Unconditional mean and variance of multivariate term structure

tigerbill · March 14th, 2011, 4:47 pm

Suppose , where z is a vector of independent brownian motion, b is a lower triangular matrix, a is a vector (let's assume it is a three dimensional model). Since b isn't a diagonal matrix, dx(1), dx(2), ... dx(n) are dependent. How can we calculate the unconditional mean and variance of x then? I got infinite mean and variance, which looks unreasonable.Many thanks in advance for your help.

Alan · March 15th, 2011, 6:52 pm

Since nobody has replied, I will guess:1. You get the unconditional mean by setting the expected drift to 0 => a + b xbar = 0 => xbar = - b^(-1) a, using the matrix inverse.2. You get the unconditional variance by googling 'vector OU process', finding the sde solution, and thenrepeating how you would do it for the scalar process.