Almost Sure

14 September 16

Failure of the Martingale Property For Stochastic Integration

If X is a cadlag martingale and {\xi} is a uniformly bounded predictable process, then is the integral

\displaystyle  Y=\int\xi\,dX (1)

a martingale? If {\xi} is elementary this is one of most basic properties of martingales. If X is a square integrable martingale, then so is Y. More generally, if X is an {L^p}-integrable martingale, any {p > 1}, then so is Y. Furthermore, integrability of the maximum {\sup_{s\le t}\lvert X_s\rvert} is enough to guarantee that Y is a martingale. Also, it is a fundamental result of stochastic integration that Y is at least a local martingale and, for this to be true, it is only necessary for X to be a local martingale and {\xi} to be locally bounded. In the general situation for cadlag martingales X and bounded predictable {\xi}, it need not be the case that Y is a martingale. In this post I will construct an example showing that Y can fail to be a martingale. (more…)

14 August 16

Purely Discontinuous Semimartingales

As stated by the Bichteler-Dellacherie theorem, all semimartingales can be decomposed as the sum of a local martingale and an FV process. However, as the terms are only determined up to the addition of an FV local martingale, this decomposition is not unique. In the case of continuous semimartingales, we do obtain uniqueness, by requiring the terms in the decomposition to also be continuous. Furthermore, the decomposition into continuous terms is preserved by stochastic integration. Looking at non-continuous processes, there does exist a unique decomposition into local martingale and predictable FV processes, so long as we impose the slight restriction that the semimartingale is locally integrable.

In this post, I look at another decomposition which holds for all semimartingales and, moreover, is uniquely determined. This is the decomposition into continuous local martingale and purely discontinuous terms which, as we will see, is preserved by the stochastic integral. This is distinct from each of the decompositions mentioned above, except for the case of continuous semimartingales, in which case it coincides with the sum of continuous local martingale and FV components. Before proving the decomposition, I will start by describing the class of purely discontinuous semimartingales which, although they need not have finite variation, do have many of the properties of FV processes. In fact, they comprise precisely of the closure of the set of FV processes under the semimartingale topology. The terminology can be a bit confusing, and it should be noted that purely discontinuous processes need not actually have any discontinuities. For example, all continuous FV processes are purely discontinuous. For this reason, the term `quadratic pure jump semimartingale’ is sometimes used instead, referring to the fact that their quadratic variation is a pure jump process. Recall that quadratic variations and covariations can be written as the sum of continuous and pure jump parts,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle [X]_t&\displaystyle=[X]^c_t+\sum_{s\le t}(\Delta X_s)^2,\smallskip\\ \displaystyle [X,Y]_t&\displaystyle=[X,Y]^c_t+\sum_{s\le t}\Delta X_s\Delta Y_s. \end{array} (1)

The statement that the quadratic variation is a pure jump process is equivalent to saying that its continuous part, {[X]^c}, is zero. As the only difference between the generalized Ito formula for semimartingales and for FV processes is in the terms involving continuous parts of the quadratic variations and covariations, purely discontinuous semimartingales behave much like FV processes under changes of variables and integration by parts. Yet another characterisation of purely discontinuous semimartingales is as sums of purely discontinuous local martingales — which were studied in the previous post — and of FV processes.

Rather than starting by choosing one specific property to use as the definition, I prove the equivalence of various statements, any of which can be taken to define the purely discontinuous semimartingales.

Theorem 1 For a semimartingale X, the following are equivalent.

  1. {[X]^c=0}.
  2. {[X,Y]^c=0} for all semimartingales Y.
  3. {[X,Y]=0} for all continuous semimartingales Y.
  4. {[X,M]=0} for all continuous local martingales M.
  5. {X=M+V} for a purely discontinuous local martingale M and FV process V.
  6. there exists a sequence {\{X^n\}_{n=1,2,\ldots}} of FV processes such that {X^n\rightarrow X} in the semimartingale topology.

(more…)

8 August 16

Purely Discontinuous Local Martingales

The previous post introduced the idea of a purely discontinuous local martingale. In the context of that post, such processes were used to construct local martingales with prescribed jumps, and enabled us to obtain uniqueness in the constructions given there. However, purely discontinuous local martingales are a very useful concept more generally in martingale and semimartingale theory, so I will go into more detail about such processes now. To start, we restate the definition from the previous post.

