Bridge.jl

ContinuousTimeProcess{T}

Types inheriting from the abstract type ContinuousTimeProcess{T} characterize the properties of a T-valued stochastic process, play a similar role as distribution types like Exponential in the package Distributions.

SamplePath{T} <: AbstractPath{T}

The struct

struct SamplePath{T}
    tt::Vector{Float64}
    yy::Vector{T}
    SamplePath{T}(tt, yy) where {T} = new(tt, yy)
end

serves as container for discretely observed ContinuousTimeProcesses and for the sample path returned by direct and approximate samplers. tt is the vector of the grid points of the observation/simulation and yy is the corresponding vector of states.

It supports getindex, setindex!, length, copy, vcat, start, next, done, endof.

valtype(::ContinuousTimeProcess) -> T

Returns statespace (type) of a ContinuousTimeProcess{T].

ODESolver

Abstract (super-)type for solving methods for ordinary differential equations.

solve!(method, F, X::SamplePath, x0, P) -> X, [err]

Solve ordinary differential equation $(d/dx) x(t) = F(t, x(t), P)$ on the fixed grid X.tt writing into X.yy

method::R3 - using a non-adaptive Ralston (1965) update (order 3).

method::BS3 use non-adaptive Bogacki–Shampine method to give error estimate.

Call _solve! to inline. "Pretty fast if x is a bitstype or a StaticArray."

R3

Ralston (1965) update (order 3 step of the Bogacki–Shampine 1989 method) to solve $y(t + dt) - y(t) = \int_t^{t+dt} F(s, y(s)) ds$.

BS3

Ralston (1965) update (order 3 step of the Bogacki–Shampine 1989 method) to solve $y(t + dt) - y(t) = \int_t^{t+dt} F(s, y(s)) ds$. Uses Bogacki–Shampine method to give error estimate.

sample(tt, P, x1=zero(T))

Sample the process P on the grid tt exactly from its transitionprob(-ability) starting in x1.

sample!(X, P, x1=zero(T))

Sample the process P on the grid X.tt exactly from its transitionprob(-ability) starting in x1 writing into X.yy.

quvar(X)

Computes quadratic variation of X.

bracket(X)
bracket(X,Y)

Computes quadratic variation process of X (of X and Y).

ito(Y, X)

Integrate a valued stochastic process with respect to a stochastic differential.

girsanov(X::SamplePath, P::ContinuousTimeProcess, Pt::ContinuousTimeProcess)

Girsanov log likelihood $\mathrm{d}P/\mathrm{d}Pt(X)$

lp(s, x, t, y, P)

Log-transition density, shorthand for logpdf(transitionprob(s,x,t,P),y).

llikelihood(X::SamplePath, P::ContinuousTimeProcess)

Log-likelihood of observations X using transition density lp.

llikelihood(X::SamplePath, Pº::LocalGammaProcess, P::LocalGammaProcess)

Log-likelihood dPº/dP. (Up to proportionality.)

llikelihood(X::SamplePath, P::LocalGammaProcess)

Bridge log-likelihood with respect to reference measure P.P. (Up to proportionality.)

euler(u, W, P) -> X

Solve stochastic differential equation $dX_t = b(t,X_t)dt + σ(t,X_t)dW_t$ using the Euler scheme.

euler!(Y, u, W, P) -> X

Solve stochastic differential equation $dX_t = b(t,X_t)dt + σ(t,X_t)dW_t$ using the Euler scheme in place.

thetamethod(u, W, P, theta=0.5)

Solve stochastic differential equation using the theta method and Newton-Raphson steps.

GammaProcess

A GammaProcess with jump rate γ and inverse jump size λ has increments Gamma(t*γ, 1/λ) and Levy measure

Here Gamma(α,θ) is the Gamma distribution in julia's parametrization with shape parameter α and scale θ.

 nu(k,P)

(Bin-wise) integral of the Levy measure $\nu(B_k)$.

endpoint!(X::SamplePath, v)

Convenience functions setting the endpoint of X tov`.

inner(x[, y])

Short-hand for quadratic form x'x (or x'y).

cumsum0

Cumulative sum starting at 0,

mat(X::SamplePath{SVector}) 
mat(yy::Vector{SVector})

Reinterpret X or yy to an array without change in memory.

outer(x[, y])

Short-hand for quadratic form xx' (or xy').

CSpline(s, t, x, y = x, m0 = (y-x)/(t-s), m1 = (y-x)/(t-s))

Cubic spline parametrized by $f(s) = x$ and $f(t) = y$, $f'(s) = m_0$, $f'(t) = m_1$.

integrate(cs::CSpline, s, t)

Integrate the cubic spline from s to t.

logpdfnormal(x, A)

logpdf of centered gaussian with covariance A

mcstart(x) -> state

Create state for random chain online statitics. The entries/value of x are ignored

mcnext(state, x) -> state

Update random chain online statistics when new chain value x was observed. Return new state.

mcband(mc)

Compute marginal 95% coverage interval for the chain from normal approximation.

mcmeanband(mc)

Compute marginal confidence interval for the chain mean using normal approximation

LinPro(B, μ::T, σ)

Linear diffusion $dX = B(X - μ)dt + σdW$.

Ptilde(cs::CSpline, σ)

Affine diffusion $dX = cs(t) dt + σdW$ with cs a cubic spline ::CSpline.

GuidedProp

General bridge proposal process

bridge(W, P, scheme! = euler!) -> Y

Integrate with scheme! and set Y[end] = P.v1.

Vs(s, T1, T2, v, P)

Time changed V for generation of U.

mdb(u, W, P)
mdb!(copy(W), u, W, P)

Euler scheme with the diffusion coefficient correction of the modified diffusion bridge.

r(t, x, T, v, P)

Returns $r(t,x) = \operatorname{grad}_x \log p(t,x; T, v)$ where $p$ is the transition density of the process $P$.

gpK!(K::SamplePath, P)

Precompute $K = H^{-1}$ from $(d/dt)K = BK + KB' + a$ for a guided proposal.

LocalGammaProcess

compensator0(kstart, P::LocalGammaProcess)

Compensator of GammaProcess approximating the LocalGammaProcess. For kstart == 1 (only choice) this is $\nu_0([b_1,\infty)$.

compensator(kstart, P::LocalGammaProcess)

Compensator of LocalGammaProcess For kstart = 1, this is $\sum_{k=1}^N \nu(B_k)$, for kstart = 0, this is $\sum_{k=0}^N \nu(B_k) - C$ (where $C$ is a constant).

θ(x, P::LocalGammaProcess)

Inverse jump size compared to gamma process with same alpha and beta.

soft(t, T1, T2)

Time change mapping s in [T1, T2](U-time) tot`in[T1, T2](X`-time).

tofs(s, T1, T2)

Time change mapping t in [T1, T2] ($X$-time) to s in [T1, T2] (U-time).

dotVs (s, T1, T2, v, P)

Time changed time derivative of V for generation of U.