Example SDE models provided by `msde`
In msde: Bayesian Inference for Multivariate Stochastic Differential Equations

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This vignette contains a complete description of the sample models found in msde::sde.examples().

Heston's stochastic volatility model

Let $S_t$ denote the value of a financial asset at time $t$. Heston's stochastic volatility model [@heston93] is given by the pair of stochastic differential equations $$ \begin{split} \ud S_t & = \alpha S_t\ud t + V_t^{1/2}S_t\ud B_{1t} \ \ud V_t & = -\gamma(V_t - \mu)\ud t + \sigma V_t^{1/2} \ud B_{2t}, \end{split} $$ where $V_t$ is a latent stochastic volatility process, and $B_{1t}$ and $B_{2t}$ are Brownian motions with $\cor(B_{1t}, B_{2t}) = \rho$. 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 $$ \begin{split} \ud X_t & = (\alpha - \tfrac 1 8 Z_t^2)\ud t + \tfrac 1 2 Z_t \ud B_{1t} \ \ud Z_t & = (\beta/Z_t - \tfrac \gamma 2 Z_t)\ud t + \sigma \ud B_{2t}, \end{split} $$ with $\cor(B_{1t}, B_{2t}) = \rho$. Thus the diffusion function on the variance scale is $$ \df_\tth(\Y_t) = \begin{bmatrix} \tfrac 1 4 Z_t^2 & \tfrac \sigma 2 Z_t \ \tfrac \sigma 2 Z_t & \sigma^2 \end{bmatrix}, $$ where $\Y_t = (X_t, Z_t)$ and $\tth = (\alpha, \gamma, \beta, \sigma, \rho)$. 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").

Bivariate Ornstein-Uhlenbeck process

This model for $\Y_t = (Y_{1t}, Y_{2t})$ is given by $$ \ud \Y_t = (\GGam \Y_t + \LLam)\ud t + \PPsi \ud \bm{B}t, $$ where $\GGam$ is a $2\times 2$ matrix, $\LLam$ is a $2 \times 1$ vector, and $\PPsi$ is a $2\times 2$ upper Choleski factor. The model parameters are thus $\tth = (\Gamma{11}, \Gamma_{21}, \Gamma_{12}, \Gamma_{22}, \Lambda_{1}, \Lambda_2, \Psi_{11}, \Psi_{21}, \Psi_{22})$, and the model restrictions are $\Psi_{11}, \Psi_{22} > 0$. This model is contained in sde.examples(model = "biou").

Lotka-Volterra predator-prey model

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-wilkinson10]: $$ \begin{bmatrix} \mathrm{d} H_t \ \mathrm{d} L_t \end{bmatrix} = \begin{bmatrix} \a H_t - \b H_tL_t \ \b H_tL_t - \g L_t \end{bmatrix}\, \mathrm{d} t + \begin{bmatrix} \a H_t + \b H_tL_t & -\b H_tL_t \ -\b H_tL_t & \b H_tL_t + \g L_t\end{bmatrix}^{1/2} \begin{bmatrix} \mathrm{d} B_{1t} \ \mathrm{d} B_{2t} \end{bmatrix}. $$ The data and parameters are all restricted to be positive. This model is contained in sde.examples(model = "lotvol").

Prokaryotic autoregulatory gene network model

Let $\Y_t = (R_t, P_t, Q_t, D_t)$ 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.wilkinson05 define an SDE describing the dynamics of $\Y_t$ with drift and (variance-scale) diffusion functions $$ \begin{split} \dr_\tth(\Y_t) & = \begin{bmatrix} \gamma_3 D_t - \gamma_7 R_t \ 2 \gamma_6 Q_t - \gamma_8P_t + \gamma_4 R_t -\gamma_5 P_t(P_t-1) \ \gamma_2(10-D_t) - \gamma_1 D_t Q_t - \gamma_6 Q_t + \tfrac 1 2 \gamma_5 P_t(P_t-1) \ \gamma_2(10-D_t) - \gamma_1 D_t Q_t \end{bmatrix} \ \df_\tth(\Y_t) & = \begin{bmatrix} \gamma_3 D_t + \gamma_7 R_t & 0 & 0 & 0 \ 0 & \gamma_8P_t + 4\gamma_6 Q_t + \gamma_4 R_t + 2 \gamma_5 P_t(P_t-1) & -2 \gamma_6 Q_t - \gamma_5 P_t(P_t-1) & 0 \ 0 & -2 \gamma_6 Q_t - \gamma_5 P_t(P_t-1) & A + \gamma_6 Q_t + \tfrac 1 2 \gamma_5 P_t(P_t-1) & A_t \ 0 & 0 & A_t & A_t \end{bmatrix}, \end{split} $$ where $A_t = \gamma_1D_tQ_t + \gamma_2(10-D_t)$ and $\tth = (\theta_1, \ldots, \theta_8)$, $\theta_i = \log(\gamma_i)$, are various reaction rates. The data and parameter restrictions for this model are $\tth \in \mathbb R^8$, $\Y_t > 1$, and $D_t < 10$. This model is contained in sde.examples(model = "pgnet").