Numerical solutions of complex stochastic differential equations driven by real Wiener processes are useful in many situations.
In this talk we will review some basics of Monte-Carlo methods for SDEs, accuracy of weak solutions, and the stability of the simulations. The difficulty for complex valued processes is that they are often very unstable in long-time integrations. Our principal example is the simple linear SDE dX = (a0 + a1 X) dt + (c0 + c1 X) dW(t), in particular the two cases: dX = (1 + i X) dt + dW(t) and dX = (1 + i X) dt + X dW(t). The mean value for both of these processes is E[X] = (X(0) - i) exp(i t) + i, but they have very different distributions. Some other examples, a Bose-Einstein quantum noise problem and processes which become stationary, will be briefly outlined.