# Skorohod embedding

Background

Suppose we are given a standard Brownian motion $(B_t)$, and a stopping time T. Then, so long as T satisfies one of the regularity conditions under which the Optional Stopping Theorem applies, we know that $\mathbb{E}[B_T]=0$. (See here for a less formal introduction to OST.) Furthermore, since $B_t^2-t$ is a martingale, $\mathbb{E}[B_T^2]=\mathbb{E}[T]$, so if the latter is finite, so is the former.

Now, using the strong Markov property of Brownian motion, we can come up with a sequence of stopping times $0=T_0, T_1, T_2,\ldots$ such that the increments $T_k-T_{k-1}$ are IID with the same distribution as T. Then $0,B_{T_1},B_{T_2},\ldots$ is a centered random walk. By taking T to be the hitting time of $\{-1,+1\}$, it is easy to see that we can embed simple random walk in a Brownian motion using this approach.

Embedding simple random walk in Brownian motion.

The Skorohod embedding question asks: can all centered random walks be constructed in this fashion, by stopping Brownian motion at a sequence of stopping time? With the strong Markov property, it immediately reduces the question of whether all centered finite-variance distributions X can be expressed as $B_T$ for some integrable stopping time T.

The answer to this question is yes, and much of what follows is drawn from, or at least prompted by Obloj’s survey paper which details the problem and rich history of the many approaches to its solution over the past seventy years.

Applications and related things

The relationship between random walks and Brownian motion is a rich one. Donsker’s invariance principle asserts that Brownian motion appears as the scaling limit of a random walk. Indeed, one can construct Brownian motion itself as the limit of a sequence of consistent random walks with normal increments on an increasingly dense set of times. Furthermore, random walks are martingales, and we know that continuous, local martingales can be expressed as a (stochastically) time-changed Brownian motion, from the Dubins-Schwarz theorem.

The Skorohod embedding theorem can be used to prove results about random walks with general distribution by proving the corresponding result for Brownian motion, and checking that the construction of the sequence of stopping times has the right properties to allow the result to be carried back to the original setting. It obviously also gives a coupling between a individual random walk and a Brownian motion which may be useful in some contexts, as well as a coupling between any pair of random walks. This is useful in proving results for random walks which are much easier for special cases of the distribution. For example, when the increments are Gaussian, or when there are combinatorial approaches to a problem about simple random walk. At the moment no aspect of this blog schedule is guaranteed, but I plan to talk about the law of the iterated logarithm shortly, whose proof is approachable in both of these settings, as well as for Brownian motion, and Skorohod embedding provides the route to the general proof.

At the end, we will briefly compare some other ways to couple a random walk and a Brownian motion.

One thing we could do is sample a copy of X independently from the Brownian motion, then declare $T= \tau_{X}:= \inf\{t\ge 0: B_t=X\}$, the hitting time of (random value) X. But recall that unfortunately $\tau_x$ has infinite expectation for all non-zero x, so this doesn’t fit the conditions required to use OST.

Skorohod’s original method is described in Section 3.1 of Obloj’s notes linked above. The method is roughly to pair up positive values taken by X appropriately with negative values taken by X in a clever way. If we have a positive value b and a negative value a, then $\tau_{a,b}$, the first hitting time of $\mathbb{R}\backslash (a,b)$ is integrable. Then we choose one of these positive-negative pairs according to the projection of the distribution of X onto the pairings, and let T be the hitting time of this pair of values. The probability of hitting b conditional on hitting {a,b} is easy to compute (it’s $\frac{-a}{b-a}$) so we need to have chosen our pairs so that the ‘probability’ of hitting b (ie the density) comes out right. In particular, this method has to start from continuous distributions X, and treat atoms in the distribution of X separately.

The case where the distribution X is symmetric (that is $X\stackrel{d}=-X$) is particularly clear, as then the pairs should be $(-x,x)$.

However, it feels like there is enough randomness in Brownian motion already, and subsequent authors showed that indeed it wasn’t necessary to introduce extra randomness to provide a solution.

One might ask whether it’s possible to generate the distribution on the set of pairs (as above) out of the Brownian motion itself, but independently from all the hitting times. It feels like it might be possible to make the distribution on the pairs measurable with respect to

$\mathcal{F}_{0+} = \bigcap\limits_{t>0} \mathcal{F}_t,$

the sigma-algebra of events determined by limiting behaviour as $t\rightarrow 0$ (which is independent of hitting times). But of course, unfortunately $\mathcal{F}_{0+}$ has a zero-one law, so it’s not possible to embed non-trivial distributions there.