Definition 1 A local martingale X is said to be purely discontinuous iff XM is a local martingale for all continuous local martingales M.

We can show that every local martingale decomposes uniquely into continuous and purely discontinuous parts. Continuous local martingales are well understood — for instance, they can always be realized as time-changed Brownian motions. On the other hand, as we will see in a moment, purely discontinuous local martingales can be realized as limits of FV processes, and arguments involving FV local martingales can often to be extended to the purely discontinuous case. So, decomposition (1) below is useful as it allows arguments involving continuous-time local martingales to be broken down into different approaches involving their continuous and purely discontinuous parts. As always, two processes are considered to be equal if they are equivalent up to evanescence.

Theorem 2 Every local martingale X decomposes uniquely as

\displaystyle  X = X^{\rm c} + X^{\rm d} (1)

where {X^{\rm c}} is a continuous local martingale with {X^{\rm c}_0=0} and {X^{\rm d}} is a purely discontinuous local martingale.

Proof: As the process {H=\Delta X} is, by definition, equal to the jump process of a local martingale then it satisfies the hypothesis of Theorem 5 of the previous post. So, there exists a purely discontinuous local martingale {X^{\rm d}} with {\Delta X^{\rm d}=H=\Delta X}. We can take {X^{\rm d}_0=X_0} so that {X^{\rm c}=X-X^{\rm d}} is a continuous local martingale starting from 0.

If {X=\tilde X^{\rm c}+\tilde X^{\rm d}} is another such decomposition, then {\tilde X^{\rm d}} and {X^{\rm d}} have the same jumps and initial value so, by Lemma 3 of the previous post, {\tilde X^{\rm d}=X^{\rm d}}. ⬜

Throughout the remainder of this post, the notation {X^{\rm c}} and {X^{\rm d}} will be used to denote the continuous and purely discontinuous parts of a local martingale X, as given by decomposition (1). Using the notation {\mathcal{M}_{\rm loc}}, {\mathcal{M}_{{\rm loc},0}^{\rm c}} and {\mathcal{M}_{\rm loc}^{\rm d} } respectively for the spaces of local martingales, continuous local martingales starting from zero and the purely discontinuous local martingales, Theorem 2 can be expressed succinctly as

\displaystyle  \mathcal{M}_{\rm loc} = \mathcal{M}_{{\rm loc},0}^{\rm c} \oplus \mathcal{M}_{\rm loc}^{\rm d}. (2)

That is, {\mathcal{M}_{\rm loc}} is the direct sum of {\mathcal{M}_{{\rm loc},0}^{\rm c}} and {\mathcal{M}_{\rm loc}^{\rm d}}. Definition 2 identifies the purely discontinuous local martingales to be, in a sense, orthogonal to the continuous local martingales. Then, (2) can be understood as the decomposition of {\mathcal{M}_{\rm loc}} into the direct sum of the closed subspace {\mathcal{M}_{{\rm loc},0}^{\rm c}} and its orthogonal complement. This does in fact give an alternative, elementary, and commonly used, method of proving decomposition (1). As we have already shown the rather strong result of Theorem 5 from the previous post, the quickest way of proving the decomposition was to simply apply this result. I’ll give more details on the more elementary approach further below.

Definition 1 used above for the class of purely discontinuous local martingales was very convenient for our purposes, as it leads immediately to the proof of Theorem 2. However, there are many alternative characterizations of such processes. For example, they are precisely the processes which are limits of FV local martingales in a strong enough sense. They can also be characterized in terms of their quadratic variations and covariations. Recall that the quadratic variation and covariation are FV processes with jumps {\Delta[X]=(\Delta X)^2} and {\Delta[X,Y]=\Delta X\Delta Y}, so that they can be decomposed into continuous and pure jump components,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle [X]_t &\displaystyle=[X]^c_t+\sum_{s\le t}(\Delta X_s)^2,\smallskip\\ \displaystyle [X,Y]_t &\displaystyle=[X,Y]^c_t+\sum_{s\le t}\Delta X_s\Delta Y_s. \end{array} (3)

The following theorem gives several alternative characterizations of the class of purely discontinuous local martingales.

