Suppose you have the following system of SDEs for two state variables X_1(t) and X_2(t), for t >= 0 :dX_1 = a_1 ( b_1 – c_11 X_ 1 – c_21 X_2 ) dt + \sigma_1 dW_1dX_2 = a_2 ( b_2 – c_12 X_ 1) dt + \sigma_2 dW_2X_1(0) and X_2(0) are known.X_1 and X_2 are state variables, W_1 and W_2 are independent standard Brownian motions, a_1, a_2, b_1, b_2, c_11, c_21, c_12, \sigma_1, \sigma_2 are, at most, deterministic functions of t.Now clearly, if c_12 = 0 and c_21 = 0, it is trivial to solve for X_1(t) and X_2(t) in terms of X_1(0) and X_2(0) because then X_1 is an OU process and X_2 is simply Brownian motion with drift. What about the general case. It seems to me that X_1(t) and X_2(t) are (I think) Gaussian and so it should be possible to solve this system.My aim is to solve this system so that I can do long-step Monte Carlo. Does anyone know if this system has a solution and, if so, what it is, please?Many thanks for any help.

I suppose it depends what you mean by 'solve' this solution and whether you want a qualitative (aka closed solution) or quantitative result. The risk is that you may not find a closed form or it might take forever. Have you tried to diagonalise the system?

I am looking for a closed form solution in the sense that you would have a closed form solution in the special case that c_12 = 0 and c_21 = 0.ie in this special case you can say, X_1(t) and X_2(t), conditional on their time zero values, are Gaussian with known mean and variance. You can then simulate X_1(t) and X_2(t) with long-steps (no need for Euler or anything like that) in the obvious way.The aim is more or less the same. I am thinking that, even in the general case, X_1(t) and X_2(t), conditional on their time zero values, are Gaussian. However, I don't know for sure and even if I knew they were Gaussian, I still don't know how to compute the mean and variance.

It looks like that you first need to consider the homogeneous system when both sigma equal to 0. The general solution will depend on constants. Then apply the variation constants method to get a solution of your system. This would be the general approach stemmed from deterministic Dif Eq. I do not know about Monte Carlo for the systems but it might work.

It depends on what you mean by closed form solution. In this linear system of SDE, X_{1,2} can be expressed in terms of integrals.