Preservation of the Local Martingale Property

Martingales as Integrators, Stochastic Calculus Notes

Now that it has been shown that stochastic integration can be performed with respect to any local martingale, we can move on to the following important result. Stochastic integration preserves the local martingale property. At least, this is true under very mild hypotheses. That the martingale property is preserved under integration of bounded elementary processes is straightforward. The generalization to predictable integrands can be achieved using a limiting argument. It is necessary, however, to restrict to locally bounded integrands and, for the sake of generality, I start with local sub and supermartingales.

Theorem 1 Let X be a local submartingale (resp., local supermartingale) and {\xi} be a nonnegative and locally bounded predictable process. Then, {\int\xi\,dX} is a local submartingale (resp., local supermartingale).

Proof: We only need to consider the case where X is a local submartingale, as the result will also follow for supermartingales by applying to -X. By localization, we may suppose that {\xi} is uniformly bounded and that X is a proper submartingale. So, {\vert\xi\vert\le K} for some constant K. Then, as previously shown there exists a sequence of elementary predictable processes {\vert\xi^n\vert\le K} such that {Y^n\equiv\int\xi^n\,dX} converges to {Y\equiv\int\xi\,dX} in the semimartingale topology and, hence, converges ucp. We may replace {\xi_n} by {\xi_n\vee0} if necessary so that, being nonnegative elementary integrals of a submartingale, {Y^n} will be submartingales. Also, {\vert\Delta Y^n\vert=\vert\xi^n\Delta X\vert\le K\vert\Delta X\vert}. Recall that a cadlag adapted process X is locally integrable if and only its jump process {\Delta X} is locally integrable, and all local submartingales are locally integrable. So,

\displaystyle \sup_n\vert\Delta Y^n_t\vert\le K\vert\Delta X_t\vert

is locally integrable. Then, by ucp convergence for local submartingales, Y will satisfy the local submartingale property. ⬜

For local martingales, applying this result to {\pm X} gives,

Theorem 2 Let X be a local martingale and {\xi} be a locally bounded predictable process. Then, {\int\xi\,dX} is a local martingale.

This result can immediately be extended to the class of local {L^p}-integrable martingales, denoted by {\mathcal{M}^p_{\rm loc}}.

Corollary 3 Let {X\in\mathcal{M}^p_{\rm loc}} for some {0< p\le\infty} and {\xi} be a locally bounded predictable process. Then, {\int\xi\,dX\in\mathcal{M}^p_{\rm loc}}.

Proof: The previous theorem shows that {Y\equiv\int\xi\,dX} is a local martingale. By localization, we may suppose that {\xi} is uniformly bounded, so {\vert\xi\vert\le K} for some constant K. As {\Delta X} is locally {L^p}-integrable,

\displaystyle \vert\Delta Y\vert\le K\vert\Delta X\vert

will also be locally {L^p}-integrable, as required. ⬜

Moving on, it seems natural to ask if Theorem 2 also applies to proper martingales. That is, if X is a cadlag martingale and {\xi} is a uniformly bounded predictable process, then is the integral {\int\xi\,dX} necessarily a martingale? Unfortunately, this is not true. Theorem 2 shows that the integral will be a local martingale, but there are examples where it is not a proper martingale. However, by placing some restriction on X, it is possible to get a positive result. In particular, the following result says that {\int\xi\,dX} will be a martingale if X is also square integrable. In fact, the result generalizes to {L^p}-integrable martingales for all {p>1}, although the proof of that more general statement will have to wait until after we have introduced the Burkholder-Davis-Gundy inequalities.

Lemma 4 Let X be a cadlag square integrable martingale and {\xi} be a bounded predictable process. Then, {\int\xi\,dX} is a square integrable martingale.

Proof: Suppose that {\vert\xi\vert\le K} for some constant K and set {Y=\int\xi\,dX}. Then, choose elementary predictable {\vert\xi^n\vert\le K} such that {Y^n\equiv\int\xi^n\,dX\rightarrow Y} in the semimartingale topology and, hence, {Y^n_t\rightarrow Y_t} in probability for each time t. Then, {Y^n} are martingales and, by the inequality for elementary integrals of square integrable martingales given in the previous post,

\displaystyle {\mathbb E}\left[(Y^n_t)^2\right]\le K^2{\mathbb E}\left[X_t^2\right].