Theorem 3 For a local martingale X, the following are equivalent.

  1. X is purely discontinuous.
  2. {[X,Y]=0} for all continuous local martingales Y.
  3. {[X,Y]^c=0} for all local martingales Y.
  4. {[X]^c=0}.
  5. there exists a sequence {\{X^n\}_{n=1,2,\ldots}} of FV local martingales such that

    \displaystyle  {\mathbb E}\left[\sup_{t\ge0}(X^n_t-X_t)^2\right]\rightarrow0.

(more…)

25 July 16

Constructing Martingales with Prescribed Jumps

In this post we will describe precisely which processes can be realized as the jumps of a local martingale. This leads to very useful decomposition results for processes — see Theorem 10 below, where we give a decomposition of a process X into martingale and predictable components. As I will explore further in future posts, this enables us to construct particularly useful decompositions for local martingales and semimartingales.

Before going any further, we start by defining the class of local martingales which will be used to match prescribed jump processes. The purely discontinuous local martingales are, in a sense, the orthogonal complement to the class of continuous local martingales.

Definition 1 A local martingale X is said to be purely discontinuous iff XM is a local martingale for all continuous local martingales M.

The class of purely discontinuous local martingales is often denoted as {\mathcal{M}_{\rm loc}^{\rm d}}. Clearly, any linear combination of purely discontinuous local martingales is purely discontinuous. I will investigate {\mathcal{M}_{\rm loc}^{\rm d}} in more detail later but, in order that we do have plenty of examples of such processes, we show that all FV local martingales are purely discontinuous.

Lemma 2 Every FV local martingale is purely discontinuous.

Proof: If X is an FV local martingale and M is a continuous local martingale then we can compute the quadratic covariation,

\displaystyle  [X,M]_t=\sum_{s\le t}\Delta X_s\Delta M_s=0.

The first equality follows because X is an FV process, and the second because M is continuous. So, {XM=XM-[X,M]} is a local martingale and X is purely discontinuous. ⬜

Next, an important property of purely discontinuous local martingales is that they are determined uniquely by their jumps. Throughout these notes, I am considering two processes to be equal whenever they are equal up to evanescence.

Lemma 3 Purely discontinuous local martingales are uniquely determined by their initial value and jumps. That is, if X and Y are purely discontinuous local martingales with {X_0=Y_0} and {\Delta X = \Delta Y}, then {X=Y}.

Proof: Setting {M=X-Y} we have {M_0=0} and {\Delta M = 0}. So, M is a continuous local martingale and {M^2= MX-MY} is a local martingale starting from zero. Hence, it is a supermartingale and we have

\displaystyle  {\mathbb E}[M_t^2]\le{\mathbb E}[M_0^2]=0.

So {M_t=0} almost surely and, by right-continuity, {M=0} up to evanescence. ⬜

Note that if X is a continuous local martingale, then the constant process {Y_t=X_0} has the same initial value and jumps as X. So Lemma 3 has the immediate corollary.

Corollary 4 Any local martingale which is both continuous and purely discontinuous is almost surely constant.

Recalling that the jump process, {\Delta X}, of a cadlag adapted process X is thin, we now state the main theorem of this post and describe precisely those processes which occur as the jumps of a local martingale.

Theorem 5 Let H be a thin process. Then, {H=\Delta X} for a local martingale X if and only if

  1. {\sqrt{\sum_{s\le t}H_s^2}} is locally integrable.
  2. {{\mathbb E}[1_{\{\tau < \infty\}}H_\tau\;\vert\mathcal{F}_{\tau-}]=0} (a.s.) for all predictable stopping times {\tau}.

Furthermore, X can be chosen to be purely discontinuous with {X_0=0}, in which case it is unique.

(more…)

3 May 11

Continuous Semimartingales

A stochastic process is a semimartingale if and only if it can be decomposed as the sum of a local martingale and an FV process. This is stated by the Bichteler-Dellacherie theorem or, alternatively, is often taken as the definition of a semimartingale. For continuous semimartingales, which are the subject of this post, things simplify considerably. The terms in the decomposition can be taken to be continuous, in which case they are also unique. As usual, we work with respect to a complete filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge0},{\mathbb P})}, all processes are real-valued, and two processes are considered to be the same if they are indistinguishable.

