Markov Processes

For a Markov process defined by a matrix T where T is the operator such that Tf = E[df]

• stationary_distribution(T) returns its stationary distribution
• feynman_kac_backward(T, t, ψ, f, V) returns the solution of the PDE u_t(x, t) + T u - V(x, t) u + f(x, t) = 0 with u(x, T) = ψ(x)

Moreoveor,

• generator(DiffusionProcess(x, μ, σ)) creates the transition matrix of a diffusive process with drift μ(x) and volatility σ(x) with reflecting boundaries.

Additive Functionals

For an additive functional m defined by a function ξ -> T(ξ) where T is the operator such that T f= E[d(e^(ξm)f)]

• cgf(f) returns the long run scaled CGF of m
• tail_index(f) returns the tail index of the stationary distribution of e^m

Moreover,

• generator(AdditiveFunctional(DiffusionProcess(x, μ, σ), μm, σm) creates the function ξ -> T(ξ) for the additive functional with drift μm(x) and volatility σm(x)

Related Packages

• SimpleDifferentialOperators contains more general tools to define operators with different boundary counditions. In contrast, InfinitesimalGenerators always assumes reflecting boundaries.