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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Semimartingale Completeness

A sequence of stochastic processes, {X^n}, is said to converge to a process X under the semimartingale topology, as n goes to infinity, if the following conditions are met. First, {X^n_0} should tend to {X_0} in probability. Also, for every sequence {\xi^n} of elementary predictable processes with {\vert\xi^n\vert\le 1},

\displaystyle  \int_0^t\xi^n\,dX^n-\int_0^t\xi^n\,dX\rightarrow 0

in probability for all times t. For short, this will be denoted by {X^n\xrightarrow{\rm sm}X}.

The semimartingale topology is particularly well suited to the class of semimartingales, and to stochastic integration. Previously, it was shown that the cadlag and adapted processes are complete under semimartingale convergence. In this post, it will be shown that the set of semimartingales is also complete. That is, if a sequence {X^n} of semimartingales converge to a limit X under the semimartingale topology, then X is also a semimartingale.

Theorem 1 The space of semimartingales is complete under the semimartingale topology.

The same is true of the space of stochastic integrals defined with respect to any given semimartingale. In fact, for a semimartingale X, the set of all processes which can be expressed as a stochastic integral {\int\xi\,dX} can be characterized as follows; it is precisely the closure, under the semimartingale topology, of the set of elementary integrals of X. This result was originally due to Memin, using a rather different proof to the one given here. The method used in this post only relies on the elementary properties of stochastic integrals, such as the dominated convergence theorem.

Theorem 2 Let X be a semimartingale. Then, a process Y is of the form {Y=\int\xi\,dX} for some {\xi\in L^1(X)} if and only if there is a sequence {\xi^n} of bounded elementary processes with {\int\xi^n\,dX\xrightarrow{\rm sm}Y}.

Writing S for the set of processes of the form {\int\xi\,dX} for bounded elementary {\xi}, and {\bar S} for its closure under the semimartingale topology, the statement of the theorem is equivalent to

\displaystyle  \bar S=\left\{\int\xi\,dX\colon \xi\in L^1(X)\right\}. (1)

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Existence of the Stochastic Integral

The principal reason for introducing the concept of semimartingales in stochastic calculus is that they are precisely those processes with respect to which stochastic integration is well defined. Often, semimartingales are defined in terms of decompositions into martingale and finite variation components. Here, I have taken a different approach, and simply defined semimartingales to be processes with respect to which a stochastic integral exists satisfying some necessary properties. That is, integration must agree with the explicit form for piecewise constant elementary integrands, and must satisfy a bounded convergence condition. If it exists, then such an integral is uniquely defined. Furthermore, whatever method is used to actually construct the integral is unimportant to many applications. Only its elementary properties are required to develop a theory of stochastic calculus, as demonstrated in the previous posts on integration by parts, Ito’s lemma and stochastic differential equations.

The purpose of this post is to give an alternative characterization of semimartingales in terms of a simple and seemingly rather weak condition, stated in Theorem 1 below. The necessity of this condition follows from the requirement of integration to satisfy a bounded convergence property, as was commented on in the original post on stochastic integration. That it is also a sufficient condition is the main focus of this post. The aim is to show that the existence of the stochastic integral follows in a relatively direct way, requiring mainly just standard measure theory and no deep results on stochastic processes.

Recall that throughout these notes, we work with respect to a complete filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},{\mathbb P})}. To recap, elementary predictable processes are of the form

\displaystyle  \xi_t=Z_01_{\{t=0\}}+\sum_{k=1}^n Z_k1_{\{s_{k}<t\le t_k\}} (1)

for an {\mathcal{F}_0}-measurable random variable {Z_0}, real numbers {s_k,t_k\ge 0} and {\mathcal{F}_{s_k}}-measurable random variables {Z_k}. The integral with respect to any other process X up to time t can be written out explicitly as,

\displaystyle  \int_0^t\xi\,dX = \sum_{k=1}^n Z_k(X_{t_k\wedge t}-X_{s_k\wedge t}). (2)

The predictable sigma algebra, {\mathcal{P}}, on {{\mathbb R}_+\times\Omega} is generated by the set of left-continuous and adapted processes or, equivalently, by the elementary predictable process. The idea behind stochastic integration is to extend this to all bounded and predictable integrands {\xi\in{\rm b}\mathcal{P}}. Other than agreeing with (2) for elementary integrands, the only other property required is bounded convergence in probability. That is, if {\xi^n\in{\rm b}\mathcal{P}} is a sequence uniformly bounded by some constant K, so that {\vert\xi^n\vert\le K}, and converging to a limit {\xi} then, {\int_0^t\xi^n\,dX\rightarrow\int_0^t\xi\,dX} in probability. Nothing else is required. Other properties, such as linearity of the integral with respect to the integrand follow from this, as was previously noted. Note that we are considering two random variables to be the same if they are almost surely equal. Similarly, uniqueness of the stochastic integral means that, for each integrand, the integral is uniquely defined up to probability one.

Using the definition of a semimartingale as a cadlag adapted process with respect to which the stochastic integral is well defined for bounded and predictable integrands, the main result is as follows. To be clear, in this post all stochastic processes are real-valued.

Theorem 1 A cadlag adapted process X is a semimartingale if and only if, for each {t\ge 0}, the set

\displaystyle  \left\{\int_0^t\xi\,dX\colon \xi{\rm\ is\ elementary}, \vert\xi\vert\le 1\right\} (3)

is bounded in probability.

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Martingales and Elementary Integrals

A martingale is a stochastic process which stays the same, on average. That is, the expected future value conditional on the present is equal to the current value. Examples include the wealth of a gambler as a function of time, assuming that he is playing a fair game. The canonical example of a continuous time martingale is Brownian motion and, in discrete time, a symmetric random walk is a martingale. As always, we work with respect to a filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},{\mathbb P})}. A process {X} is said to be integrable if the random variables {X_t} are integrable, so that {{\mathbb E}[\vert X_t\vert]<\infty}.

Definition 1 A martingale, {X}, is an integrable process satisfying

\displaystyle  X_s={\mathbb E}[X_t\mid\mathcal{F}_s]

for all {s<t\in{\mathbb R}_+}.

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