Lecture 20. Local martingales, Semimartingales and Integrators

The goal of this Lecture is to extend the domain of definition of the Itō integral with respect to Brownian motion. The idea is to use the fruitful concept of localization. We will then be interested in the wider class of processes for which it is possible to define a stochastic integral satisfying natural probabilistic properties. This will lead to the natural notion of semimartingales.

As before, we consider here a Brownian motion $(B_t)_{t\ge 0}$ that is defined on a filtered probability space $(\Omega, (\mathcal{F}_t)_{t \ge 0},\mathcal{F},\mathbb{P})$ that satisfies the usual conditions.

Definition. We define the space $L_{loc}^2 (\Omega, (\mathcal{F}_t)_{t \ge 0},\mathbb{P})$ , as the set of the processes $(u_t)_{t \ge 0}$ that are progressively measurable with respect to the filtration $(\mathcal{F}_t)_{t \ge 0}$ and such that for every $t \ge 0$ ,
$\mathbb{P} \left( \int_0^{t} u_s^2 ds <+\infty\right)=1.$

We first have the following fact:

Lemma. Let $u \in L_{loc}^2 (\Omega, (\mathcal{F}_t)_{t \ge 0},\mathbb{P})$ . There exists an increasing family of stopping times $(T_n)_{n \ge 0}$ for the filtration $(\mathcal{F}_t)_{t \ge 0}$ such that:

Almost surely, $\lim_{n \rightarrow +\infty} T_n=+\infty;$

$\mathbb{E} \left( \int_0^{T_n} u_s^2 ds \right)<+\infty.$

Thanks to this Lemma, it is now easy to naturally define $\int_0^t u_s dB_s$ for $u \in L_{loc}^2 (\Omega, (\mathcal{F}_t)_{t \ge 0},\mathbb{P})$ . Indeed, let $u \in L_{loc}^2 (\Omega,(\mathcal{F}_t)_{t \ge 0},\mathbb{P})$ and let $t \ge 0$ . According to the previous lemma, let us now consider an increasing sequence of stopping times $(T_n)_{n \ge 0}$ such that:

Almost surely, $\lim_{n \rightarrow +\infty} T_n=+\infty;$
$\mathbb{E} \left( \int_0^{T_n} u_s^2 ds \right)< +\infty.$

Since $\mathbb{E} \left( \int_0^{T_n} u_s^2 ds \right)<+\infty,$ the stochastic integral
$\int_0^{T_n} u_s dB_s=\int_0^{+\infty} u_s 1_{[0,T_n]} (s)dB_s$
exists. We may therefore define in a unique way a stochastic process $\left(\int_0^{t} u_s dB_s \right)_{t \ge 0}$ such that:

$\left(\int_0^{t} u_s dB_s \right)_{t \ge 0}$ is a continuous stochastic process adapted to the filtration $(\mathcal{F}_t)_{t \ge 0}$ ;
The stochastic process $\left(\int_0^{t\wedge T_n} u_s dB_s \right)_{t \ge 0}$ is a uniformly integrable martingale with respect to the filtration $(\mathcal{F}_t)_{t \ge 0}$ (because it is bounded in $L^2$ ).

This leads to the following definition:

Definition. A stochastic process $(M_t)_{t \geq 0}$ is called a local martingale (with respect to the filtration $(\mathcal{F}_t)_{t \geq 0}$ ) if there is a sequence of stopping times $(T_n)_{n \geq 0}$ such that:

The sequence $(T_n)_{n\geq 0}$ is increasing and almost surely satisfies $\lim_{n \rightarrow +\infty} T_n =+\infty$ ;

For $n \geq 1$ , the process $(M_{t \wedge T_n})_{t \geq 0}$ is a uniformly integrable martingale with respect to the filtration $(\mathcal{F}_t)_{t \geq 0}$ .

Thus, as an example, if $u \in L_{loc}^2 (\Omega, (\mathcal{F}_t)_{t \ge 0},\mathbb{P})$ then the process $\left(\int_0^{t} u_s dB_s \right)_{t \ge 0}$ is a local martingale. Of course, any martingale turns out to be a local martingale. But, as we will see it later, in general the converse is not true. The following Exercise gives a useful criterion to prove that a given local martingale is actually martingale.

Exercise.
Let $(M_t)_{t \geq 0}$ be a continuous local martingale such that for $t \geq 0$ ,
$\mathbb{E} \left( \sup_{s \leq t} \mid M_s \mid \right) < +\infty.$
Show that $(M_t)_{t \geq 0}$ is a martingale. As a consequence, bounded local martingales necessarily are martingales.

It is interesting to observe that if $(M_t)_{t \ge 0}$ is a local martingale, then the sequence of stopping times may explicitly be chosen so that the resulting stopped martingales enjoy nice properties.

