The Martingale Representation Theorem

The martingale representation theorem states that any martingale adapted with respect to a Brownian motion can be expressed as a stochastic integral with respect to the same Brownian motion.

Theorem 1 Let B be a standard Brownian motion defined on a probability space {(\Omega,\mathcal{F},{\mathbb P})} and {\{\mathcal{F}_t\}_{t\ge 0}} be its natural filtration.

Then, every {\{\mathcal{F}_t\}}–local martingale M can be written as

\displaystyle M = M_0+\int\xi\,dB

for a predictable, B-integrable, process {\xi}.

As stochastic integration preserves the local martingale property for continuous processes, this result characterizes the space of all local martingales starting from 0 defined with respect to the filtration generated by a Brownian motion as being precisely the set of stochastic integrals with respect to that Brownian motion. Equivalently, Brownian motion has the predictable representation property. This result is often used in mathematical finance as the statement that the Black-Scholes model is complete. That is, any contingent claim can be exactly replicated by trading in the underlying stock. This does involve some rather large and somewhat unrealistic assumptions on the behaviour of financial markets and ability to trade continuously without incurring additional costs. However, in this post, I will be concerned only with the mathematical statement and proof of the representation theorem.

In more generality, the martingale representation theorem can be stated for a d-dimensional Brownian motion as follows.

Theorem 2 Let {B=(B^1,\ldots,B^d)} be a d-dimensional Brownian motion defined on the filtered probability space {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},{\mathbb P})}, and suppose that {\{\mathcal{F}_t\}} is the natural filtration generated by B and {\mathcal{F}_0}.

\displaystyle \mathcal{F}_t=\sigma\left(\{B_s\colon s\le t\}\cup\mathcal{F}_0\right)

Then, every {\{\mathcal{F}_t\}}-local martingale M can be expressed as

\displaystyle M=M_0+\sum_{i=1}^d\int\xi^i\,dB^i (1)

for predictable processes {\xi^i} satisfying {\int_0^t(\xi^i_s)^2\,ds<\infty}, almost surely, for each {t\ge0}.

Continue reading “The Martingale Representation Theorem” →

SDEs Under Changes of Time and Measure

The previous two posts described the behaviour of standard Brownian motion under stochastic changes of time and equivalent changes of measure. I now demonstrate some applications of these ideas to the study of stochastic differential equations (SDEs). Surprisingly strong results can be obtained and, in many cases, it is possible to prove existence and uniqueness of solutions to SDEs without imposing any continuity constraints on the coefficients. This is in contrast to most standard existence and uniqueness results for both ordinary and stochastic differential equations, where conditions such as Lipschitz continuity is required. For example, consider the following SDE for measurable coefficients {a,b\colon{\mathbb R}\rightarrow{\mathbb R}} and a Brownian motion B

\displaystyle dX_t=a(X_t)\,dB_t+b(X_t)\,dt. (1)

If a is nonzero, {a^{-2}} is locally integrable and b/a is bounded then we can show that this has weak solutions satisfying uniqueness in law for any specified initial distribution of X. The idea is to start with X being a standard Brownian motion and apply a change of time to obtain a solution to (1) in the case where the drift term b is zero. Then, a Girsanov transformation can be used to change to a measure under which X satisfies the SDE for nonzero drift b. As these steps are invertible, every solution can be obtained from a Brownian motion in this way, which uniquely determines the distribution of X.

A standard example demonstrating the concept of weak solutions and uniqueness in law is provided by Tanaka’s SDE

\displaystyle dX_t={\rm sgn}(X_t)\,dB_t (2)

Continue reading “SDEs Under Changes of Time and Measure” →

Girsanov Transformations

Girsanov transformations describe how Brownian motion and, more generally, local martingales behave under changes of the underlying probability measure. Let us start with a much simpler identity applying to normal random variables. Suppose that X and {Y=(Y^1,\ldots,Y^n)} are jointly normal random variables defined on a probability space {(\Omega,\mathcal{F},{\mathbb P})}. Then {U\equiv\exp(X-\frac{1}{2}{\rm Var}(X)-{\mathbb E}[X])} is a positive random variable with expectation 1, and a new measure {{\mathbb Q}=U\cdot{\mathbb P}} can be defined by {{\mathbb Q}(A)={\mathbb E}[1_AU]} for all sets {A\in\mathcal{F}}. Writing {{\mathbb E}_{\mathbb Q}} for expectation under the new measure, then {{\mathbb E}_{\mathbb Q}[Z]={\mathbb E}[UZ]} for all bounded random variables Z. The expectation of a bounded measurable function {f\colon{\mathbb R}^n\rightarrow{\mathbb R}} of Y under the new measure is

\displaystyle {\mathbb E}_{\mathbb Q}\left[f(Y)\right]={\mathbb E}\left[f\left(Y+{\rm Cov}(X,Y)\right)\right], (1)

where {{\rm Cov}(X,Y)} is the covariance. This is a vector whose i’th component is the covariance {{\rm Cov}(X,Y^i)}. So, Y has the same distribution under {{\mathbb Q}} as {Y+{\rm Cov}(X,Y)} has under {{\mathbb P}}. That is, when changing to the new measure, Y remains jointly normal with the same covariance matrix, but its mean increases by {{\rm Cov}(X,Y)}. Equation (1) follows from a straightforward calculation of the characteristic function of Y with respect to both {{\mathbb P}} and {{\mathbb Q}}.

