Category: Martingales as Integrators

The Burkholder-Davis-Gundy Inequality

The Burkholder-Davis-Gundy inequality is a remarkable result relating the maximum of a local martingale with its quadratic variation. Recall that [X] denotes the quadratic variation of a process X, and {X^*_t\equiv\sup_{s\le t}\vert X_s\vert} is its maximum process.

Theorem 1 (Burkholder-Davis-Gundy) For any {1\le p<\infty} there exist positive constants {c_p,C_p} such that, for all local martingales X with {X_0=0} and stopping times {\tau}, the following inequality holds.

\displaystyle c_p{\mathbb E}\left[ [X]^{p/2}_\tau\right]\le{\mathbb E}\left[(X^*_\tau)^p\right]\le C_p{\mathbb E}\left[ [X]^{p/2}_\tau\right]. (1)

Furthermore, for continuous local martingales, this statement holds for all {0<p<\infty}.

A proof of this result is given below. For {p\ge 1}, the theorem can also be stated as follows. The set of all cadlag martingales X starting from zero for which {{\mathbb E}[(X^*_\infty)^p]} is finite is a vector space, and the BDG inequality states that the norms {X\mapsto\Vert X^*_\infty\Vert_p={\mathbb E}[(X^*_\infty)^p]^{1/p}} and {X\mapsto\Vert[X]^{1/2}_\infty\Vert_p} are equivalent.

The special case p=2 is the easiest to handle, and we have previously seen that the BDG inequality does indeed hold in this case with constants {c_2=1}, {C_2=4}. The significance of Theorem 1, then, is that this extends to all {p\ge1}.

One reason why the BDG inequality is useful in the theory of stochastic integration is as follows. Whereas the behaviour of the maximum of a stochastic integral is difficult to describe, the quadratic variation satisfies the simple identity {\left[\int\xi\,dX\right]=\int\xi^2\,d[X]}. Recall, also, that stochastic integration preserves the local martingale property. Stochastic integration does not preserve the martingale property. In general, integration with respect to a martingale only results in a local martingale, even for bounded integrands. In many cases, however, stochastic integrals are indeed proper martingales. The Ito isometry shows that this is true for square integrable martingales, and the BDG inequality allows us to extend the result to all {L^p}-integrable martingales, for {p> 1}.

Theorem 2 Let X be a cadlag {L^p}-integrable martingale for some {1<p<\infty}, so that {{\mathbb E}[\vert X_t\vert^p]<\infty} for each t. Then, for any bounded predictable process {\xi}, {Y\equiv\int\xi\,dX} is also an {L^p}-integrable martingale.

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Continuous Local Martingales

Continuous local martingales are a particularly well behaved subset of the class of all local martingales, and the results of the previous two posts become much simpler in this case. First, the continuous local martingale property is always preserved by stochastic integration.

Theorem 1 If X is a continuous local martingale and {\xi} is X-integrable, then {\int\xi\,dX} is a continuous local martingale.

Proof: As X is continuous, {Y\equiv\int\xi\,dX} will also be continuous and, therefore, locally bounded. Then, by preservation of the local martingale property, Y is a local martingale. ⬜

Next, the quadratic variation of a continuous local martingale X provides us with a necessary and sufficient condition for X-integrability.

Theorem 2 Let X be a continuous local martingale. Then, a predictable process {\xi} is X-integrable if and only if

\displaystyle \int_0^t\xi^2\,d[X]<\infty

for all {t>0}.

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Quadratic Variations and the Ito Isometry

As local martingales are semimartingales, they have a well-defined quadratic variation. These satisfy several useful and well known properties, such as the Ito isometry, which are the subject of this post. First, the covariation [X,Y] allows the product XY of local martingales to be decomposed into local martingale and FV terms. Consider, for example, a standard Brownian motion B. This has quadratic variation {[B]_t=t} and it is easily checked that {B^2_t-t} is a martingale.

Lemma 1 If X and Y are local martingales then XY-[X,Y] is a local martingale.

In particular, {X^2-[X]} is a local martingale for all local martingales X.

Proof: Integration by parts gives

\displaystyle XY-[X,Y] = X_0Y_0+\int X_-\,dY+\int Y_-\,dX

which, by preservation of the local martingale property, is a local martingale. ⬜

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Preservation of the Local Martingale Property

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}}.

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Martingales are Integrators

A major foundational result in stochastic calculus is that integration can be performed with respect to any local martingale. In these notes, a semimartingale was defined to be a cadlag adapted process with respect to which a stochastic integral exists satisfying some simple desired properties. Namely, the integral must agree with the explicit formula for elementary integrands and satisfy bounded convergence in probability. Then, the existence of integrals with respect to local martingales can be stated as follows.

Theorem 1 Every local martingale is a semimartingale.

This result can be combined directly with the fact that FV processes are semimartingales.

Corollary 2 Every process of the form X=M+V for a local martingale M and FV process V is a semimartingale.

Working from the classical definition of semimartingales as sums of local martingales and FV processes, the statements of Theorem 1 and Corollary 2 would be tautologies. Then, the aim of this post is to show that stochastic integration is well defined for all classical semimartingales. Put in another way, Corollary 2 is equivalent to the statement that classical semimartingales satisfy the semimartingale definition used in these notes. The converse statement will be proven in a later post on the Bichteler-Dellacherie theorem, so the two semimartingale definitions do indeed agree.

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