arbitrage free valuation of European options in Black-Scholes and Merton markets, where the general structure of the market is known, however the specific parameters are not known. In order to reflect this subjective uncertainty…