Theorem 1 A continuous stochastic process X is a semimartingale if and only if it decomposes as

\displaystyle  X=M+A (1)

for a continuous local martingale M and continuous FV process A. Furthermore, assuming that {A_0=0}, decomposition (1) is unique.

Proof: As sums of local martingales and FV processes are semimartingales, X is a semimartingale whenever it satisfies the decomposition (1). Furthermore, if {X=M+A=M^\prime+A^\prime} were two such decompositions with {A_0=A^\prime_0=0} then {M-M^\prime=A^\prime-A} is both a local martingale and a continuous FV process. Therefore, {A^\prime-A} is constant, so {A=A^\prime} and {M=M^\prime}.

It just remains to prove the existence of decomposition (1). However, X is continuous and, hence, is locally square integrable. So, Lemmas 4 and 5 of the previous post say that we can decompose {X=M+A} where M is a local martingale, A is an FV process and the quadratic covariation {[M,A]} is a local martingale. As X is continuous we have {\Delta M=-\Delta A} so that, by the properties of covariations,

\displaystyle  -[M,A]_t=-\sum_{s\le t}\Delta M_s\Delta A_s=\sum_{s\le t}(\Delta A_s)^2. (2)

We have shown that {-[M,A]} is a nonnegative local martingale so, in particular, it is a supermartingale. This gives {\mathbb{E}[-[M,A]_t]\le\mathbb{E}[-[M,A]_0]=0}. Then (2) implies that {\Delta A} is zero and, hence, A and {M=X-A} are continuous. \Box

Using decomposition (1), it can be shown that a predictable process {\xi} is X-integrable if and only if it is both M-integrable and A-integrable. Then, the integral with respect to X breaks down into the sum of the integrals with respect to M and A. This greatly simplifies the construction of the stochastic integral for continuous semimartingales. The integral with respect to the continuous FV process A is equivalent to Lebesgue-Stieltjes integration along sample paths, and it is possible to construct the integral with respect to the continuous local martingale M for the full set of M-integrable integrands using the Ito isometry. Many introductions to stochastic calculus focus on integration with respect to continuous semimartingales, which is made much easier because of these results.

Theorem 2 Let {X=M+A} be the decomposition of the continuous semimartingale X into a continuous local martingale M and continuous FV process A. Then, a predictable process {\xi} is X-integrable if and only if

\displaystyle  \int_0^t\xi^2\,d[M]+\int_0^t\vert\xi\vert\,\vert dA\vert < \infty (3)

almost surely, for each time {t\ge0}. In that case, {\xi} is both M-integrable and A-integrable and,

\displaystyle  \int\xi\,dX=\int\xi\,dM+\int\xi\,dA (4)

gives the decomposition of {\int\xi\,dX} into its local martingale and FV terms.

(more…)

31 August 10

Zero-Hitting and Failure of the Martingale Property

For nonnegative local martingales, there is an interesting symmetry between the failure of the martingale property and the possibility of hitting zero, which I will describe now. I will also give a necessary and sufficient condition for solutions to a certain class of stochastic differential equations to hit zero in finite time and, using the aforementioned symmetry, infer a necessary and sufficient condition for the processes to be proper martingales. It is often the case that solutions to SDEs are clearly local martingales, but is hard to tell whether they are proper martingales. So, the martingale condition, given in Theorem 4 below, is a useful result to know. The method described here is relatively new to me, only coming up while preparing the previous post. Applying a hedging argument, it was noted that the failure of the martingale property for solutions to the SDE {dX=X^c\,dB} for {c>1} is related to the fact that, for {c<1}, the process hits zero. This idea extends to all continuous and nonnegative local martingales. The Girsanov transform method applied here is essentially the same as that used by Carlos A. Sin (Complications with stochastic volatility models, Adv. in Appl. Probab. Volume 30, Number 1, 1998, 256-268) and B. Jourdain (Loss of martingality in asset price models with lognormal stochastic volatility, Preprint CERMICS, 2004-267).

