Recovering a Family of Two-Dimensional Gaussian Variables from the Minimum Process

Suppose that {(X t, Y t): t>}0 is a family of two independent Gaussian random variables with means m 1(t) and m 2(t) and variances (sigma)2 1(t) and (sigma)2 2(t). If at every time t>0 the first and second moment of the minimum process X t^Y t are known, are the parameters governing these four moment functions uniquely determined ? We answer this question in the negative for a large class of Gaussian families including the "Brownian" case. Except for some degenerate situation where one variance function dominates the other, in which case the recovery of the parameters is fully successful, the second moment of the minimum process does not provide any additional clues on identifying the parameters.