Local Martingales

Recall from the previous post that a cadlag adapted process {X} is a local martingale if there is a sequence {\tau_n} of stopping times increasing to infinity such that the stopped processes {1_{\{\tau_n>0\}}X^{\tau_n}} are martingales. Local submartingales and local supermartingales are defined similarly.

An example of a local martingale which is not a martingale is given by the `double-loss’ gambling strategy. Interestingly, in 18th century France, such strategies were known as martingales and is the origin of the mathematical term. Suppose that a gambler is betting sums of money, with even odds, on a simple win/lose game. For example, betting that a coin toss comes up heads. He could bet one dollar on the first toss and, if he loses, double his stake to two dollars for the second toss. If he loses again, then he is down three dollars and doubles the stake again to four dollars. If he keeps on doubling the stake after each loss in this way, then he is always gambling one more dollar than the total losses so far. He only needs to continue in this way until the coin eventually does come up heads, and he walks away with net winnings of one dollar. This therefore describes a fair game where, eventually, the gambler is guaranteed to win.

Of course, this is not an effective strategy in practise. The losses grow exponentially and, if he doesn’t win quickly, the gambler must hit his credit limit in which case he loses everything. All that the strategy achieves is to trade a large probability of winning a dollar against a small chance of losing everything. It does, however, give a simple example of a local martingale which is not a martingale.

The gamblers winnings can be defined by a stochastic process {\{Z_n\}_{n=1,\ldots}} representing his net gain (or loss) just before the n’th toss. Let {\epsilon_1,\epsilon_2,\ldots} be a sequence of independent random variables with {{\mathbb P}(\epsilon_n=1)={\mathbb P}(\epsilon_n=-1)=1/2}. Here, {\epsilon_n} represents the outcome of the n’th toss, with 1 referring to a head and -1 referring to a tail. Set {Z_1=0} and

\displaystyle  Z_{n}=\begin{cases} 1,&\text{if }Z_{n-1}=1,\\ Z_{n-1}+\epsilon_n(1-Z_{n-1}),&\text{otherwise}. \end{cases}

This is a martingale with respect to its natural filtration, starting at zero and, eventually, ending up equal to one. It can be converted into a local martingale by speeding up the time scale to fit infinitely many tosses into a unit time interval

\displaystyle  X_t=\begin{cases} Z_n,&\text{if }1-1/n\le t<1-1/(n+1),\\ 1,&\text{if }t\ge 1. \end{cases}

This is a martingale with respect to its natural filtration on the time interval {[0,1)}. Letting {\tau_n=\inf\{t\colon\vert X_t\vert\ge n\}} then the optional stopping theorem shows that {X^{\tau_n}_t} is a uniformly bounded martingale on {t<1}, continuous at {t=1}, and constant on {t\ge 1}. This is therefore a martingale, showing that {X} is a local martingale. However, {{\mathbb E}[X_1]=1\not={\mathbb E}[X_0]=0}, so it is not a martingale. Alternatively, an example of a continuous local martingale which is not a martingale can be constructed as follows. Let {B} be a standard Brownian motion and {\tau} be the first time at which it hits one, which is almost surely finite. Then, by optional stopping, {B^\tau} is a martingale starting at 0 and ending up at 1. Rescaling the time index of the Brownian motion,

\displaystyle  X_t=\begin{cases} B^\tau_{t/(1-t)},&\text{if }t<1,\\ 1,&\text{if }t\ge 1 \end{cases}

defines a local martingale with respect to its natural filtration, in a similar way as above. Again, however, {{\mathbb E}[X_1]=1\not={\mathbb E}[X_0]=0}, so {X} is not a martingale.

Martingale and submartingale criteria

The first question we might ask is, when is a local martingale actually a martingale?

Theorem 1 A local martingale {X} is a martingale if and only if it is of class (DL).

This result is a simple application of uniform integrability to the limits of {X^{\tau_n}} for some localizing sequence {\tau_n}. However, it also follows from Theorem 4 below applied to both {X} and {-X}.

Even though the martingale property can fail, nonnegative local martingales are, at least, supermartingales. In particular, if {X} is one of the examples of local martingales given above, then {1-X} is nonnegative and, hence, a supermartingale. Consequently, the local martingale examples above are, in fact, submartingales. Furthermore, as they are not proper martingales, Theorem 1 shows that they are examples of submartingales which are not of class (DL).

Lemma 2 A nonnegative local supermartingale {X} such that {X_0} is integrable is a supermartingale.

Proof: Let {\tau_n\uparrow\infty} be a localizing sequence for {X} (w.r.t. the supermartingale property). Then, for any times {s<t} and {A\in\mathcal{F}_s} the supermartingale property gives

\displaystyle  {\mathbb E}\left[1_{A\cap\{X^{\tau_n}_s<K\}}X^{\tau_n}_s\right]\ge{\mathbb E}\left[1_{A\cap\{X^{\tau_n}_s<K\}}X^{\tau_n}_t\right]

for all {K>0}. Letting {n} go to infinity, bounded convergence on the left hand side and Fatou’s lemma on the right gives,

\displaystyle  {\mathbb E}\left[1_{A\cap\{X_s<K\}}X_s\right]\ge\liminf_{n\rightarrow\infty}{\mathbb E}\left[1_{A\cap\{X^{\tau_n}_s<K\}}X^{\tau_n}_t\right]\ge{\mathbb E}\left[1_{A\cap\{X_s<K\}}X_t\right].

