By Ludwig Arnold
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Np=0. 9) Theorem. 1) hold uniformly in s 44 2. Markov Processes and Diffusion Processes and x and which possesses a transition density p (s, x, t, y). 10) d a 1 d d a2 (l, (t, y) P) - 1 at + E ay, 2 i-I an ay; (b41(t, y) P) = 0. For proof, we again refer to Gikhman and Skorokhod (51, p. 375. If we define the distribution of X,O in terms of the initial probability P,Q, we obtain from p (s, x, t, y) the probability density p (t, y) of X, itself: P (t, y) = f P (to, x, t, y) P O (dx). 10), we see that p (t, y) also satisfies the forward equation.
For every 21(Y)-measurable random variable Z, there exists a measurable function h such that Z = h (Y); that is, the value of Z is fixed by the value of Y at w. This h is uniquely defined up to a set N of images of Y such that P [Y E NI = 0. For the conditional expectation E (X I Y), this means the existence of a measurable function h defined on Q' such that E(XIY)=h(Y). We write suggestively h(y)=E(XIY=y) In the special case of the conditional probability P (X E B I Y), we go on to the conditional distribution p (w, B), for which there is now an almost certainly unique q such that p (w, B) = q (Y (w), B).
7) Example. The transition density of the Wiener process p (s, x, t, y) = (2 it (t-s))-d/2 e-ly-xl212(9-s) is, for fixed t and y, a fundamental solution of the backward equation ap+l as a2p=0. E ax? 8) Remark. If X, is a homogeneous process, then the coefficients f (s, x) I (x) and B (s, x) = B (x) (and hence the operators) are independent of s. np=0. 9) Theorem. 1) hold uniformly in s 44 2. Markov Processes and Diffusion Processes and x and which possesses a transition density p (s, x, t, y). 10) d a 1 d d a2 (l, (t, y) P) - 1 at + E ay, 2 i-I an ay; (b41(t, y) P) = 0.