
294 Modeling and Inverse Problems in the Presence of Uncertainty
the SDE and its pointwise equivalent RDE have the same probability density
function at each time t, these two processes may not be the s ame, as they may
have different mth (m ≥ 2) probability density functions (recall the discussion
in Section 7.1.1). Below we give an example to illustrate this by comparing
the covariance functions of the two stochastic processes resulting from an SDE
and its pointwise equivalent RDE.
Based on the res ults in Section 7.4.2.2 we find that the SDE
dX(t) = b
0
(X(t) + c
0
)dt +
√
2tσ
0
(X(t) + c
0
)dW (t), X(0) = X
0
(7.161)
is p ointwise equivalent to the RDE
dx(t; X