Now consider a standard Brownian motion B and fix a time {T>0} and a constant {\mu}. Then, for all times {t\ge 0}, the covariance of {B_t} and {B_T} is {{\rm Cov}(B_t,B_T)=t\wedge T}. Applying (1) to the measure {{\mathbb Q}=\exp(\mu B_T-\mu^2T/2)\cdot{\mathbb P}} shows that

\displaystyle B_t=\tilde B_t + \mu (t\wedge T)

where {\tilde B} is a standard Brownian motion under {{\mathbb Q}}. Under the new measure, B has gained a constant drift of {\mu} over the interval {[0,T]}. Such transformations are widely applied in finance. For example, in the Black-Scholes model of option pricing it is common to work under a risk-neutral measure, which transforms the drift of a financial asset to be the risk-free rate of return. Girsanov transformations extend this idea to much more general changes of measure, and to arbitrary local martingales. However, as shown below, the strongest results are obtained for Brownian motion which, under a change of measure, just gains a stochastic drift term. Continue reading “Girsanov Transformations” →

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(t–s) 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. Continue reading “Time-Changed Brownian Motion” →

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 t–s 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}.

Continue reading “Lévy’s Characterization of Brownian Motion” →

Special Processes

The point in these stochastic calculus notes has been reached where the theory of stochastic integration is sufficiently well developed to apply in a wide range of situations.

Results such as Ito’s lemma, properties of quadratic variations and existence and uniqueness of solutions to stochastic differential equations followed quite directly from the definition of stochastic integration. Then, once it was shown that integration with respect to martingales is well-defined, results such as preservation of the local martingale property and Ito’s isometry also followed without too much effort.

Over the next few posts, I will take a break from further development of the general theory. Instead, I look at certain special processes, applying the calculus developed so far and gaining a few examples to motivate further development of the theory.

This will include properties of Brownian motion, such as Lévy’s characterization, Girsanov transforms, stochastic time changes and martingale representation. Other important processes which I take a brief look at include Bessel processes, the Poisson process and the Cauchy process. We will also derive a general description of processes with independent increments, including the Lévy-Khintchine formula characterizing Lévy processes.

The Burkholder-Davis-Gundy Inequality

The Burkholder-Davis-Gundy inequality is a remarkable result relating the maximum of a local martingale with its quadratic variation. Recall that [X] denotes the quadratic variation of a process X, and {X^*_t\equiv\sup_{s\le t}\vert X_s\vert} is its maximum process.

Theorem 1 (Burkholder-Davis-Gundy) For any {1\le p<\infty} there exist positive constants {c_p,C_p} such that, for all local martingales X with {X_0=0} and stopping times {\tau}, the following inequality holds.

\displaystyle c_p{\mathbb E}\left[ [X]^{p/2}_\tau\right]\le{\mathbb E}\left[(X^*_\tau)^p\right]\le C_p{\mathbb E}\left[ [X]^{p/2}_\tau\right]. (1)

Furthermore, for continuous local martingales, this statement holds for all {0<p<\infty}.

A proof of this result is given below. For {p\ge 1}, the theorem can also be stated as follows. The set of all cadlag martingales X starting from zero for which {{\mathbb E}[(X^*_\infty)^p]} is finite is a vector space, and the BDG inequality states that the norms {X\mapsto\Vert X^*_\infty\Vert_p={\mathbb E}[(X^*_\infty)^p]^{1/p}} and {X\mapsto\Vert[X]^{1/2}_\infty\Vert_p} are equivalent.

The special case p=2 is the easiest to handle, and we have previously seen that the BDG inequality does indeed hold in this case with constants {c_2=1}, {C_2=4}. The significance of Theorem 1, then, is that this extends to all {p\ge1}.

One reason why the BDG inequality is useful in the theory of stochastic integration is as follows. Whereas the behaviour of the maximum of a stochastic integral is difficult to describe, the quadratic variation satisfies the simple identity {\left[\int\xi\,dX\right]=\int\xi^2\,d[X]}. Recall, also, that stochastic integration preserves the local martingale property. Stochastic integration does not preserve the martingale property. In general, integration with respect to a martingale only results in a local martingale, even for bounded integrands. In many cases, however, stochastic integrals are indeed proper martingales. The Ito isometry shows that this is true for square integrable martingales, and the BDG inequality allows us to extend the result to all {L^p}-integrable martingales, for {p> 1}.

Theorem 2 Let X be a cadlag {L^p}-integrable martingale for some {1<p<\infty}, so that {{\mathbb E}[\vert X_t\vert^p]<\infty} for each t. Then, for any bounded predictable process {\xi}, {Y\equiv\int\xi\,dX} is also an {L^p}-integrable martingale.

Continue reading “The Burkholder-Davis-Gundy Inequality” →

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}.

Continue reading “Continuous Local Martingales” →

Quadratic Variations and the Ito Isometry

As local martingales are semimartingales, they have a well-defined quadratic variation. These satisfy several useful and well known properties, such as the Ito isometry, which are the subject of this post. First, the covariation [X,Y] allows the product XY of local martingales to be decomposed into local martingale and FV terms. Consider, for example, a standard Brownian motion B. This has quadratic variation {[B]_t=t} and it is easily checked that {B^2_t-t} is a martingale.

Lemma 1 If X and Y are local martingales then XY-[X,Y] is a local martingale.

In particular, {X^2-[X]} is a local martingale for all local martingales X.

Proof: Integration by parts gives

\displaystyle XY-[X,Y] = X_0Y_0+\int X_-\,dY+\int Y_-\,dX

which, by preservation of the local martingale property, is a local martingale. ⬜

Continue reading “Quadratic Variations and the Ito Isometry” →

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}}.

Continue reading “Preservation of the Local Martingale Property” →