Dubins solution

The exemplar for solutions without extra randomness is due to Dubins, shortly after Skorohod’s original argument. The idea is to express the distribution X as the almost sure limit of a martingale. We first use the hitting time of a pair of points to ‘decide’ whether we will end up positive or negative, and then given this information look at the hitting time (after this first time) of two subsequent points to ‘decide’ which of four regions of the real interval we end up in.

I’m going to use different notation to Obloj, corresponding more closely with how I ended up thinking about this method. We let

$a_+:= \mathbb{E}[X \,|\, X>0], \quad a_- := \mathbb{E}[X\,|\, X<0],$ (*)

and take $T_1 = \tau_{\{a_-,a_+\}}$. We need to check that

$\mathbb{P}\left( B_{T_1}=a_+\right) = \mathbb{P}\left(X>0\right),$

for this to have a chance of working. But we know that

$\mathbb{P}\left( B_{T_1}=a_+\right) = \frac{a_+}{a_+-a_-},$

and we can also attack the other side using (*) and the fact that $\mathbb{E}[X]=0$, using the law of total expectation:

$0=\mathbb{E}[X]=\mathbb{E}[X\,|\, X>0] \mathbb{P}(X>0) + \mathbb{E}[X\,|\,X<0]\mathbb{P}(X<0) = a_+ \mathbb{P}(X>0) + a_- \left(1-\mathbb{P}(X>0) \right),$

$\Rightarrow\quad \mathbb{P}(X>0)=\frac{a_+}{a_+-a_-}.$

Now we define

$a_{++}=\mathbb{E}[X \,|\, X>a_+],\quad a_{+-}=\mathbb{E}[X\,|\, 0

and similarly $a_{-+},a_{--}$. So then, conditional on $B_{T_1}=a_+$, we take

$T_2:= \inf_{t\ge T_1}\left\{ B_t\not\in (a_{+-},a_{++}) \right\},$

and similarly conditional on $B_{T_1}=a_-$. By an identical argument to the one we have just deployed, we have $\mathbb{E}\left[B_{T_2} \,|\,\mathcal{F}_{T_1} \right] = B_{T_1}$ almost surely. So, although the $a_{+-+}$ notation now starts to get very unwieldy, it’s clear we can keep going in this way to get a sequence of stopping times $0=T_0,T_1,T_2,\ldots$ where $B_{T_n}$ determines which of the $2^n$ regions of the real line any limit $\lim_{m\rightarrow\infty} B_{T_m}$ should lie in.

A bit of work is required to check that the almost sure limit $T_n\rightarrow T$ is almost surely finite, but once we have this, it is clear that $B_{T_n}\rightarrow B_T$ almost surely, and $B_T$ has the distribution required.

We want to know how close we can make this coupling between a centered random walk with variance 1, and a standard Brownian motion. Here, ‘close’ means uniformly close in probability. For large times, the typical difference between one of the stopping times $0,T_1,T_2,\ldots$ in the Skorohod embedding and its expectation (recall $\mathbb{E}[T_k]=k$) is $\sqrt{n}$. So, constructing the random walk $S_0,S_1,S_2,\ldots$ from the Brownian motion via Skorohod embedding leads to

$\left |S_k - B_k \right| = \omega(n^{1/4}),$

for most values of $k\le n$. Strassen (1966) shows that the true scale of the maximum

$\max_{k\le n} \left| S_k - B_k \right|$

is slightly larger than this, with some extra powers of $\log n$ and $\log\log n$ as one would expect.

The Komlos-Major-Tusnady coupling is a way to do a lot better than this, in the setting where the distribution of the increments has a finite MGF near 0. Then, there exists a coupling of the random walk and the Brownian motion such that

$\max_{k\le n}\left|S_k- B_k\right| = O(\log n).$

That is, there exists C such that

$\left[\max_{k\le n} \left |S_k-B_k\right| - C\log n\right] \vee 0$

is a tight family of distributions, indeed with uniform exponential tail. To avoid digressing infinitely far from my original plan to discuss the proof of the law of iterated logarithm for general distributions, I’ll stop here. I found it hard to find much coverage of the KMT result apart from the challenging original paper, and many versions expressed in the language of empirical processes, which are similar to random walks in many ways relevant to convergence and this coupling, but not for Skorohod embedding. So, here is a link to some slides from a talk by Chatterjee which I found helpful in getting a sense of the history, and some of the modern approaches to this type of normal approximation problem.

