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 . A process is said to be integrable if the random variables are integrable, so that .
Definition 1 A martingale, , is an integrable process satisfying
for all .
A closely related concept is that of a submartingale, which is a process which increases on average. This could represent the wealth of a gambler who is involved in games where the odds are in his favour. Similarly, a supermartingale decreases on average.
Definition 2 A process is a
- submartingale if it is integrable and
for all .
- supermartingale if it is integrable and
for all .
This terminology can perhaps be a bit confusing, but it is related to the result that if is an n-dimensional Brownian motion and then is a submartingale or supermartingale when is a subharmonic or, respectively, a superharmonic function.
Clearly, is a submartingale if and only if is a supermartingale, and is a martingale if and only if it is both a submartingale and a supermartingale.
Lemma 3 If is a martingale and is a convex function such that is integrable, then, is a submartingale.
Proof: This is a direct consequence of Jensen’s inequality,
In particular, if is any -integrable martingale for then is a submartingale. A similar result holds for submartingales, although the additional hypothesis that the function is increasing is required.
Lemma 4 Suppose that is a submartingale and is an increasing convex function such that is integrable. Then, is a submartingale.
Proof: Again, Jensen’s inequality gives the result,
where the final inequality follows from the monotonicity of . ⬜
So, the positive part of a submartingale is itself a submartingale.
Martingales and submartingales are especially well behaved under stochastic integration. An elementary or elementary predictable process is of the form
for , times , -measurable random variable and -measurable random variables . Alternatively, is elementary if it is left-continuous and adapted, and there are times such that is constant on each of the intervals and zero on . In particular, these are predictable processes and, in fact, generate the predictable sigma-algebra. It is also clear that the set of elementary processes is closed under linear combinations and products, and is elementary for any elementary and measurable function .
Stochastic integrals of such elementary processes can be written out explicitly. The integral of the process given by (1) with respect to a stochastic process is
The integral over a finite range is,
Note that, for this expression to make sense, it is only strictly necessary for to be elementary for each . Equivalently, is given by expression (1) for and . Alternatively, is left-continuous and adapted, and there is a sequence of times such that it is constant over each of the intervals .
Letting run over the domain , the integral defines a new process. I shall often write , dropping the limits, to express the stochastic integral as a process. This can equivalently be written in the differential form , which is just a shorthand for the integral expression.
These elementary integrals satisfy some basic properties which follow directly from the definitions. Linearity in both the integrand and integrator is clear. Furthermore, associativity holds. If are elementary and then
In differential notation, this is simply . Stopping an integral process at a random time is the same as stopping the integrator,
The full theory of stochastic calculus extends these elementary integrals to arbitrary predictable integrands. However, just the elementary case defined above is enough to get going with. First, considering expectations of stochastic integrals leads to the following alternative definition of martingales and submartingales.
Theorem 5 An adapted integrable process is
- a martingale if and only if
for all bounded elementary processes .
- a submartingale if and only if
for all nonnegative bounded elementary processes .
Proof: It is enough to prove the second statement, because the first one follows immediately from applying this to both and . So, suppose that is a submartingale. An elementary and nonnegative elementary process can be written in the form (1) for nonnegative. Then,
Conversely, suppose that inequality (3) holds. Choosing any and . The process is elementary and
Any non-positive random variable whose expectation is zero must itself be equal to zero, almost surely. So, as required. ⬜
Elementary integrals preserve the martingale property.
Lemma 6 Let be a process and be a bounded elementary process. Define . Then
- If is a martingale then so is .
- If is a submartingale and is non-negative then is a submartingale.
Proof: The first statement follows from the second applied to both and . So, it is enough to consider the case where is a submartingale. If is a nonnegative elementary process then associativity of the integral gives
This inequality follows from the previous lemma and, again by the previous lemma, it shows that is a submartingale. ⬜
Finally, optional stopping of martingales follows from the properties of elementary integrals. This extends the martingale property to random stopping times. The value of an arbitrary process at the random time need not be well behaved, or even measurable, unless we restrict to nice versions of processes. So, for this post, we shall call a stopping time simple if it only takes finitely many values in . If is a bounded simple stopping time taking values in then the process
is elementary and, furthermore, equation (2) can be extended to give
The optional stopping theorem states that the class of martingales is closed under stopping at arbitrary stopping times.
Lemma 7 Let be a martingale (resp. submartingale, supermartingale) and be a simple stopping time. Then, the stopped process is also a martingale (resp. submartingale, supermartingale).
Proof: This follows from applying Lemma 6 to the following identity