msdeThis vignette contains a complete
description of the sample models found in
msde::sde.examples().
Let St
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 Vt is a latent
stochastic volatility process, and B1t and B2t are Brownian
motions with cor(B1t, B2t) = ρ.
To improve the accuracy of the numerical discretization scheme used for
inference, the variables are transformed to Xt = log (St)
and Zt = 2Vt1/2,
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 cor(B1t, B2t) = ρ.
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 Yt = (Xt, Zt)
and θ = (α, γ, β, σ, ρ).
The data and parameter restrictions are Zt, γ, σ > 0,
|ρ| < 1, and $\beta > \tfrac 1 2 \sigma^2$, with the
final restriction ensuring that Zt > 0 with
probability 1. This model is contained in
sde.examples(model = "hest").
This model for Yt = (Y1t, Y2t)
is given by dYt = (ΓYt + Λ)dt + ΨdBt,
where Γ is a 2 × 2 matrix, Λ is a 2 × 1 vector, and Ψ is a 2 × 2 upper Choleski factor. The model
parameters are thus θ = (Γ11, Γ21, Γ12, Γ22, Λ1, Λ2, Ψ11, Ψ21, Ψ22),
and the model restrictions are Ψ11, Ψ22 > 0.
This model is contained in
sde.examples(model = "biou").
Let Ht
and Lt
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 Yt = (Rt, Pt, Qt, Dt)
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 Yt 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 At = γ1DtQt + γ2(10 − Dt)
and θ = (θ1, …, θ8),
θi = log (γi),
are various reaction rates. The data and parameter restrictions for this
model are θ ∈ ℝ8, Yt > 1,
and Dt < 10. This
model is contained in sde.examples(model = "pgnet").