So, {\{Y^n\colon n=1,2,\ldots\}} is {L^2}-bounded and, hence, is uniformly integrable. As convergence in probability of a uniformly integrable sequence implies {L^1}-convergence, {{\mathbb E}\vert Y^n_t-Y_t\vert\rightarrow 0} as n goes to infinity. Therefore, Y is a martingale. Finally, passing to a subsequence so that {Y^n_t\rightarrow Y_t} almost surely, Fatou’s lemma can be applied

\displaystyle {\mathbb E}\left[Y^2_t\right] = {\mathbb E}\left[\lim_{n\rightarrow\infty} (Y^n_t)^2\right]\le\liminf_{n\rightarrow\infty}{\mathbb E}\left[(Y^n_t)^2\right]\le K^2{\mathbb E}\left[X_t^2\right].

So Y is square integrable as required. ⬜

Finally, let us consider how Theorem 2 can be extended to arbitrary X-integrable integrands. In order for {Y\equiv\int\xi\,dX} to be a local martingale it must, at the very least, be locally integrable. In fact, it only needs to be shown that either the positive part {Y^+} or negative part {Y^-} of Y is locally integrable, as the following result shows.

Recall that we say that a process Y is locally integrable if there is a sequence of stopping times {\tau_n\uparrow\infty} such that the stopped processes {1_{\{\tau_n>0\}}(Y^*)^{\tau_n}} are all integrable, where {Y^*_t\equiv\sup_{s\le t}\vert Y_s\vert} is the maximum process of Y. The standard definition is often taken to apply only to nonnegative increasing processes, for which {Y^*=Y}. The more general definition is used here, as it seems to give slightly cleaner statements and proofs.

Theorem 5 Let {Y=\int\xi\,dX} for a local martingale X and X-integrable process {\xi}. Then, the following are equivalent.

  1. Y is a local martingale.
  2. Y is locally integrable.
  3. {Y^+} is locally integrable.
  4. {Y^-} is locally integrable.

Proof: Property 1 implies 2, because all local martingales are locally integrable. Then, 3 and 4 follow directly from 2. It only remains to show that 3 implies 1 because, applying the same result to -Y, 4 would then also imply 1, showing that all the conditions are equivalent.

So, suppose that {Y^+} is locally integrable. Consider the bounded predictable processes {\xi^n\equiv (\xi\wedge n)\vee(-n)}. These satisfy {\vert \xi^n\vert\le\vert\xi\vert} and have the same sign as {\xi} at all times. Therefore, setting {Y^n=\int\xi^n\,dX} gives {\Delta Y^n=\xi^n\Delta X} and {\Delta Y=\xi\Delta X}. So, {\vert\Delta Y^n\vert\le\vert\Delta Y\vert} and {\Delta Y^n}, {\Delta Y} have the same sign. In particular, {(\Delta Y^n)^+\le(\Delta Y)^+}. Furthermore, by Theorem 2, {Y^n} are local martingales and, by dominated convergence, tend ucp to Y. Passing to a subsequence, we may suppose that {Y^n} converge uniformly to Y on bounded intervals, with probability one.

Set

\displaystyle M_t\equiv\sup_n(Y^n_t)^+

which is a cadlag adapted process. Being left-continuous, {Y_{t-}} is locally integrable. So {(\Delta Y)^+\le Y^+ - Y_{\cdot-}} is also locally integrable. Then,

\displaystyle \Delta M_t \le \sup_n (\Delta Y^n)^+ \le (\Delta Y )^+

is locally integrable, and so is M. By localizing, we may suppose that {M^*\equiv\sup_t M_t} has finite expectation.

As convex functions of martingales are submartingales, {a\vee Y^n} are local submartingales, for any constant a. Furthermore, as {\vert a\vee Y^n\vert} is bounded by the integrable random variable {\vert a\vert\vee M^*}, they are proper submartingales. Then, by the dominated convergence theorem for convergence of random variables, {{\mathbb E}\vert a\vee Y^n_t- a\vee Y_t\vert\rightarrow 0} as n goes to infinity. It follows that {a\vee Y} is a submartingale. For any {s\le t}, monotone convergence gives

\displaystyle Y_s = \lim_{a\rightarrow-\infty}a\vee Y_s\le\lim_{a\rightarrow-\infty}{\mathbb E}[a\vee Y_t\mid\mathcal{F}_s] ={\mathbb E}[Y_t\mid\mathcal{F}_s].

This shows that, after localizing, Y becomes a submartingale. Therefore, in general, Y is a local submartingale.

As local submartingales are locally integrable, it follows that Y and, in particular, {Y^-} are locally integrable. Then, the same argument as above can be applied to –Y to show that it is also a local submartingale. Finally, we have shown that locally, Y and –Y are both submartingales, so Y is a local martingale. ⬜

Example: Loss of the Local Martingale Property

Theorem 5 is as far as we can go in establishing the preservation of the local martingale property. There do exist integrals, with respect to local martingales, which are not locally integrable and, hence, cannot themselves be local martingales. To demonstrate this, consider the following simple example.

