Stochastic calculus interview questions9/18/2023 ![]() ![]() This result depends on choosing a good version of the stochastic integral, simultaneously for all values of x, which is a bit tricky, so is left until later. Then, there exists processes such thatĪlmost surely, for each t and x, and such that Lemma 4 Let be a measurable space and be uniformly bounded processes satisfying the measurablity requirement of lemma 3. This is the old problem of choosing good versions of stochastic processes except, now, we are concerned with the path as the variable x varies, rather than the time index t. Furthermore, the arbitrary choice of the value of the integral on an uncountable collection of zero probability events, one for each x, could affect the value of the integral over x. Therefore, asking if it is measurable with respect to x does not even make sense. The next technical difficulty in giving a stochastic version of Fubini’s theorem is that if is a bounded predictable process and X is a semimartingale, then the integral As is -measurable, the first part of Fubini’s theorem as stated above says that Proof: It is clear that is bounded, so it only needs to be shown to be predictable. Lemma 3 Let be a finite measure space and be a uniformly bounded collection of processes such that As usual, we work with respect to a filtered probability space. So, we require a slightly stronger measurability condition than in theorem 2, but this is not too difficult. ![]() That is, it should be measurable with respect to the predictable sigma-algebra. First, it is necessary that the integrand is predictable. Generalizing to semimartingales does introduce some technical problems though. The result stated in theorem 2 only applies to FV processes, whereas stochastic integration is defined more generally for semimartingales. The monotone class theorem says that all uniformly bounded satisfying the requirements of the theorem are in, so ( 3) is measurable as stated. By linearity, this is clearly closed under taking linear combinations and, by monotone convergence, is closed under taking limits of uniformly bounded and nonnegative increasing sequences in. So, let denote the collection of all jointly measurable and uniformly bounded functions such that ( 3) has the stated measurability property. Instead, we go back to basics and apply the functional monotone class theorem. Measurability of ( 3) is a bit more tricky, and the dependence of X on stops us from applying Fubini’s theorem as stated above. This is -measurable by the first part of Fubini’s theorem, as required. It only remains to prove measurability of the maps in ( 2) and ( 3), which are slightly stronger statements than that given by our application of Fubini’s theorem here. Hence, ( 4) is simply a restatement of Fubini’s theorem ( 1) with. Proof: For each individual value of, the integral with respect to with s varying over the interval is a finite signed measure. Theorem 2 Let X be an FV process, be a finite measure space, and be a uniformly bounded collection of processes such that I start with the simple case of FV processes, which can be proved as a corollary of Fubini’s theorem. Here, we work with respect to a probability space, and a process is said to be FV if it is cadlag with finite variation over each finite time interval, and locally bounded if it is almost surely bounded over each finite time interval. Alternatively, by monotone convergence, we can extend to sigma-finite measure spaces and nonnegative measurable functions, which need not be bounded.Ī slight reformulation of Fubini’s theorem is useful for applications to stochastic calculus. ![]() By simple linearity, it extends to finite signed measure spaces. There are various straightforward ways in which this base statement can be generalized. Note that the first two statements regarding measurability of the single integrals are necessary to ensure that the double integral ( 1) is well-defined. I previously gave a proof of this as a simple corollary of the functional monotone class theorem. Theorem 1 (Fubini) Let and be finite measure spaces, and be a bounded -measurable function. To start, recall the classical Fubini theorem. Here, I will consider the situation where one integral is of the standard kind with respect to a finite measure, and the other is stochastic. To help with such cases, we could do with a new stochastic version of Fubini’s theorem. However, since these can involve stochastic integration rather than the usual deterministic case, the classical results are not always applicable. In the theory of stochastic calculus, we also encounter double integrals and would like to be able to commute their order. Fubini’s theorem states that, subject to precise conditions, it is possible to switch the order of integration when computing double integrals. ![]()
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