SDE

Miscellaneous

SDE.syl(a, b, c)

Solves the Sylvester equation AX + XB = C, where C is symmetric and A and -B have no common eigenvalues using (inefficient) algebraic approach via the Kronecker product, see http://en.wikipedia.org/wiki/Sylvester_equation

Stochastic Processes

SDE.mu(t, x, T, P)

Expectation \(E_(t,x)(X_{T})\)

SDE.K(t, T, P)

Covariance matrix \(Cov(X_{T}-x_t)\)

SDE.H(t, T, P)

Negative Hessian of \(\log p(t,x; T, v)\) as a function of x.

SDE.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.

SDE.bstar(t, x, T, v, P::MvPro)

Returns the drift function of a vector linear process bridge which end at time T in point v.

SDE.bcirc(t, x, T, v, Pt::Union(MvLinPro, MvAffPro), P::MvPro)

Drift for guided proposal derived from a vector linear process bridge which end at time T in point v.

SDE.lp(t, x, T, y, P)

Returns \(log p(t,x; T, y)\), the log transition density of the process P

SDE.samplep(t, x, T, P)

Samples from the transition density of the process P.

SDE.exact(u, tt, P)

Simulate process P starting in u on a discrete grid tt from its transition probability.

SDE.ll(X, P)

Compute log likelihood evaluated in B, beta and Lyapunov matrix lambda for a observed linear process on a discrete grid dt from its transition density.

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

Returns \(log p(t,x; T, y)\), the log transition density

SDE.euler(u, W::CTPath, P::CTPro)

Multivariate euler scheme for U, starting in u using the same time grid as the underlying Wiener process W.

SDE.llikeliXcirc(t, T, Xcirc, b, a, B, beta, lambda)

Loglikelihood (log weights) of Xcirc with respect to Xstar.

t, T – timespan Xcirc – bridge proposal (drift Bcirc and diffusion coefficient sigma) b, sigma – diffusion coefficient sigma target B, beta – drift b(x) = Bx + beta of Xtilde lambda – solution of the lyapunov equation for Xtilde

SDE.tofs(s, tmin, T)

SDE.soft(t, tmin, T)

Time change mapping s in [0, T=t_2 - t_1] (U-time) to t in [t_1, t_2] (X-time), and inverse.

SDE.Vs(s, T, v, B, beta)

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

Time changed V and time changed time derivative of V for generation of U

SDE.XofU(UU, tmin, T, v, P)

U is the scaled and time changed process

U(s)= exp(s/2.)*(v(s) - X(tofs(s)))

XofU transforms entire process U sampled at time points ss to X at tt.

SDE.stable(Y, d, ep)

Return real stable d-dim matrix with real eigenvalues smaller than -ep parametrized with a vector of length d*d,

For maximum likelihood estimation we need to search the maximum over all stable matrices. These are matrices with eigenvalues with strictly negative real parts. We obtain a dxd stable matrix as difference of a antisymmetric matrix and a positive definite matrix.