# Tightness in Skorohod Space

This post continues the theme of revising topics in the analytic toolkit relevant to proving convergence of stochastic processes. Of particular interest is the question of how to prove that families of Markov chains might have a process scaling limit converging to a solution of some stochastic differential equation, in a generalisation of Donsker’s theorem for Brownian motion. In this post, however, we address more general aspects of convergence of stochastic processes, with particular reference to Skorohod space.

Topological Background

I’ve discussed Skorohod space in a previous post. For now, we focus attention on compactly supported functions, D[0,T]. Some of what follows can be extended to the infinite-time setting easily, and some requires more work. Although we can define a metric on the space of cadlag functions in lots of ways, it is more useful to think topologically, or at least with a more vague sense of metric. We say two cadlag functions are close to one another if there is a reparameterisation of the time-axis, (a function [0,T] to itself) that is uniformly close to the identity function, and when applied to one of the cadlag functions, brings it close to the other cadlag function. Heuristically, two cadlag functions are close if their large jumps are close to one another and of similar size, and if they are uniformly close elsewhere. It is worth remembering that a cadlag function on even an unbounded interval can have only countably many jumps, and only finitely many with magnitude greater than some threshold on any compact interval.

For much of the theory one would like to use, it is useful for the spaces under investigation to be separable. Recall a topological space is separable if there exists a countable dense subset. Note in particular that D[0,T] is not separable under the uniform metric, since we can define $f_x(\cdot)=\mathbf{1}_{(\cdot \ge x)}$ for each $x\in[0,T]$, then $||f_x-f_y||_\infty=1$ whenever $x\ne y$. In particular, we have an uncountable collection of disjoint open sets given by the balls $\mathcal{B}(f_x,\frac12)$, and so the space is not countable. Similarly, $C[0,\infty)$ is not separable. A counterexample might be given by considering functions which take the values {0,1} on the integers. Thus we have a map from $\{0,1\}^{\mathbb{N}}\rightarrow C[0,\infty)$, where the uniform distance between any two distinct image points is at least one, hence the open balls of radius 1/2 around each image point give the same contradiction as before. However, the Stone-Weierstrass theorem shows that C[0,T] is separable, as we can approximate any such function uniformly well by a polynomial, and thus uniformly well by a polynomial with rational coefficients.

In any case, it can be shown that D[0,T] is separable with respect to the natural choice of metric. It can also be shown that there is a metric which gives the same open sets (hence is a topologically equivalent metric) under which D[0,T] is complete, and hence a Polish space.

Compactness in C[0,T] and D[0,T]

We are interested in tightness of measures on D[0,T], so first we need to address compactness for sets of deterministic functions in D[0,T]. First, we consider C[0,T]. Here, the conditions for a set of functions to be compact is given by the celebrated Arzela-Ascoli theorem. We are really interested in compactness as a property of size, so we consider instead relative compactness. A set is relatively compact (sometimes pre-compact) if its closure is compact. For the existence of subsequential limits, this is identical to compactness, only now we allow the possibility of the limit point lying outside the set.

We note that the function $C[0,T]\rightarrow \mathbb{R}$ given by $||f||_\infty$ is continuous, and hence uniform boundedness is certainly a required condition for compactness in C[0,T]. Arzela-Ascoli states that uniform boundedness plus equicontinuity is sufficient for a set of such functions to be compact. Equicontinuity should be thought of as uniform continuity that is uniform among all the functions in the set, rather than just within the argument of an individual particular function.

