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 distributionfeynman_kac_backward(T, t, ψ, f, V)
returns the solution of the PDEu_t(x, t) + T u - V(x, t) u + f(x, t) = 0
withu(x, T) = ψ(x)
Moreoveor,
generator(DiffusionProcess(x, μ, σ))
creates the transition matrix of a diffusive process with driftμ(x)
and volatilityσ(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 ofm
tail_index(f)
returns the tail index of the stationary distribution ofe^m
Moreover,
generator(AdditiveFunctional(DiffusionProcess(x, μ, σ), μm, σm)
creates the functionξ -> T(ξ)
for the additive functional with driftμm(x)
and volatilityσm(x)
Related Packages
- SimpleDifferentialOperators contains more general tools to define operators with different boundary counditions. In contrast, InfinitesimalGenerators always assumes reflecting boundaries.