Let {U, \tau} be independent random variables such that {\tau} is uniformly distributed over the interval [0,1] and U has the discrete distribution {{\mathbb P}(U=1)={\mathbb P}(U=-1)=1/2}. Then define the process

\displaystyle X_t = 1_{\{t\ge\tau\}}U.

With respect to its (completed) natural filtration {\{\mathcal{F}_t\}_{t\ge 0}}, X is a martingale. The only stopping times {\sigma\le\tau} defined on this filtration are of the form {\sigma=\tau\wedge t} for a fixed time {t\in\bar{\mathbb R}_+}. To see this, let t be the supremum of all times s satisfying {{\mathbb P}(\sigma>s)>0}. Then, for any {s<t}, the random variables {\{X_u\colon u\le s\}} are all zero when restricted to the set {\{\tau>s\}} and, therefore, any set {A\in\mathcal{F}_s} satisfies {{\mathbb P}(A\mid\tau>s)=0} or 1. In particular, {{\mathbb P}(\sigma>s\mid\tau>s)=1} and, therefore, {\{\tau>s\ge\sigma\}} has zero probability. This holds for all {s<t}, giving {\sigma\ge\tau\wedge t}. By construction, {\sigma\le t} almost surely, giving {\sigma=\tau\wedge t}.

Now consider the integral,

\displaystyle Y_t = \int_0^t s^{-1}\,dX_s = 1_{\{t\ge\tau\}}\tau^{-1}U.

If {\sigma} is any stopping time with a positive probability of being non-zero, then {\sigma\wedge\tau=t\wedge\tau} for some fixed positive time t. So,

\displaystyle {\mathbb E}\vert Y_\sigma\vert = {\mathbb E}\left[1_{\{\tau\le t\}}\tau^{-1}\vert U\vert\right] =\int_0^{t\wedge 1} s^{-1}\,ds = \infty.

Therefore Y is not locally integrable, and cannot be a local martingale.

Published by George Lowther

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12 thoughts on “Preservation of the Local Martingale Property”

  1. locally bounded predictable process

    Does this just mean that the predictable process is finite?

    Or in other words, what is an example for a predictable process that is not locally bounded?

    Thanks a lot. Not just for a prospective answer but for the blog in general. It’s the best textbook on stochastic calculus that I’ve ever seen.

    Reply

    1. Hi. For a process \xi to be locally bounded means that there is a sequence of stopping times \tau_n increasing to infinity such that the stopped processes 1_{\{\tau_n > 0\}}\xi^{\tau_n} are uniformly bounded (as defined here). Equivalently, 1_{(0,\tau_n]}\xi are uniformly bounded.

      This certainly implies that \xi is finite. In fact, it implies that \xi^*_t\equiv\sup_{s\le t}\vert\xi_s\vert is almost surely finite for each time t. The converse does not hold in general, but for a right-continuous filtration and predictable process \xi, the converse does hold (its not easy to prove this without using some advanced results though).

      An example of a predictable but not locally bounded processes is \xi_t = 1_{\{t > 0\}}t^{-1}. An example of a predictable process for which \xi^* is finite but \xi is not locally bounded can be constructed as follows. Let U be an unbounded random variable (e.g., standard normal). Set \xi_t=1_{\{t > 0\}}\min(\vert U\vert,1/t). With respect to its natural filtration, this is not locally bounded. As \mathcal{F}_0 is trivial, any stopping time \tau with positive probability of being positive is almost-surely positive. Then, the supremum of 1_{\{\tau > 0\}}\xi^\tau is U, which is not locally bounded. Note that if we passed to the right-continuous filtration \mathcal{F}_{t+} then \xi will be locally bounded. As |U| is \mathcal{F}_{0+}-measurable, you can choose \tau_n to be equal to zero whenever |U| is greater than n and infinity otherwise.
      Examples of non-predictable processes (but adapted to a right-continuous filtration) which are not locally bounded but for which \xi^* is finite are given by Lévy processes with unbounded jump size.

      Reply

  2. It is not clear what your last example is an “example of”. You write:
    “There do exist integrals, with respect to local martingales, which are not locally integrable”
    — however, I do not recall you defining any “abstract integral” — as opposed to “an integral of X-integrable process”.
    Moreover, the preceding Th.5 assumes X-integrability.