Then, increasing {K} to infinity, monotone convergence gives {{\mathbb E}[1_AX_s]\ge{\mathbb E}[1_AX_t]}. In particular, putting {t=0} gives {{\mathbb E}[X_t]\le{\mathbb E}[X_0]<\infty}, so {X} is a supermartingale. ⬜

The class (DL) property also gives a criterion for a nonnegative local submartingale to be a proper submartingale. Considering the examples of submartingales which are not of class (DL) above, we see that the nonnegativity condition is required here.

Lemma 3 A nonnegative local submartingale is a submartingale if and only if it is of class (DL).

Proof: Nonnegative cadlag submartingales are of class (DL), so only the converse statement is required. Suppose that {X} is a class (DL) local submartingale. Then there is a localizing sequence {\tau_n\uparrow\infty} such that {X^{\tau_n}_s\le{\mathbb E}[X^{\tau_n}_t\mid\mathcal{F}_s]} for times {s<t}. As {X} is of class (DL), uniform integrability can be used to take the limit {n\rightarrow\infty} on both sides of this inequality, showing that {X_t} is integrable and {X_s\le{\mathbb E}[X_t\mid\mathcal{F}_s]}. ⬜

Finally, the following gives a criterion for a general local submartingale to be a proper martingale.

Theorem 4 A local submartingale {X} is a submartingale if and only if {X_0} is integrable and {X^+} is of class (DL).
Similarly, a local supermartingale is a supermartingale if and only if {X_0} is integrable and {X^-} is of class (DL).

Proof: By applying the result to {-X}, only the supermartingale case needs to be proven. If {X} is a cadlag supermartingale then {X_0} is integrable by definition and, {X^-} is a nonnegative submartingale and hence of class (DL).

Conversely, suppose that {X_0} is integrable and {X^-} is of class (DL). Then, {X^+} is a nonnegative supermartingale and, by Lemma 2 above, is a supermartingale. Similarly, {X^-} is a class (DL) local submartingale and, by Lemma 3, is a submartingale. Therefore, {X=X^+-X^-} is a supermartingale. ⬜

Limits of martingales

One way in which local martingales arise is as limits of local martingales. In general, limits of martingales are not martingales. Consider, for example, any local martingale {X} which is not a proper martingale, and let {\tau_n\uparrow\infty} be a localizing sequence. Then, the martingales {1_{\{\tau_n>0\}}X^{\tau_n}} converge uniformly on compacts to the non-martingale {X}. So, the local conditions cannot be dropped from the following.

Theorem 5 Let {\{X^n\}_{n=1,2,\ldots}} be a sequence of continuous local martingales converging ucp to a limit {X}. Then, {X} is a continuous local martingale.

In general, ucp limits of cadlag martingales need not even be local martingales. However, such limits will indeed be local martingales if a local integrability condition is applied to the jumps of the martingales. In particular, recalling that ucp limits of continuous processes are themselves continuous, Theorem 5 above is an immediate consequence of the following.

Theorem 6 Let {\{X^n\}_{n=1,2,\ldots}} be a sequence of local martingales (resp. local submartingales, local supermartingales) converging ucp to a limit {X}. If

\displaystyle  \sup_n\sup_{s\le t}\vert\Delta X^n_s\vert

is locally integrable then {X} is a local martingale (resp. local submartingale, local supermartingale).

Proof: It is enough to prove the submartingale case, as the martingale and supermartingale cases follow from applying this to {-X}.

First, as it is a ucp limit of cadlag adapted processes, {X} will be cadlag and adapted. Passing to a subsequence if necessary, we may suppose that {X^n} converges to {X} uniformly on compacts. Then,

\displaystyle  M_t\equiv\sup_n\sup_{s\le t}\vert X^n_t\vert

is cadlag, adapted, and increasing. It has jumps {\vert\Delta M\vert\le\sup_n\vert \Delta X^n\vert} which, by the condition of the theorem, is locally integrable. Therefore, {M} is locally integrable. Let {\tau_k} be a localizing sequence, so that {1_{\{\tau_k>0\}}M^{\tau_k}} is integrable. Then, {1_{\{\tau_k>0\}}(X^n)^{\tau_k}} are local submartingales bounded by {1_{\{\tau_k>0\}}M^{\tau_k}} and, in particular, are of class (DL). So, they are proper submartingales converging to {1_{\{\tau_k>0\}}X^{\tau_k}} and, applying bounded convergence to this limit, {1_{\{\tau_k>0\}}X^{\tau_k}} is a submartingale. Therefore, {\tau_k} is a localizing sequence for {X}, showing that it is a local submartingale. ⬜

6 thoughts on “Local Martingales

  1. Dear George

    Could you please explain how you optain in the proof of Theorem 4 that $\latex X^+$ is a non-negative supermartingale, given that $\latex X$ is a supermartingale?

    1. Dear Eric,

      Argh, well spotted. That statement is wrong. The theorem still holds, but the proof needs to be fixed. I’ll get back to this and update the proof. Thanks!

  2. Dear George,

    For the example of local martingale in the text, I wonder whether the stop time \tau would go to infinity (is it bounded by 1)? If the stopping time sequence is bounded, does this define a local martingale? Thanks!

  3. For the example of local martingale can we used yo stopping times such that the property of martingale is not satisfied

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s