Theory of Statistics - George Mason University

764 0 Statistical Mathematics
Multidimensional Wiener Processes
If we have two Wiener processes B1 and B2, with V(dB1) = V(dB2) = dt
and cov(dB1, dB2) = ρdt (that is, Cor(dB1, dB2) = ρ), then by a similar
argument as before, we have dB1dB2 = ρdt, almost surely.
Again, this is deterministic.
The results of course extend to any vector of Wiener processes (B1, . . ., Bd).
If (B1, . . ., Bd) arise from
√
∆Bi = Xi ∆t,
where the vector of Xs has a multivariate normal distribution with mean
0 and variance-covariance matrix Σ, then the variance-covariance matrix of
(dB1, . . ., dBd) is Σdt, which is deterministic.
iid
Starting with (Z1, . . ., Zd ∼ N(0, 1) and forming the Wiener processes
B = (B1, . . ., Bd) beginning with
√
∆Bi = Zi ∆t,
we can form a vector of Wiener processes B = (B1, . . ., Bd) with variancecovariance
matrix Σdt for dB = (dB1, . . ., dBd) by the transformation
or equivalently by
B = Σ 1/2 B,
B = ΣCB,
where ΣC is a Cholesky factor of Σ, that is, Σ T C ΣC = Σ.
Recall, for a fixed matrix A,
so from above, for example,
V(AY ) = A T V(Y )A,
V(dB) = Σ T CV(dB)ΣC = Σ T Cdiag(dt)ΣC = Σdt. (0.2.13)
The stochastic differentials such as dB naturally lead us to consider integration
with respect to stochastic differentials, that is, stochastic integrals.
Stochastic Integrals with Respect to Wiener Processes
If B is a Wiener process on [0, T], we may be interested in an integral of the
form T
g(Y (t), t)dB,
0
where Y (t) is a stochastic process (that is, Y is a random variable) and g
is some function. First, however, we must develop a definition of such an
integral. We will return to this problem in Section 0.2.2. Before doing that,
let us consider some generalizations of the Wiener process.
Theory of Statistics c○2000–2013 James E. Gentle