For identical reasons, we need uniform boundedness for relative compactness in D[0,T], but obviously uniform continuity won’t work as a criterion for discontinuous functions! We seek some analogue of the modulus of continuity that ignores jumps. We define

$\omega'_\delta(f):=\inf_{\{t_i\}} \max_i \sup_{s,t\in[t_{i-1},t_i)} |f(s)-f(t)|,$

where the infimum is taken over all meshes $0=t_0 with $t_i-t_{i-1}>\delta$. Note that as $\delta\downarrow 0$, we can, if we want, place the $t_i$ so that large jumps of the function f take place over the boundaries between adjacent parts of the mesh. In particular, for a given cadlag function, it can be shown fairly easily that $\omega'_\delta(f)\downarrow 0$ as $\delta\rightarrow 0$. Then, unsurprisingly, in a similar fashion to the Arzela-Ascoli theorem, it follows that a set of functions $A\subset D[0,T]$ is relatively compact if it is uniformly bounded, and

$\lim_{\delta\rightarrow 0} \sup_{f\in A}\omega'_\delta(f)=0.$

Note that this ‘modulus of continuity’ needs to decay uniformly across the set of functions, but that we do not need to choose the mesh at level $\delta$ uniformly across all functions. This would obviously not work, as then the functions $\mathbf{1}_{(\cdot\ge x_n)}$ for any sequence $x_n\rightarrow x$ would not be compact, but they clearly converge in Skorohod space!

Tightness in C[0,T] and D[0,T]

Naturally, we are mainly interested in (probability) measures on D[0,T], and in particular conditions for tightness on this space. Recall a family of measures is tight if for any $\epsilon>0$, there exists some compact set A such that

$\pi(A)>1-\epsilon,\quad \forall \pi\in\Pi.$

So, for measures $(\mu_n)$ on D[0,T], the sequence is tight precisely if for any $\epsilon>0$, there exists $M,\delta$ and some N such that for any n>N, both

$\mu_n(||f||_\infty >M)\le \epsilon,\quad \mu_n(\omega'_\delta(f)>\epsilon)\le \epsilon$

hold. In fact, the second condition controls variation sufficiently strongly, that we can replace the first condition with

$\mu_n(|f(0)|>M)\le \epsilon.$

Often we might be taking some sort of scaling limit of these processes in D[0,T], where the jumps become so small in the limit that we expect the limit process to be continuous, perhaps an SDE or diffusion. If we can replace $\omega'_\delta$ by $\omega_\delta$, the standard modulus of continuity, then we have the additional that any weak limit lies in C[0,T].

In general, to prove convergence of some stochastic processes, we will want to show that the processes are tight, by demonstrating the properties above, or something equivalent. Then Prohorov’s theorem (which I tend to think of as a probabilistic functional version of Bolzano-Weierstrass) asserts that the family of processes has a weak subsequential limit. Typically, one then shows that any weak subsequential limit must have the law of some particular random process. Normally this is achieved by showing some martingale property (eg for an SDE) in the limit, often by using the Skorohod representation theorem to use almost sure subsequential convergence rather than merely weak convergence. Then one argues that there is a unique process with this property and a given initial distribution. So since all weak subsequential limits are this given process, in fact the whole family has a weak limit.

# Ornstein-Uhlenbeck Process

A large part of my summer has been spent proving some technical results pertaining to the convergence of some functionals of a critical Frozen Percolation process. This has been worthwhile, but hasn’t involved a large amount of reading around anything in particular, which has probably contributed to the lack of posts in recent months. Perhaps a mixture of that and general laziness?

Anyway, it turns out that the limit of the discrete processes under consideration is the Ornstein-Uhlenbeck process. The sense in which this limit holds (or at least, for now, is conjectured to hold) is something for another article. However, I thought it would be worth writing a bit about this particular process and why it is interesting.

The O-U process is described by the SDE

$dX_t=-\beta (X_t-\mu)dt+\sigma dW_t,$

where W is a standard Brownian motion. We think of $\mu$ as the ‘mean’. The extent to which this behaves as a mean will be discussed shortly. The process is then mean-reverting, in the sense that the drift is directed against deviations of the process away from this mean. The parameter $\beta$ measures the extent of this mean reversion, while as usual $\sigma$ controls the magnitude of the Brownian noise.

The motivation for considering mean-reverting processes is considerable. One measure of this is how many equations with articles on Wikipedia turn out to be precisely this Ornstein-Uhlenbeck process with different context or notation. In most cases, the motivation arises because Brownian motion is for some reason unsuitable to take as a canonical random process. We will see why the O-U process is somehow the next most canonical choice for a random process.

In physics, it is sometimes unsatisfactory to model the trajectory of a particle with Brownian motion (even though this motivated the name…) as the velocities are undefined (see this post from ages ago), or infinite, depending on your definition of velocity. Using the Ornstein-Uhlenbeck process to model the velocity of a particle is often a satisfactory alternative. It is not unreasonable that there should be a mean velocity, presumably zero. The mean reversion models a frictional force from the underlying medium, while the Brownian noise describes random collisions with similar particles.