Consider nonnegative solutions to the stochastic differential equation

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle dX=a(X)X\,dB,\smallskip\\ &\displaystyle X_0=x_0, \end{array} (1)

where {a\colon{\mathbb R}_+\rightarrow{\mathbb R}}, B is a Brownian motion and the fixed initial condition {x_0} is strictly positive. The multiplier X in the coefficient of dB ensures that if X ever hits zero then it stays there. By time-change methods, uniqueness in law is guaranteed as long as a is nonzero and {a^{-2}} is locally integrable on {(0,\infty)}. Consider also the following SDE,

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle dY=\tilde a(Y)Y\,dB,\smallskip\\ &\displaystyle Y_0=y_0,\smallskip\\ &\displaystyle \tilde a(y) = a(y^{-1}),\ y_0=x_0^{-1} \end{array} (2)

Being integrals with respect to Brownian motion, solutions to (1) and (2) are local martingales. It is possible for them to fail to be proper martingales though, and they may or may not hit zero at some time. These possibilities are related by the following result.

Theorem 1 Suppose that (1) and (2) satisfy uniqueness in law. Then, X is a proper martingale if and only if Y never hits zero. Similarly, Y is a proper martingale if and only if X never hits zero.

(more…)

16 August 10

Failure of the Martingale Property

In this post, I give an example of a class of processes which can be expressed as integrals with respect to Brownian motion, but are not themselves martingales. As stochastic integration preserves the local martingale property, such processes are guaranteed to be at least local martingales. However, this is not enough to conclude that they are proper martingales. Whereas constructing examples of local martingales which are not martingales is a relatively straightforward exercise, such examples are often slightly contrived and the martingale property fails for obvious reasons (e.g., double-loss betting strategies). The aim here is to show that the martingale property can fail for very simple stochastic differential equations which are likely to be met in practice, and it is not always obvious when this situation arises.

Consider the following stochastic differential equation

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle dX = aX^c\,dB +b X dt,\smallskip\\ &\displaystyle X_0=x, \end{array}

(1)

for a nonnegative process X. Here, B is a Brownian motion and a,b,c,x are positive constants. This a common SDE appearing, for example, in the constant elasticity of variance model for option pricing. Now consider the following question: what is the expected value of X at time t?

The obvious answer seems to be that {{\mathbb E}[X_t]=xe^{bt}}, based on the idea that X has growth rate b on average. A more detailed argument is to write out (1) in integral form

\displaystyle  X_t=x+\int_0^t\,aX^c\,dB+ \int_0^t bX_s\,ds.

(2)

The next step is to note that the first integral is with respect to Brownian motion, so has zero expectation. Therefore,

\displaystyle  {\mathbb E}[X_t]=x+\int_0^tb{\mathbb E}[X_s]\,ds.

This can be differentiated to obtain the ordinary differential equation {d{\mathbb E}[X_t]/dt=b{\mathbb E}[X_t]}, which has the unique solution {{\mathbb E}[X_t]={\mathbb E}[X_0]e^{bt}}.

In fact this argument is false. For {c\le1} there is no problem, and {{\mathbb E}[X_t]=xe^{bt}} as expected. However, for all {c>1} the conclusion is wrong, and the strict inequality {{\mathbb E}[X_t]<xe^{bt}} holds.

The point where the argument above falls apart is the statement that the first integral in (2) has zero expectation. This would indeed follow if it was known that it is a martingale, as is often assumed to be true for stochastic integrals with respect to Brownian motion. However, stochastic integration preserves the local martingale property and not, in general, the martingale property itself. If {c>1} then we have exactly this situation, where only the local martingale property holds. The first integral in (2) is not a proper martingale, and has strictly negative expectation at all positive times. The reason that the martingale property fails here for {c>1} is that the coefficient {aX^c} of dB grows too fast in X.

In this post, I will mainly be concerned with the special case of (1) with a=1 and zero drift.

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} &\displaystyle dX=X^c\,dB,\smallskip\\ &\displaystyle X_0=x. \end{array}

(3)

The general form (1) can be reduced to this special case, as I describe below. SDEs (1) and (3) do have unique solutions, as I will prove later. Then, as X is a nonnegative local martingale, if it ever hits zero then it must remain there (0 is an absorbing boundary).

The solution X to (3) has the following properties, which will be proven later in this post.

