In the previous two posts I gave a definition of stochastic integration. This was achieved via an explicit expression for elementary integrands, and extended to all bounded predictable integrands by bounded convergence in probability. The extension to unbounded integrands was done using dominated convergence in probability. Similarly, semimartingales were defined as those cadlag adapted processes for which such an integral exists.
The current post will show how the basic properties of stochastic integration follow from this definition. First, if is a cadlag process whose sample paths are almost surely of finite variation over an interval , then can be interpreted as a Lebesgue-Stieltjes integral on the sample paths. If the process is also adapted, then it will be a semimartingale and the stochastic integral can be used. Fortunately, these two definitions of integration do agree with each other. The term FV process is used to refer to such cadlag adapted processes which are almost surely of finite variation over all bounded time intervals. The notation represents the Lebesgue-Stieltjes integral of with respect to the variation of . Then, the condition for to be -integrable in the Lebesgue-Stieltjes sense is precisely that this integral is finite.
Lemma 1 Every FV process is a semimartingale. Furthermore, let be a predictable process satisfying
almost surely, for each . Then, and the stochastic integral agrees with the Lebesgue-Stieltjes integral, with probability one.
Proof: First, for bounded predictable integrands, the stochastic integral can simply be defined as the Lebesgue-Stieltjes integral on the sample paths. This clearly agrees with the explicit expression for elementary integrands and, by the bounded convergence theorem, satisfies bounded convergence in probability, as required.
Now, suppose that the predictable process satisfies (1), so it is integrable with respect to in the Lebesgue-Stieltjes sense. If is a sequence of bounded predictable processes tending to a limit then, by the dominated convergence theorem for Lebesgue integration, the following limit holds
with probability one and, therefore, also under convergence in probability. Choosing shows that . Then, choosing shows that the Lebesgue-Stieltjes integral agrees with the stochastic integral. ⬜
Next, associativity of integration can be shown. This is easiest to understand in the differential form, in which case, equation (2) below simply says that .
Theorem 2 (Associativity) Suppose that for a semimartingale and -integrable process . Then, is a semimartingale and a predictable process is -integrable if and only if is -integrable, in which case
Proof: That is a semimartingale, and equation (2) is satisfied for bounded has already been shown in the previous post, in the proof of existence of cadlag versions of integrals. Now, suppose that , and choose a sequence of bounded predictable processes tending to a limit . As is dominated by the -integrable process ,
in probability, as . Taking gives zero for the right hand side so, by definition, . Then, taking , dominated convergence in probability shows that the left hand side tends to , and equation (2) follows.
Conversely, suppose that and let be a sequence of bounded predictable processes satisfying . Writing , the above argument shows that and
By dominated convergence, this tends to zero in probability as . So, as required. ⬜
Note that, as Theorem 2 gives an `if and only if’ condition for to be -integrable, the definition of -integrable processes as good dominators given in these notes is precisely the correct set of processes to make this theorem hold. In fact, noting that and are bounded for any process , associativity gives the following alternative criterion for -integrability and definition of stochastic integration for unbounded integrands.
Corollary 3 Let be a semimartingale and be a predictable process. Then, is -integrable if and only if there exists a semimartingale satisfying and
in which case .
Proof: First, if and then equation (3) follows from associativity of the stochastic integral. Conversely, suppose that is a semimartingale satisfying (3) and that . Letting equal the integral on the left hand side of (3), associativity of integration shows that is -integrable and
Similarly, as is also equal to the right hand side of (3), associativity shows that is -integrable and
Comparing these equalities gives as required. ⬜
Stochastic integrals behave particularly well under stopping. Recall that represents a process stopped at the time .
Lemma 4 Let be a semimartingale, be -integrable and be a stopping time. Then, the stopped process is also a semimartingale, is -integrable and
Proof: Approximating by the discrete stopping times gives
As , bounded convergence in probability can be applied,
Associativity of the stochastic integral shows that is a semimartingale, and, by integrating , gives the right hand equality of (4).
Similarly, integrating with respect to gives,
as required. ⬜
Next, the class of -integrable processes is unchanged under localization.