In financial applications, the Ornstein-Uhlenbeck model has been applied, apparently under the title of the Vasicek model since the 70s to describe quantities such as interest rates where there is some underlying reason to ban indefinite growth, and require mean reversion. Another setting might be a commodity which, because of external driving factors, has over the relevant time-scale well-defined mean value, around which mean-reverting fluctuations on the observed time-scale can be described. As with other financial models, it is undesirable for a process to take negative values. This can be fixed by taking a positive mean, then setting the volatility to be state dependent, decaying to zero as the state tends to zero, so for small values, the positive drift dominates. I don’t fully understand why patching this aspect is significantly more important than patching any other non-realistic properties of the model, but the resulting SDE is, at least in one particular case where the volatility is $\sqrt{X_t}$, called the Cox-Ingersoll-Ross model.

Anyway, a mathematical reason to pay particular attention to this Ornstein-Uhlenbeck process is the following. It is the unique family of continuous Markov processes to have a stationary Gaussian distribution. It is the mean-reverting property that is key. There is no chance of Brownian motion having any stationary distribution, let alone a Gaussian one. If this isn’t clear, you can convince yourself by thinking of the stationary distribution of SRW on $\mathbb{Z}$. Since the process is space-homogeneous, the only stationary measure is the uniform measure.

I want to focus on one particular property of the O-U process, through which some other aspects will be illuminated. If we take $\sigma=\beta$ and let $\beta\rightarrow\infty$, then the stationary processes converge to white noise.

First though, we should note this is perhaps the easiest SDE to solve explicitly. We consider $X_t e^{\theta t}$, and applying Ito’s lemma rapidly gives

$X_t=\mu + (X_0-\mu)e^{-\beta t}+\sigma\int_0^t e^{-\beta(t-s)}dW_s.$

W is Gaussian so the distribution of $X_t$ conditional on $X_0=x_0$ is also Gaussian, and since W is centred we can read off the expectation. Applying the Ito isometry then gives the variance. In conclusion:

$X_t\stackrel{d}{=}\mathcal{N}(\mu+(x_0-\mu)e^{-\beta t}, \frac{\sigma^2}{\beta}(1-e^{-2\beta t})).$

In particular, note that the variation has no dependence on $x_0$. So as t grows to infinity, this converges to $\mathcal{N}(\mu, \frac{\sigma^2}{\beta})$. This is, unsurprisingly, the stationary distribution of the process.

To address the white noise convergence, we need to consider $\text{Cov}(X_0,X_t)$ in stationarity. Let’s assume WLOG that $\mu=0$ so most of the expectations will vanish. We obtain

$\text{Cov}(X_0,X_t)=\mathbb{E}[X_0X_t]=\mathbb{E}_{x_0}\left[\mathbb{E}[X_t| X_0=x_0]\right]=\mathbb{E}[X_0^2 e^{-\beta t}]= \frac{sigma^2}{2\beta}2^{-\beta t}.$

If we want, the Chapman-Kolmogorov equations work particularly nicely here, and we are able to derive a PDE for the evolution of the density function, though obviously this is very related to the result above. This PDE is known as the Fokker-Planck equation.

So, in particular, when $\sigma=\beta\rightarrow \infty$, this covariance tends to 0. I’m not purporting that this constitutes a proof that the Ornstein-Uhlenbeck processes converge as processes to white noise. It’s not obvious how to define process convergence, not least because there’s flexibility about how to view white noise as a process. One doesn’t really want to define the value of white noise at a particular time, but you can consider the covariance of integrals of white noise over disjoint intervals as a limit, in similar way to convergence of finite dimensional distributions.

The fact that taking $\beta=0$ gives Brownian motion, and this case gives white noise, intermediate versions of the Ornstein-Uhlenbeck process are sometimes referred to as coloured noise.

Finally, the Ornstein-Uhlenbeck process emerges as the scaling limit of mean-reverting discrete Markov chains, analogous to Brownian motion as the scaling limit of simple random walk. One particularly nice example is the Ehrenfest Urn model. We have two urns, and 2N balls. In each time step one of the 2N balls is chosen uniformly at random, and it is moved to the other urn. So a ball is more likely to be removed from an urn with more than N balls. We can view this as a model for molecules in, say a room, with a slightly porous division between them, eg a small hole. More complicated interface models in higher dimensions lead to fascinating PDEs, such as the famous KPZ equation, which are the subject of much ongoing interest in this area.

