msdeThis vignette contains a complete
description of the sample models found in
msde::sde.examples().
Let \(S_t\) denote the value of a
financial asset at time \(t\). Heston’s
stochastic volatility model (Heston 1993) is given by the pair of
stochastic differential equations \[
\begin{split}
\mathrm{d}S_t & = \alpha S_t\mathrm{d}t +
V_t^{1/2}S_t\mathrm{d}B_{1t} \\
\mathrm{d}V_t & = -\gamma(V_t - \mu)\mathrm{d}t + \sigma V_t^{1/2}
\mathrm{d}B_{2t},
\end{split}
\] where \(V_t\) is a latent
stochastic volatility process, and \(B_{1t}\) and \(B_{2t}\) are Brownian motions with \(\mathrm{cor}(B_{1t}, B_{2t}) = \rho\). To
improve the accuracy of the numerical discretization scheme used for
inference, the variables are transformed to \(X_t = \log(S_t)\) and \(Z_t = 2 V_t^{1/2}\), for which Heston’s SDE
becomes \[
\begin{split}
\mathrm{d}X_t & = (\alpha - \tfrac 1 8 Z_t^2)\mathrm{d}t + \tfrac 1
2 Z_t \mathrm{d}B_{1t} \\
\mathrm{d}Z_t & = (\beta/Z_t - \tfrac \gamma 2 Z_t)\mathrm{d}t +
\sigma \mathrm{d}B_{2t},
\end{split}
\] with \(\mathrm{cor}(B_{1t}, B_{2t})
= \rho\). Thus the diffusion function on the variance scale is
\[
\boldsymbol{\Sigma}_\boldsymbol{\theta}(\boldsymbol{Y}_t) =
\begin{bmatrix} \tfrac 1 4 Z_t^2 & \tfrac \sigma 2 Z_t \\ \tfrac
\sigma 2 Z_t & \sigma^2 \end{bmatrix},
\] where \(\boldsymbol{Y}_t = (X_t,
Z_t)\) and \(\boldsymbol{\theta}=
(\alpha, \gamma, \beta, \sigma, \rho)\). The data and parameter
restrictions are \(Z_t, \gamma, \sigma >
0\), \(|\rho| < 1\), and
\(\beta > \tfrac 1 2 \sigma^2\),
with the final restriction ensuring that \(Z_t
> 0\) with probability 1. This model is contained in
sde.examples(model = "hest").
This model for \(\boldsymbol{Y}_t =
(Y_{1t}, Y_{2t})\) is given by \[
\mathrm{d}\boldsymbol{Y}_t = (\boldsymbol{\Gamma}\boldsymbol{Y}_t +
\boldsymbol{\Lambda})\mathrm{d}t +
\boldsymbol{\Psi}\mathrm{d}\boldsymbol{B}_t,
\] where \(\boldsymbol{\Gamma}\)
is a \(2\times 2\) matrix, \(\boldsymbol{\Lambda}\) is a \(2 \times 1\) vector, and \(\boldsymbol{\Psi}\) is a \(2\times 2\) upper Choleski factor. The
model parameters are thus \(\boldsymbol{\theta}= (\Gamma_{11}, \Gamma_{21},
\Gamma_{12}, \Gamma_{22}, \Lambda_{1}, \Lambda_2, \Psi_{11}, \Psi_{21},
\Psi_{22})\), and the model restrictions are \(\Psi_{11}, \Psi_{22} > 0\). This model
is contained in sde.examples(model = "biou").
Let \(H_t\) and \(L_t\) denote the number of Hare and Lynx at
time \(t\) coexisting in a given
habitat. The Lotka-Volterra SDE describing the interactions between
these two animal populations is given by (Golightly and Wilkinson
2010): \[
\begin{bmatrix} \mathrm{d} H_t \\ \mathrm{d} L_t \end{bmatrix} =
\begin{bmatrix} \alpha H_t - \beta H_tL_t \\ \beta H_tL_t - \gamma L_t
\end{bmatrix}\, \mathrm{d} t + \begin{bmatrix} \alpha H_t + \beta H_tL_t
& -\beta H_tL_t \\ -\beta H_tL_t & \beta H_tL_t + \gamma
L_t\end{bmatrix}^{1/2} \begin{bmatrix} \mathrm{d} B_{1t} \\ \mathrm{d}
B_{2t} \end{bmatrix}.
\] The data and parameters are all restricted to be positive.
This model is contained in
sde.examples(model = "lotvol").
Let \(\boldsymbol{Y}_t = (R_t, P_t, Q_t,
D_t)\) denote the number of molecules at time \(t\) of four different compounds in an
autoregulatory gene network: RNA (\(R\)); a functional protein (\(P\)); protein dimmers (\(Q\)); and DNA (\(D\)). Then Golightly
and Wilkinson (2005)
define an SDE describing the dynamics of \(\boldsymbol{Y}_t\) with drift and
(variance-scale) diffusion functions \[
\begin{split}
\boldsymbol{\Lambda}_\boldsymbol{\theta}(\boldsymbol{Y}_t) & =
\begin{bmatrix}
\gamma_3 D_t - \gamma_7 R_t \\
2 \gamma_6 Q_t - \gamma_8P_t + \gamma_4 R_t -\gamma_5 P_t(P_t-1) \\
\gamma_2(10-D_t) - \gamma_1 D_t Q_t - \gamma_6 Q_t + \tfrac 1 2 \gamma_5
P_t(P_t-1) \\
\gamma_2(10-D_t) - \gamma_1 D_t Q_t
\end{bmatrix} \\
\boldsymbol{\Sigma}_\boldsymbol{\theta}(\boldsymbol{Y}_t) & =
\begin{bmatrix}
\gamma_3 D_t + \gamma_7 R_t & 0 & 0 & 0 \\
0 & \gamma_8P_t + 4\gamma_6 Q_t + \gamma_4 R_t + 2 \gamma_5
P_t(P_t-1) & -2 \gamma_6 Q_t - \gamma_5 P_t(P_t-1) & 0 \\
0 & -2 \gamma_6 Q_t - \gamma_5 P_t(P_t-1) & A + \gamma_6 Q_t +
\tfrac 1 2 \gamma_5 P_t(P_t-1) & A_t \\
0 & 0 & A_t & A_t
\end{bmatrix},
\end{split}
\] where \(A_t = \gamma_1D_tQ_t +
\gamma_2(10-D_t)\) and \(\boldsymbol{\theta}= (\theta_1, \ldots,
\theta_8)\), \(\theta_i =
\log(\gamma_i)\), are various reaction rates. The data and
parameter restrictions for this model are \(\boldsymbol{\theta}\in \mathbb R^8\), \(\boldsymbol{Y}_t > 1\), and \(D_t < 10\). This model is contained in
sde.examples(model = "pgnet").