Can I apply the standard Runge Kutta 4th order method to the Langevin Equation? If I have a Langevin Equation with an external force term (which may be time dependent), is it possible for me to apply the standard 4th order Runge Kutta algortihm to solve it numerically?
Edit: 
I would like to mention the use case for the question asked! I am trying to simulate the motion of a Brownian Particle, which is Harmonically bounded but driven with a sinusoidal force (apart from the noise and the friction)!
 A: You can, but you need to be careful.
The external force is a probabilistic function, but in the simulation you need to make sure you only sample one function value for any particular time, and use that consistently. In RK4 you do several sub-steps for each time step, and if you use different a "random" value of the force at the same time in different substeps, either the solution won't converge at all, or at best you will not be using the probability function that you intended to use.
Also, be aware that using finite size time steps is equivalent to using a low-pass filter on your probabilistic forcing function. You need to consider whether or not that is important for what you are trying to do.
It might be better to derive the Fokker-Planck equation for your system and solve that numerically for the probability distribution of the system behaviour, rather than looking at individual solutions from the Langevin equation with different (random) external forces. In any case, this would be a good check that your numerical calculations using random forcing are in fact using the correct probability distribution for the randomness!
A: No, you cannot directly apply a deterministic method such as 4th order Runge-Kutta to the integration of stochastic differential equations, in general. This is only possible, in the way described by alephzero, if the stochastic Langevin force can be considered a very weak perturbation compared to the deterministic part of the dynamics. The reason for this is that the impulse generated by the stochastic force changes at each time step by a Wiener increment ${\rm d}W \sim {\rm d} t^{1/2}$, where ${\rm d} t$ is the time step. So, a first-order method to integrate a stochastic differential equation needs to include terms of second order in ${\rm d}W$, or more precisely in $\int_t^{t+{\rm d}t}{\rm d} W(t)$, which can be achieved with the Milstein algorithm. A fourth order method would need such terms at eighth order, which becomes extremely complicated (see Kloeden and Platen). If possible, integrating the equivalent Fokker-Planck equation is probably preferable. 