Lemma 5 Let be a semimartingale. Then, a predictable process is -integrable if and only if it is locally -integrable.
Proof: If is locally -integrable then there exist stopping times such that . Choose bounded predictable processes which tend to zero as goes to infinity. Then, for any fixed , are dominated by . So, equation (4) and dominated convergence in probability give
in probability as . By choosing large, can be made as close to 1 as required, showing that goes to zero in probability. So, by definition, . ⬜
A consequence of this result is the following important statement, which applies to all semimartingales; all locally bounded predictable processes are -integrable. In particular, if is a cadlag adapted process then its left limits give a left-continuous and adapted, hence predictable, process. It is, furthermore, locally bounded and the integral is well defined.
A similar result as above shows that the class of semimartingales is closed under localization.
Lemma 6 A stochastic process is a semimartingale if and only if it is locally a semimartingale.
Proof: By definition, a process is locally a semimartingale if there are stopping times such that are semimartingales. Then, are semimartingales. For any bounded predictable process and , Lemma 4 gives
In particular whenever . So, we can define the integral with respect to by
where the limit on the right hand side is eventually constant, with probability one. This clearly satisfies the explicit expression for elementary integrands. To show that is a semimartingale, it only remains to prove bounded convergence in probability. So, suppose that is a uniformly bounded sequence of predictable processes. Dominated convergence in probability can be applied to the semimartingale ,
Then, by choosing large enough, can be made as close to 1 as required, showing that does indeed converge in probability . ⬜
Let us now move on to the dominated convergence theorem. Although dominated convergence in probability was required by the definition of stochastic integration, convergence also holds in a much stronger sense. A sequence of processes converges ucp to a limit if tends to zero in probability for each .
Theorem 7 (Dominated Convergence) Let be a semimartingale and be a sequence of predictable processes converging to a limit . If the sequence is dominated by some -integrable process , so that , then
and, furthermore, convergence holds in the semimartingale topology.
Proof: As semimartingale convergence implies ucp convergence, it is enough to show that converges to zero in the semimartingale topology. Choose a sequence of elementary predictable processes. Then, , and dominated convergence in probability gives
in probability as . By definition, this proves semimartingale convergence. ⬜
The dominated convergence theorem can be used to prove the following result stating that the jumps of a stochastic integral behave in the expected way. Recall that the jump of a cadlag process is equal to . In these notes, when two processes are shown to be equal, this is always taken to mean that they agree up to evanescence.
Corollary 8 If is a semimartingale and then
Proof: Let be the set of all -integrable processes such that equation (5) is satisfied. This contains all elementary predictable processes, by the explicit expression for the integral. Consider a sequence which is dominated by some and suppose that . If it can be shown that then the functional monotone class theorem will imply that , as required.
By Theorem 7, converges ucp to . Passing to a subsequence if necessary, we may suppose that, almost surely, the sample paths converge uniformly on compacts. Then, the following limits hold uniformly on compacts,
Stochastic integration preserves certain properties of processes, such as continuity and predictability.
Corollary 9 Suppose that for a semimartingale and . If is continuous (resp. predictable) then so is .
Proof: If is continuous then equation (5) shows that , so is continuous.
For any cadlag process , its left-limit is a left-continuous and adapted process which, by definition, is predictable. So, is predictable if and only if is. So, suppose that is predictable. Then, (5) shows that is also predictable. ⬜
Finally, the set of semimartingales is a vector space, and stochastic integration is linear in the integrator .
Lemma 10 Let be semimartingales and be real numbers. Then, is a semimartingale. Furthermore, any process which is both -integrable and -integrable is also -integrable and,
Proof: The integral of any bounded predictable process with respect to can be defined by (6). This clearly satisfies the explicit expression for elementary integrands, and satisfies bounded convergence in probability, as required. So, is a semimartingale. Now, suppose that . If is a sequence of bounded predictable processes tending to a limit , then dominated convergence in probability with respect to and gives
in probability as . Taking , the right hand side is zero and, by definition, . Then, taking , dominated convergence shows that the left hand side tends to , giving (6). ⬜