ABSTRACT
Condition (b) is referred to by saying that the process has independent increments. It is stronger than the Markov property. It implies that the position of the process at time tn, say, depends on what has happened up to time tn1 < tn only through the position at time tn1 (which is the Markov property) and moreover, the displacement Wtn Wtn1 between tn1 and tn, is independent of the position at time tn1. This enables us to write down the transition probabilities explicitly as
P ( Wtn 6 xn j Wti D xi ; 0 6i 6 n 1
) D P(Wtn Wtn1 6 xn xn1) D ∫ xnxn1 1
.u; tn tn1/du (4.1)
where .x; t/ D ex2=2t=p2t , is the probability density function of the normal distribution with mean 0 and variance t . We may also write down the joint probability density function of Wt1 ; : : : ; Wtn as
f .x1; : : : ; xn/ D n∏ 1
.xi xi1; ti ti1/: (4.2)
Notice that condition (b) is consistent with condition (a) by the property of the normal distribution that the sum of independent random variables each having a normal distribution again has a normal distribution. Conditions (a) and (b) also imply that the process is spatially homogeneous so that the distribution of the increment WtCs Ws does not depend on the position, Ws , at time s for s; t > 0. For any t > s > 0, since EWs D EWt D 0 and Wt Ws is independent of Ws , it follows that the covariance of Ws and Wt is
Cov .Ws;Wt / D E .WsWt / D E ŒWs.Wt Ws CWs/ D E ŒWs.Wt Ws/CE.W 2s / D E.Ws/E.Wt Ws/C s D sI
we then have for any s; t > 0 that the covariance is given by
Cov .Ws;Wt / D s ^ t; (4.3) where s ^ t D min.s; t/. Since the multivariate normal distribution is determined by its means and covariances and normally-distributed random variables are independent if and only if their covariances are zero, it is immediate that, (when 2 D 1) (a) (b) and (c) are equivalent to requiring that for any n > 1 and t1; : : : ; tn, the joint distribution ofWt1 ; : : : ; Wtn is normal with zero means and covariances specified by (4.3). The joint distributions of Wt1 ; : : : ; Wtn for each n > 1 and all t1; : : : ; tn are known as the finite-dimensional distributions of the process.
