# Stability and accuracy of simulations of complex Ito stochastic differential equations

GFM seminar

IIIUL, Room B3-01

2014-07-18 14:30
2014-07-18 15:30
2014-07-18
14:30
..
15:30

Wesley Petersen (ETH Zurich)

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.

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.