This result can be an application of the theory of convergence of Markov chains to SDEs pioneered by Stroock and Varadhan, about which more may follow very soon. In any case, it turns out that the fluctuations in the Ehrenfest Urn model are on the scale of $\sqrt{n}$, unsurprisingly, and are given by a centred Ornstein-Uhlenbeck process.

Investigating this has reminded me how much I’ve forgotten, or perhaps how little I ever knew, about the technicalities of stochastic processes are their convergence results, so next up will probably be a summary of all the useful definitions and properties for this sort of analysis.

# Sticky Brownian Motion

This follows on pretty much directly from the previous post about reflected Brownian motion. Recall that this is a process defined on the non-negative reals which looks like Brownian motion away from 0. We consider whether RBM is the only such process, and how any alternative might be constructed as a limit of discrete-time Markov processes.

One of the alternatives is called Sticky Brownian motion. This process spends more time at 0 than reflected Brownian motion. In fact it spends some positive proportion of time at 0. My main aim here is to explain why some intuitive ideas I had about how this might arise are wrong.

The first thought was to ensure that each visit to 0 last some positive measure of time. This could be achieved by staying at 0 for an Exp(1) duration, every time the process visited it. It doesn’t seem unreasonable that this might appear as the limit of a standard SRW adjusted so that on each visit to 0 the walker waits for a time given by independent geometric distributions. These distributions are memoryless, so that is fine, but by Blumenthal’s 0-1 Law, starting from 0 a Brownian motion hits zero infinitely many times before any small time t. So in fact the process described above would be identically zero as before it gets anywhere it would have to spend some amount of time at 0 given by an infinite sum of Exp(1) RVs.

We will return later to the question of why the proposed discrete-time model will still converge to reflected BM rather than anything more exotic. First though, we should discount the possibility of any intermediate level of stickiness, where the set of times spent at 0 still has measure zero, but the local time at 0 grows faster than for standard reflected BM. We can define the local time at 0 through a limit

$L_t=\lim_{\epsilon\downarrow 0}\frac{1}{2\epsilon}\text{Leb}(\{0\le s \le t: |B_t|<\epsilon\})$

of the measure of time spent very near 0, rescaled appropriately. So if the measure of the times when the process is at 0 is zero, then the local time is determined by behaviour near zero rather than by behaviour at zero. More precisely, on the interval $[-\epsilon,\epsilon]$, the process behaves like Brownian motion, except on a set of measure zero, so the local time process should look the same as that of BM itself. Note I don’t claim this as a formal proof, but I hope it is a helpful heuristic for why you can’t alter the local time process without altering the whole process.

At this stage, it seems sensible to define Sticky Brownian motion. For motivation, note that we are looking for a process which spends a positive measure of time at 0. So let’s track this as a process, say $C_t$. Then the set of times when C is increasing is sparse, as it coincides with the process being 0, but we know we cannot wait around at 0 for some interval of time without losing the Markov property. So C shares properties with the local time of a reflected BM. The only difference is that the measure of times when C is increasing is positive here, but zero for the local time.

So it makes sense to construct the extra time spent at zero from the local time of a standard reflected BM. The heuristic is that we slow down the process whenever it is at 0, so that local time becomes real time. We can also control the factor by which this slowing-down happens, so define

$\sigma(s)=\rho L(s)+s,$

where L is the local time process of an underlying reflected BM, and $\rho>0$ is a constant. So $\sigma$ is a map giving a random time-change. Unsurprisingly, we now define Sticky BM as the reflected BM with respect to this time-change. To do this formally, it is easiest to define a family of stopping times $\{\tau_t\}$, such that $\sigma(\tau_t)=t, \tau_{\sigma(s)}=s$, then if X is the reflected BM, define $Y_t=X_{\tau_t}$ for the sticky BM.