    So (since your 1/s is not X-integrable according to your definition) it seems that your example is not only not-related to Th.5, but it is also not related to notions of integrability you discuss here… (Finally, I’m not yet in the position to be sure about the match of definitions, but it SEEMS that Th.2.2.5 of Liptser–Shiryaev claims that the necessary amplification of your Th.5 actually holds…)

    Reply

    1. If a predictable process \xi is X-integrable, then the integral \int\xi dX is well-defined. If X is a local martingale, then you might think that the integral is necessarily a local martingale (which, in particular, means that it is locally integrable). This is often the case, but not always, which the example shows.
      Maybe you are confusing the statements that \xi is X-integrable and that \int\xi dX is locally integrable. These are different concepts. For the definition of local integrability, see here: https://almostsuremath.com/2009/12/23/localization/
      For the definition of X-integrability, see here: https://almostsuremath.com/2010/01/04/extending-the-stochastic-integral/
      For the example given, the process X is constant except for a single jump of size U at time \tau. Hence, *every* (real-valued) predictable process \xi is X-integrable with \int_0^t\xi dX=1_{\{t\ge\tau\}}\xi_\tau U. I used \xi_t=1_{\{t > 0\}}t^{-1} showing that, even though X is a martingale, the integral \int\xi dX is not a local martingale (since it is not even locally integrable. All local martingales are locally integrable, so this cannot be a local martingale).

      Reply

      1. Basically, you say that what I wrote makes no sense. NOW I agree — but to understand this, I needed to write a paragraph DEFENDING what I wrote before. (Only after this I found where was my misunderstanding — I was using &Escr; instead of honest convergence in probability.)

        So: a lot of thanks for this blog: it (when taken together with other sources) clarifies things a lot! However, X-integrability is an exception: while it is a majorly beautiful approach, I think it is not covered clear enough (especially if one takes into account that this notion seems to be different from what everybody else uses). One indication of this is that I do not recollect seeing ANY example of a non-bounded X-integrable process (except your answer above — but I’ve read maybe only 1/4 of your pages).

        I also think that it would be beneficial to mention the relationship to Th.2 of “Properties of the Stochastic Integral” here: as “not being a local martingale” does not stop it from being a semimartingale (“a spontaneous generation” of a FV term!).

        (I think I may have more remarks, but they better belong to “Further Properties of the Stochastic Integral”.)

        Reply

        1. On the comment “this notion seems to be different from what everybody else uses”. While different authors use different notation and different concepts of X-integrability, I don’t think what I use is unique to these notes. I will have to check my references when I have time to do that, to back this up. The exact definitions and proofs are unique to these notes, but the resulting classes of semimartingales and of X-integrable processes are standard.

        2. I revisited the 1st paragraph of the end-notes in “Further Properties of the Stochastic Integral”, and NOW I can see that it INDEED (essentially) says that your notion of X-integrability “is the same as the usual one”. However, I want to stress that it is not 100%-explicitly stated, and on my first reading this (in the context of discussing the differences!) lead to my confusion (indicated above).

          A lot of thanks, and I appreciate your patience with me!

  3. Do you have a reference for the result that X_t = I(t \leq \tau)U is a martingale wrt its natural filtration? I wonder such a result holds more generally, e.g. if U is integrable and F_\tau-measurable. Thank you for maintaining this excellent blog!

    Reply

    1. That was just an example I came up with but, checking the literature, I see that it is mentioned in Protter, Stochastic Integration and Differential Equations (Second Edition), Chapter IV, in the example preceding theorem 34.
      The example is due to M. Emery and has \tau being exponentially distributed rather than uniform, but that is not important.

      The example in Protter only considers the same U as I use here, but I think more generally you just need it to be integrable and to satisfy \mathbb E[U \vert\mathcal F_\tau]=0 in order that X is a martingale.

      Reply

      1. Thank you for the prompt response! I checked Protter earlier but I missed Emery’s example then.

        I see now, your example of X being a martingale is rather easy to verify. The question I specifically had in mind is as follows:

        If \tau is a bounded stopping time and M is a mean zero local martingale, is then M_\tau I(\tau \leq t) a (mean zero) local martingale?

        (under ‘regularity conditions’; usual hypotheses, cadlag processes, etc).

        Do you think such a result might hold, or do you by chance know about any counterexamples?

        Thanks again and sorry for the trouble.

        Reply

        1. That can’t be a local martingale in general. E.g., suppose M is standard Brownian motion and \tau is the first time at which it hits 1, or the minimum of this time and 1 (to keep it bounded).
          Then X_t = 1_{\{\tau\le t\}}M_\tau is a process which starts at 0 and jumps to 1 (if \tau < 1), so cannot be a local martingale.

        2. Thank you very much for your last response, George, that makes perfect sense to me. I wasn’t able to reply to your comment as the ‘Reply’ button under your comment is hidden for some reason. So I post it here

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