Lemma. Let $(M_t)_{t \geq 0}$ be a continuous local martingale on $(\Omega, (\mathcal{F}_t)_{t \ge 0},\mathcal{F},\mathbb{P})$ such that $M_0=0$ . Let
$T_n= \inf \{ t \ge 0, | M_t | \ge n \}.$
Then, for $n \in \mathbb{N}$ , the process $(M_{t \wedge T_n})_{t \geq 0}$ is a bounded martingale.

Proof. Let $(S_n)_{n \geq 0}$ be a sequence of stopping times such that:

The sequence $(S_n)_{n\geq 0}$ is increasing and almost surely $\lim_{n \rightarrow +\infty} S_n =+\infty$ ;
For every $n \geq 1$ , the process $(M_{t \wedge s_n})_{t \geq 0}$ is a uniformly integrable martingale with respect to the filtration $(\mathcal{F}_t)_{t \geq 0}$ .

For $t \ge s$ and $k,n \ge 0$ , we have:
$\mathbb{E} \left( M_{t \wedge S_k \wedge T_n} \mid \mathcal{F}_s \right)=M_{s \wedge S_k \wedge T_n}.$
Letting $k \rightarrow +\infty$ leads then to the expected result $\square$

Since bounded martingales are of course square integrable, we easily deduce from the previous Lemma that the following result holds:

Theorem. Let $(M_t)_{t \geq 0}$ be a continuous local martingale on $(\Omega, (\mathcal{F}_t)_{t \ge 0},\mathcal{F},\mathbb{P})$ such that $M_0=0$ . Then, there is a unique continuous increasing process $(\langle M \rangle_t)_{t \geq 0}$ such that:

$\langle M \rangle_0=0$ ;

The process $(M_t^2 - \langle M \rangle_t)_{t \geq 0}$ is a local martingale.

Furthermore, for every $t \ge 0$ and every sequence of subdivisions $\Delta_n [0,t]$ such that $\lim_{n \rightarrow +\infty}\mid\Delta_n [0,t]\mid=0$ , the following limit holds in probability:
$\lim_{n \rightarrow +\infty} \sum_{k=1}^{n} \left( M_{t^n_k} -M_{t^n_{k-1}}\right)^2=\langle M \rangle_t.$
Moreover, if $u$ is a progressively measurable process such that for every $t \ge 0$ , $\mathbb{P} \left( \int_0^t u_s^2 d\langle M \rangle_s <+\infty \right)=1,$ then we may define a stochastic integral $\left( \int_0^t u_s dM_s \right)_{t \ge 0}$ such that the stochastic process $\left( \int_0^t u_s dM_s \right)_{t \ge 0}$ is a continuous local martingale.

At that point, we already almost found the widest class of stochastic processes with respect to which it was possible to naturally construct a stochastic integral. To go further in that direction, let us first observe that if we add a bounded variation process to a local martingale, then we obtain a process with respect to which a stochastic integral is naturally defined.

More precisely, if $(X_t)_{t \ge 0}$ may be written under the form:
$X_t=X_0+A_t+M_t$ , where $(A_t)_{t \ge 0}$ is a bounded variation process and where
$(M_t)_{t \geq 0}$ is a continuous local martingale on $(\Omega, (\mathcal{F}_t)_{t \ge 0},\mathcal{F},\mathbb{P})$ such that $M_0=0$ , then if $u$ is a progressively measurable process such that for $t \ge 0$ ,
$\mathbb{P} \left( \int_0^t u_s^2 d\langle M \rangle_s <+\infty \right)=1,$ we may define a stochastic integral as
$\left( \int_0^t u_s dX_s \right)_{t \ge 0}=\left( \int_0^t u_s dA_s + \int_0^t u_s dM_s \right)_{t \ge 0},$
where $\int_0^t u_s dA_s$ is simply understood as the Riemann-Stieltjes integral with respect to the process $(A_t)_{t \ge 0}$ .

The class of stochastic processes that we obtained is called the class of semimartingales and, as we will see it later, is the most relevant one:
Definition.
Let $(X_t)_{t \ge 0}$ be an adapted continuous stochastic process on the filtered probability space $(\Omega, (\mathcal{F}_t)_{t \ge 0},\mathcal{F},\mathbb{P})$ . We say that $(X_t)_{t \ge 0}$ is a semimartingale with respect to the filtration $(\mathcal{F}_t)_{t \ge 0}$ if $(X_t)_{t \ge 0}$ may be written as:
$X_t=X_0+A_t+M_t$
where $(A_t)_{t \ge 0}$ is a bounded variation process and $(M_t)_{t \geq 0}$ is a continuous local martingale such that $M_0=0$ . If it exists, the previous decomposition is unique.

