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We introduce an approximation strategy for the discounted moments of a stochastic process that can, for a large class of problems, approximate the true moments. These moments appear in pricing formulas of financial products such as bonds…
An effective method to obtain exact analytical solutions of equations describing the coherent dynamics of multilevel systems is presented. The method is based on the usage of orthogonal polynomials, integral transforms and their discrete…
An ensemble of classical subsystems interacting with surrounding particles has been considered. In general case, a phase volume of the subsystems ensemble was shown to be a function of time. The evolutional equations of the ensemble are…
The two-parameter Macdonald polynomials are a central object of algebraic combinatorics and representation theory. We give a Markov chain on partitions of k with eigenfunctions the coefficients of the Macdonald polynomials when expanded in…
We study linear chance-constrained problems where the coefficients follow a Gaussian mixture distribution. We provide mixed-binary quadratic programs that give inner and outer approximations of the chance constraint based on piecewise…
We analyze a class of nonlinear partial differential equations (PDEs) defined on $\mathbb{R}^d \times \mathcal{P}_2(\mathbb{R}^d),$ where $\mathcal{P}_2(\mathbb{R}^d)$ is the Wasserstein space of probability measures on $\mathbb{R}^d$ with…
The probability distribution describing the state of a Stochastic Reaction Network evolves according to the Chemical Master Equation (CME). It is common to estimated its solution using Monte Carlo methods such as the Stochastic Simulation…
Gaussian mixture distributions are commonly employed to represent general probability distributions. Despite the importance of using Gaussian mixtures for uncertainty estimation, the entropy of a Gaussian mixture cannot be calculated…
The existence and analyticity of solutions to linear systems of moment differential equations with analytic coefficients is studied. The relation of solutions of such systems with respect to linear moment differential equations is…
A widely used approach to describe the dynamics of gene regulatory networks is based on the chemical master equation, which considers probability distributions over all possible combinations of molecular counts. The analysis of such models…
This paper introduces a novel method for approximating the dynamics of a large autonomous system projected onto a fixed subspace. The core contribution is a novel recursive algorithm to construct an effective time-dependent generator that…
We present a technique for constructing suitable posterior probability distributions in situations for which the sampling distribution of the data is not known. This is very useful for modern scientific data analysis in the era of "big…
We propose a family of "exactly solvable" probability distributions to approximate partition functions of two-dimensional statistical mechanics models. While these distributions lie strictly outside the mean-field framework, their free…
The polygonal distributions are a class of distributions that can be defined via the mixture of triangular distributions over the unit interval. The class includes the uniform and trapezoidal distributions, and is an alternative to the beta…
Probabilistic programming is related to a compositional approach to stochastic modeling by switching from discrete to continuous time dynamics. In continuous time, an operator-algebra semantics is available in which processes proceeding in…
The classical problem of moments is addressed by the maximum entropy approach for one-dimensional discrete distributions. The numerical technique of adaptive support approximation is proposed to reconstruct the distributions in the region…
Approximate solutions of the chemical master equation and the chemical Fokker-Planck equation are an important tool in the analysis of biomolecular reaction networks. Previous studies have highlighted a number of problems with the…
A simple and transparent derivation of the formally exact probability distribution for classical non-equilibrium systems is given. The corresponding stochastic, dissipative equations of motion are also derived.
As Gaussian processes are used to answer increasingly complex questions, analytic solutions become scarcer and scarcer. Monte Carlo methods act as a convenient bridge for connecting intractable mathematical expressions with actionable…
We derive a simple expression for the probability of trajectories of a master equation. The expression is particularly useful when the number of states is small and permits the calculation of observables that can be defined as functionals…