Ito's formula

1. In (6.6) one should change k by i.

2. On page 360. one should say that on the random interval up to the stopping time the Wiener process remains in a compact subset of U and not only in a bounded subset. The problem is that the function f and its derivative are unbounded in the origin.

3. On page 368 one can notice that for every t the process has a Gaussian distribution. The same argument is valid for the restarted processes as well, so every increment has a Gaussian distribution. As the process has independent increments the whole process is a Gaussian process, that is every finite distribution of the process is Gaussian. On the other hand of course there are uncorrelated Gaussian variables which are not independent. To have the independence one needs, of course, that the joint distribution is Gaussian. Using Ito’s formula and the assumption of Lévy’s theorem one can show as in the other cases above that the conditional Fourier transform of the increments are deterministic and the conditional increments are also Gaussian. To do this one needs to take the conditional expectation and not the real expectation in the previous calculations. Using this one can easily show that if the quadratic variation is just t then the joint distributions are Gaussian, so the process is a Wiener process with respect to the filtration generated by the process.

4. On page 385 Example 6.36 one should use the fact that if there is an additive measure on an algebra which is regular with respect to a sigma compact system then it is sigma additive on the algebra therefore this measure has a sigma additive extension to the generated sigma algebra. One should observe that the cilinder sets which are compact when restricted to C([0,t]) form a sigma compact system of sets in the space of continuous functions over the real line. The reason for this is that these cilinder sets are closed in the greater space, so one can use the usual diagonal procedure to find a continuous function in the intersection. If every finite intersection is non-empty one can assume that the sets are decreasing and one can select a function from every finite intersection. If the set of endpoints of the supporting time intervals is bounded then we can assume that after the supremum of the right endpoints all the selected functions are constant.

5. On page 394. in Theorem 6.46 one should assume that U contains the left limits X(t-) as well.

6. On page 430. in formula (6.59) there is a latex error. One should delete [4pt].

7. On page 448. in the proof of Proposition 6.94 one can use Fubini’s theorem as w is continuous in the time variable, and as it is adapted it is product measurable so the set in the double integral is product measurable. The same argument is valid for every level set of w.

 
ch6.txt · Utolsó módosítás: 2012/06/19 16:39 szerkesztette: medvegyev
 
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