Do Stochastic Differential Equation models conserve energy? I have recently started looking into stochastic models of chemical reaction systems, particularly the Chemical Langevin Equation (CLE) SDE model (e.g. here). One thing I'm trying to understand is whether a stochastic model is able to conserve energy at all points in time.
For example, take the reversible reaction $ A \rightleftharpoons B$. The CLE-SDE model is:
$$
\begin{align}
dA(t) =& \left(- k_1 A\left( t \right) + k_2 B\left( t \right)\right)dt - \sqrt{k_1 A\left( t \right)}dW_1(t)  + \sqrt{k_2 B\left( t \right)}dW_2(t) \\
dB(t) =& \left(k_1 A\left( t \right) - k_2 B\left( t \right)\right)dt + \sqrt{k_1 A\left( t \right)}dW_1(t) - \sqrt{k_2 B\left( t \right)}dW_2(t) 
\end{align}
$$
The (deterministic) rate of change of each species is determined by the chemical potential of the species $A$ and $B$ at time $t$ and the reaction rate parameters $k_1$ and $k_2$.
From what I understand (and I could be wrong here), the noise terms represent the randomness of a reaction occurring, with the noise $dW(t)$ coming from the thermal energy in the system. In the SDE model, this will mean the rate of change of a species will be slightly "faster" or "slower" than the deterministic rate due to more (or less) energy.
My question is this: Does this model conserve energy at all points in time? Alternatively, how do stochastic models like this deal with energy conservation?
To me, it seems like energy is not conserved, because at a random point in time there will be more (or less) species produced and energy spent than expected. As I understand it $dW(t)$ is normally distributed with mean zero, so over a long enough timespan the noise, and therefore the extra energy, will on average be zero. Yet this still seems like it is violating energy conservation somehow, but I'm not sure how exactly. Is my intuition right here?
 A: Physically motivated Langevin equations usually contain dissipative terms alongside the noise terms, whose parameters are related via the fluctuation-dissipation theorem (of which the Einstein relation between the diffusion coefficient and the mobility is an example).
From purely mathematical viewpoint, energy conservation means existence of the first integral for the differential equation in question - for a general stochastic equation such an integral does not exist. However, Stochastic equations are often derived by reduction of a more complex system, where the subsystem of interest interacts with a bath - in such a setting the total energy of the subsystem of interest and the bath is usually conserved.
A: 
My question is this: Does this model conserve energy at all points in time? Alternatively, how do stochastic models like this deal with energy conservation?

You actually answer this question in the paragraph right before asking it:

From what I understand (and I could be wrong here), the noise terms represent the randomness of a reaction occurring, with the noise $()$ coming from the thermal energy in the system.

This is precisely the case: the noise terms come from the process interacting with the background, hence the "system" is the CLE + background which means that, when considering both of these as the total system, energy is indeed conserved.
