Cox process

In probability theory, a Cox process, also known as a doubly stochastic Poisson process is a point process which is a generalization of a Poisson process where the time-dependent intensity is itself a stochastic process. The process is named after the statistician David Cox, who first published the model in 1955. In probability theory, a Cox process, also known as a doubly stochastic Poisson process is a point process which is a generalization of a Poisson process where the time-dependent intensity is itself a stochastic process. The process is named after the statistician David Cox, who first published the model in 1955. Cox processes are used to generate simulations of spike trains (the sequence of action potentials generated by a neuron), and also in financial mathematics where they produce a 'useful framework for modeling prices of financial instruments in which credit risk is a significant factor.' Let ξ {displaystyle xi } be a random measure. A random measure η {displaystyle eta } is called a Cox process directed by ξ {displaystyle xi } , if L ( η ∣ ξ = μ ) {displaystyle {mathcal {L}}(eta mid xi =mu )} is a Poisson process with intensity measure μ {displaystyle mu } . Here, L ( η ∣ ξ = μ ) {displaystyle {mathcal {L}}(eta mid xi =mu )} is the conditional distribution of η {displaystyle eta } , given { ξ = μ } {displaystyle {xi =mu }} . If ξ {displaystyle xi } is a Cox process directed by η {displaystyle eta } , then ξ {displaystyle xi } has the Laplace transform for any positive, measurable function f {displaystyle f} .