# Example SDE models provided by msde

#### 2019-01-11

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 (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").

## Bivariate Ornstein-Uhlenbeck process

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").

## 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 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").

## Prokaryotic autoregulatory gene network model

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").