Exercise. Let $(M_t)_{t \ge 0}$ be a continuous local martingale on the filtered probability space $(\Omega, (\mathcal{F}_t)_{t \ge 0},\mathcal{F},\mathbb{P})$ . Show that $(M^2_t)_{t \ge 0}$ is a semimartingale.

Since a bounded variation process has a zero quadratic variation, it is easy to prove the following theorem:
Proposition. Let
$X_t=X_0+A_t+M_t, \quad t \ge 0,$
be a continuous adapted semimartingale. For every $t \ge 0$ and every sequence of subdivisions $\Delta_n [0,t]$ such that $\lim_{n \rightarrow +\infty}\mid\Delta_n[0,t]\mid=0,$ the following limit holds in probability:
$\lim_{n \rightarrow +\infty} \sum_{k=1}^{n} \left( X_{t^n_k} -X_{t^n_{k-1}}\right)^2=\langle M\rangle_t.$
We therefore call $\langle M \rangle$ the quadratic variation of $X$ and denote $\langle X \rangle=\langle M \rangle$ .

Exercise. Let $(X_t)_{t \geq 0}$ be a continuous semimartingale on the filtered probability space $(\Omega, (\mathcal{F}_t)_{t \ge 0},\mathcal{F},\mathbb{P})$ . If $\Delta [0,T]$ is a subdivision of the time interval $[0,T]$, we denote $S_t^{\Delta [0,T]}(X)=\sum_{i=0}^{k-1}\left( X_{t_{i+1}} -X_{t_i} \right)^2 +(X_t-X_{t_k})^2$ , where $k$ is such that $t_k \le t <t_{k+1}$ . Let $\Delta_n [0,T]$ be a sequence of subdivisions of $[0,T]$ such that $\lim_{n \rightarrow +\infty}\mid\Delta_n [0,T]\mid=0.$ Show that the following limit holds in probability, $\lim_{n \rightarrow +\infty} \sup_{0\le t \le T} \left| S_t^{\Delta [0,T]}(X) - \langle X \rangle_t \right|=0.$

Exercise. Let $(X_t)_{t \geq 0}$ be a continuous semimartingale on $(\Omega, (\mathcal{F}_t)_{t \ge 0},\mathcal{F},\mathbb{P})$ . Let $u^n$ be a sequence of locally bounded and adapted processes almost surely converging toward 0 such that $u^n \le u$ , where $u$ is a locally bounded process. Show that for $T \ge 0$ , the following limit holds in probability
$\lim_{n \rightarrow +\infty} \sup_{0\le t \le T} \left| \int_0^t u^n_s dX_s \right|=0.$

It already has been observed that in the Brownian case, though the stochastic integral is not an almost sure limit of Riemann sums, it is however a limit in probability of such sums. This may extended to semimartingales in the following way.

Proposition. Let $u$ be a continuous and adapted process, let $(X_t)_{t \ge 0}$ be a continuous and adapted semimartingale and let $t \ge 0$ . For every sequence of subdivisions $\Delta_n [0,t]$ such that
$\lim_{n \rightarrow +\infty}\mid\Delta_n [0,t]\mid=0,$
the following limit holds in probability:
$\lim_{n \rightarrow +\infty} \sum_{k=0}^{n-1}u_{t^n_{k}} \left( X_{t^n_{k+1}} X_{t^n_{k}}\right)=\int_0^t u_s dX_s.$

As we already suggested it, the class of semimartingales is actually the wider class of stochastic processes with respect to which we may define a stochastic integral that enjoys natural properties. Let us more precisely explain what the previous statement means.

Let us denote by $\mathcal{E}_b$ the set of processes $(u_t)_{t \ge 0}$ such that:
$u_t=\sum_{i=1}^{N} F_i 1_{(S_i,T_{i}]} (t),$ where $0\le S_1 \le T_1 \le ... \le S_N \le T_N$ are bounded stopping times and where the $F_i$ ‘s are random variable that are bounded and measurable with respect to $\mathcal{F}_{S_i}$ . If $(X_t)_{t \ge 0}$ is a continuous and adapted process and if $u \in \mathcal{E}_b$ , then we naturally define
$\int_0^t u_s dX_s =\sum_{i=1}^{N} F_i ( X_{T_i \wedge t}-X_{S_i \wedge t}).$
We have the following theorem that we shall admit without proof:

Proposition. Let $(X_t)_{t \ge 0}$ be a continuous and adapted process. The process $(X_t)_{t \ge 0}$ is a semimartingale if and only if for every sequence $u^n$ in $\mathcal{E}_b$ that almost surely converges to 0, we have for every $t \ge 0$ and $\varepsilon >0$ ,
$\lim_{n \rightarrow + \infty} \mathbb{P} \left( \left| \int_0^t u^n_s dX_s \right| > \varepsilon \right)=0.$

Lecture 20. Local martingales, Semimartingales and Integrators

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