Give a direct proof of the formula (3.5) using the joint probability distribution (left(Wleft(t_{0} ight), W(t), Wleft(t_{1}

Give a direct proof of the formula (3.5) using the joint probability distribution \(\left(W\left(t_{0}\right), W(t), W\left(t_{1}\right)\right)\) of the Brownian motion \(W(t)\).

Data From Formula 3.5

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3.9 Rigorous proofs of (3.5). There are several possibilities to prove Lvy's formula (3.5). First, one could use a direct attack based on the joint law of (Bs, Bt, Bu) and the Ex. 3.9 familiar formula for conditional probabilities. Although this is straightforward, it is a bit tedious. A second possibility is to use the behaviour of Gaussian random variables under conditioning: if (G, I) is an R x Rd-valued Gaussian random variable, then the condi- tional expectation E(G | I) is of the form a + (b, I). Thus, G - E(G | I) is centred and normal, e.g. - N(0, 2). The conditional law is P(Ge dx | I = y) = N(a + (b, y), ), see [235, Theorem 36.4] or [233, Satz 16.11]. Statisticians know this fact from linear models. Let us add yet another proof. Observe that for all 0 = a and n/a = {t. Then E (en W(a) | W(b) = exp [a (1) n] exp [in w(b)] a - = = exp [-ban] exp [in wb)]. (3.6)