Stochastic Differential Equations have the form: \begin{equation} dX = f(t,X) \, dt + g(t,X) \, dW \end{equation} Where the regular ODE is given by $f(t,X) \, dt$ and the noise is given by $g(t,X) \, dW$. Please note that $f(t,X)$ and $dW$ are vectors and $g(t,X)$ is a matrix. In general the noise terms are independent stochastic processes with mean 0, i.e., we have one stochastic variable for each integration variable.
Let us look at a chemical reaction system in which the deterministic part is given by: \begin{equation} f(t,X) = N\, v(t,X) \end{equation} where $N$ is the $m$ x $n$ stoichiometry matrix, $m$ the number of species and $n$ the number of reactions.
To assure that the stochastic term does not violate mass conservation the random variables may no longer be independent. We introduce the following stochastic term where $\tilde{g}(t,X)$ is a $n$ x $n$ matrix of real valued functions and thus $\tilde{g}(t,X) \, d\widetilde{W}$ is a vector of dimension $n$. \begin{equation} g(t,X) \, dW = N \, \tilde{g}(t,X) \, d\widetilde{W} \end{equation} Here $\tilde{g}(t,X) \, d\widetilde{W}$ are independent stochastic processes with mean 0. Since the linear combination of independent mean 0 processes leads to a mean 0 process $g(t,X) \, dW$ has still mean 0 as required.
This leads us to the following SDE for chemical reaction systems which obviously does not violate the mass conservation. \begin{equation} dX = N \left( v(t,X) \, dt + \tilde{g}(t,X) \, d\widetilde{W} \right) \end{equation}
The above SDE for a chemical reaction system can be augmented with additional variables Y which are also determined by stochastic differential equations leading to: \begin{equation} \begin{pmatrix}dX\\dY\end{pmatrix} = \begin{pmatrix}N \, v(t,X,Y) \, dt + N \,\tilde{g}(t,X,Y) \, d\widetilde{W}\\ f(t,X,Y) \, dt + g(t,X,Y) \, dW\end{pmatrix} \end{equation} Equation (5) is the form to which any model represented in COPASI can be transformed.
To fully specify the stochastic differential equation we need to define the functions $\tilde{g}(t,X,Y)$ and $g(t,X,Y)$. As a default we choose: \begin{equation} \begin{matrix} \tilde{g}_{i,j}(t,X,Y) & = & \begin{cases} k_{i} \, \mbox{sign}(v_{i}(t,X,Y)) \, \sqrt{|v_{i}(t,X,Y)|} & \mbox{for } i = j \\ 0 & \mbox{otherwise} \end{cases} \\ g_{i}(t,X,Y) & = & l_{i} \, \mbox{sign}(f_{i}(t,X,Y)) \, \sqrt{|f_{i}(t,X,Y)|} \end{matrix} \end{equation} Please note the above choice is the Itô SDE representation of the Chemical Langevin Equation (CLE) if we choose $k_i = 1$ in equation (6). The $k_i$ can be interpreted as the standard variation of the reaction rate $v_i$ and $l_i$ as the standard variation of the rate of the variable $Y_i$. The user will be able to override the default chosen above. Please note that the user will not be able to create arbitrary noise terms. The matrix $g(t,X)$ will always be of diagonal form.
Contrary to ODEs it is possible for SDEs to generate negative values for species or compartment sizes. We will use the root finding mechanism which is necessary for processing events to detect negative values. The step size will be reduced so that no negative (within tolerance) values will be created.