  • If {c\le1} then X is a martingale and, for {c<1}, it eventually hits zero with probability one.
  • If {c>1} then X is a strictly positive local martingale but not a martingale. In fact, the following inequality holds

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

    (4)

    (almost surely) for times {s<t}. Furthermore, for any positive constant {p<2c-1}, {{\mathbb E}[X_t^p]} is bounded over {t\ge0} and tends to zero as {t\rightarrow\infty}.

(more…)

20 April 10

Time-Changed Brownian Motion

From the definition of standard Brownian motion B, given any positive constant c, {B_{ct}-B_{cs}} will be normal with mean zero and variance c(ts) for times {t>s\ge 0}. So, scaling the time axis of Brownian motion B to get the new process {B_{ct}} just results in another Brownian motion scaled by the factor {\sqrt{c}}.

This idea is easily generalized. Consider a measurable function {\xi\colon{\mathbb R}_+\rightarrow{\mathbb R}_+} and Brownian motion B on the filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},{\mathbb P})}. So, {\xi} is a deterministic process, not depending on the underlying probability space {\Omega}. If {\theta(t)\equiv\int_0^t\xi^2_s\,ds} is finite for each {t>0} then the stochastic integral {X=\int\xi\,dB} exists. Furthermore, X will be a Gaussian process with independent increments. For piecewise constant integrands, this results from the fact that linear combinations of joint normal variables are themselves normal. The case for arbitrary deterministic integrands follows by taking limits. Also, the Ito isometry says that {X_t-X_s} has variance

\displaystyle  \setlength\arraycolsep{2pt} \begin{array}{rl} \displaystyle{\mathbb E}\left[\left(\int_s^t\xi\,dB\right)^2\right]&\displaystyle={\mathbb E}\left[\int_s^t\xi^2_u\,du\right]\smallskip\\ &\displaystyle=\theta(t)-\theta(s)\smallskip\\ &\displaystyle={\mathbb E}\left[(B_{\theta(t)}-B_{\theta(s)})^2\right]. \end{array}

So, {\int\xi\,dB=\int\sqrt{\theta^\prime(t)}\,dB_t} has the same distribution as the time-changed Brownian motion {B_{\theta(t)}}.

With the help of Lévy’s characterization, these ideas can be extended to more general, non-deterministic, integrands and to stochastic time-changes. In fact, doing this leads to the startling result that all continuous local martingales are just time-changed Brownian motion. (more…)

13 April 10

Lévy’s Characterization of Brownian Motion

Standard Brownian motion, {\{B_t\}_{t\ge 0}}, is defined to be a real-valued process satisfying the following properties.

  1. {B_0=0}.
  2. {B_t-B_s} is normally distributed with mean 0 and variance ts independently of {\{B_u\colon u\le s\}}, for any {t>s\ge 0}.
  3. B has continuous sample paths.

As always, it only really matters is that these properties hold almost surely. Now, to apply the techniques of stochastic calculus, it is assumed that there is an underlying filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},{\mathbb P})}, which necessitates a further definition; a process B is a Brownian motion on a filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},{\mathbb P})} if in addition to the above properties it is also adapted, so that {B_t} is {\mathcal{F}_t}-measurable, and {B_t-B_s} is independent of {\mathcal{F}_s} for each {t>s\ge 0}. Note that the above condition that {B_t-B_s} is independent of {\{B_u\colon u\le s\}} is not explicitly required, as it also follows from the independence from {\mathcal{F}_s}. According to these definitions, a process is a Brownian motion if and only if it is a Brownian motion with respect to its natural filtration.

The property that {B_t-B_s} has zero mean independently of {\mathcal{F}_s} means that Brownian motion is a martingale. Furthermore, we previously calculated its quadratic variation as {[B]_t=t}. An incredibly useful result is that the converse statement holds. That is, Brownian motion is the only local martingale with this quadratic variation. This is known as Lévy’s characterization, and shows that Brownian motion is a particularly general stochastic process, justifying its ubiquitous influence on the study of continuous-time stochastic processes.

Theorem 1 (Lévy’s Characterization of Brownian Motion) Let X be a local martingale with {X_0=0}. Then, the following are equivalent.

  1. X is standard Brownian motion on the underlying filtered probability space.
  2. X is continuous and {X^2_t-t} is a local martingale.
  3. X has quadratic variation {[X]_t=t}.

(more…)

1 April 10

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}.

(more…)

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