Bridge docs

Bridge.md

Bridge.jl

Documentation for Bridge.jl

Important concepts

# Bridge.ContinuousTimeProcess — Type.

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 ``Exponentialin the packageDistributions`.

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# Bridge.SamplePath — Type.

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.

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# Base.valtype — Function.

valtype(::ContinuousTimeProcess) -> T

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

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Ordinary differential equations

solve
solve!
R3
BS3

Brownian motion

Pages = ["/wiener.jl"]

Stochastic differential equations

sample
sample!
quvar
bracket
ito
girsanov
lp
llikelihood

Miscallenious

endpoint!

Unsorted

# Bridge.CSpline — Method.

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'(x) = m0, f'(t) = m1

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# Bridge.GuidedProp — Type.

GuidedProp

General bridge proposal process

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# Bridge.LinPro — Type.

LinPro(B, μ::T, σ)

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

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# Bridge.GammaProcess — Type.

GammaProcess

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

$$ ν(x)=γ x^{-1}\exp(-λ x), $$

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

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# Bridge.LocalGammaProcess — Type.

LocalGammaProcess

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# Bridge.bracket — Method.

bracket(X)
bracket(X,Y)

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

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# Bridge.bridge — Function.

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

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

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# Bridge.euler! — Method.

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.

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# Bridge.euler — Method.

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.

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# Bridge.girsanov — Method.

girsanov{T}(X::SamplePath{T}, P::ContinuousTimeProcess{T}, Pt::ContinuousTimeProcess{T})

Girsanov log likelihood $dP/dPt(X)$

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# Bridge.ito — Method.

ito(Y, X)

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

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# Bridge.llikelihood — Function.

Bridge log-likelihood with respect to reference measure P.P

Up to proportionality

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# Bridge.llikelihood — Function.

Log-likelihood with respect to reference measure P.P

Up to proportionality

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# Bridge.llikelihood — Method.

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

Log-likelihood of observations X using transition density lp

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# Bridge.lp — Method.

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

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

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# Bridge.mcband — Method.

mcband(mc)

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

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# Bridge.mcbandmean — Method.

mcmeanband(mc)

Compute marginal confidence interval for the chain mean using normal approximation

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# Bridge.mcnext — Method.

mcnext(state, x) -> state

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

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# Bridge.mcstart — Method.

mcstart(x) -> state

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

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# Bridge.quvar — Method.

quvar(X)

Computes quadratic variation of X.

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# Bridge.solve! — Method.

solve!(method, 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."

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# Bridge.thetamethod — Function.

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

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

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# StatsBase.sample — Method.

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

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

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# Bridge.Ptilde — Method.

Ptilde(cs::CSpline, σ)

Affine diffusion $dX = cs(t) dt + σdW$ with cs a

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# Bridge.BS3 — Type.

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.

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# Bridge.ODESolver — Type.

ODESolver

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

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# Bridge.R3 — Type.

R3()

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

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# Base.valtype — Method.

valtype(::ContinuousTimeProcess) -> T

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

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# Bridge.Vs — Function.

Vs (s, T1, T2, v, B, beta)

Time changed V for generation of U

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# Bridge.compensator — Method.

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)

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# Bridge.compensator0 — Method.

compensator0(kstart, P::LocalGammaProcess)

Compensator of GammaProcess approximating the LocalGammaProcess. For kstart == 1 (only choice) this is nu([b1,Inf], P0)

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# Bridge.cumsum0 — Method.

cumsum0

Cumulative sum starting at 0,

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# Bridge.dotVs — Function.

dotVs (s, T, v, B, beta)

Time changed time derivative of V for generation of U

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# Bridge.endpoint! — Method.

endpoint!(X::SamplePath, v)

Convenience functions setting the endpoint of X``tov`.

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# Bridge.gpK! — Method.

gpK!(K::SamplePath, P)

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

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# Bridge.integrate — Method.

integrate(cs::CSpline, s, t)

Integrate the cubic spline from s to t

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# Bridge.logpdfnormal — Method.

logpdfnormal(x, A)

logpdf of centered gaussian with covariance A

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# Bridge.mat — Method.

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

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

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# Bridge.mdb! — Method.

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

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

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# Bridge.mdb — Method.

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

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

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# Bridge.nu — Method.

(Bin-wise) integral of the Levy measure

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# Bridge.outer — Method.

outer(x[, y])

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

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# Bridge.r — Method.

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 $P$.

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# Bridge.soft — Function.

soft(t, T1, T2)

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

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# Bridge.tofs — Method.

tofs(s, T1, T2)
soft(t, T1, T2)

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

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# Bridge.θ — Method.

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

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