It is worth thinking about what the generator of this process should be. In particular, why should it be different to reflected BM? The key observation is that the drift of the underlying reflected BM is essentially infinite at 0. By slowing down the process at 0, this drift becomes finite. So the martingale associated with sticky BM is precisely a time-changed version of the martingale associated with the underlying reflected BM, but this time-change is precisely what is required to give a generator. We get:

$(\mathcal{L}f)(x)=\begin{cases}\frac12f''(x)&\quad x>0\\ \rho^{-1}f'(0) &\quad x=0.\end{cases}$

Now that we have the generator, it starts to become apparent how sticky BM might appear as a limit of discrete-time walks. The process must look like mean-zero, unit-variance RW everywhere except near 0, where the limiting drift should be $\rho^{-1}$. Note that when considering the limiting drift near zero, we are taking a joint limit in x and h. The order of this matters. As explained at the end of the previous article, we only need to worry about the limiting drift along sequences of $x,h$ such that $a_h(x)\rightarrow 0$. If no such sequences exist, or the limiting drift along any of these is infinite, then we actually have a reflected boundary condition.

This highlights one confusing matter about convergence of reflected processes. The boundary of the discrete-time process should converge to the boundary of the reflected process, but we also have to consider where reflective behaviour happens. Can we get sticky BM with reflection only at the boundary in the discrete-time processes? The answer turns out to be no. At the start of this article, I proposed a model of SRW with geometric waiting times whenever the origin was visiting. What is the limit of this?

The trick is to consider how long the discrete process spends near 0, after rescaling. It will spend a multiple 1/p more time at 0 itself, where p is the parameter of the geometric distribution, but no more time than expected at any point $x\in(0,\epsilon)$. But time spent in $(0,\epsilon)$ dominates time spent at 0 before this adjustment, so must also dominate it after the adjustment, so in the limit, the proportion of time spent at 0 is unchanged, and so in particular it cannot be positive.

Because of all of this, in practice it seems that most random walks we might be interested in converge (if they converge to a process at all) to a reflected SDE/diffusion etc, rather than one with sticky boundary conditions. I feel I’ve been talking a lot about Markov processes converging, so perhaps next, or at least soon, I’ll write some more technical things about exactly what conditions and methods are required to prove this.

REFERENCES

S. Varadhan – Chapter 16 from a Lecture Course at NYU can be found here.

# Reflected Brownian Motion

A standard Brownian motion is space-homogeneous, meaning that the behaviour of $B_{T+t}-B_T$ does not depend on the value of $B_T$. By Donsker’s Theorem, such a Brownian motion is also the limit in a process space of any homogeneous random walk with zero-drift and constant variance, after suitable rescaling.

In many applications, however, we are interested in real-valued continuous-time Markov processes that are defined not on the whole of the real line, but on the half-line $\mathbb{R}_{\ge 0}$. So as BM is the fundamental real-valued continuous-time Markov process, we should ask how we might adjust it so that it stays non-negative. In particular, we want to clarify uniqueness, or at least be sure we have found all the sensible ways to make this adjustment, and also to consider how Donsker’s Theorem might work in this setting.

We should consider what properties we want this non-negative BM to have. Obviously, it should be non-negative, but it is also reasonable to demand that it looks exactly like BM everywhere except near 0. But since BM has a scale-invariance property, it is essentially meaningful to say ‘near 0’, so we instead demand that it looks exactly like BM everywhere except at 0. Apart from this, the only properties we want are that it is Markov and has continuous sample paths.

A starting point is so-called reflected Brownian motion, defined by $X_t:=|B_t|$. This is very natural and very convenient for analysis, but there are some problems. Firstly, this has the property that it looks like Brownian motion everywhere except 0 only because BM is space-homogeneous but also symmetric, in the sense that $B_t\stackrel{d}{=}-B_t$. This will be untrue for essentially any other process, so as a general method for how to keep stochastic processes positive, this will be useless. My second objection is a bit more subtle. If we consider this as an SDE, we get

$dX_t=\text{sign}(B_t)dB_t.$

This is a perfectly reasonable SDE but it is undesirable, because we have a function of B as coefficient on the RHS. Ideally, increments of X would be a function of X, and the increments of B, rather than the values of B. That is, we would expect $X_{t+\delta t}-X_t$ to depend on $X_t$ and on $(B_{t+s}-B_t, 0\le s\le \delta t)$, but not on $B_t$ itself, as that means we have to keep track of extra information while constructing X.

So we need an alternative method. One idea might be to add some non-negative process to the BM so that the sum stays non-negative. If this process is deterministic and finite, there there is some positive probability that the sum will eventually be negative, so this won’t do. We are looking therefore so a process which depends on the BM. Obviously we could take $\max(-B_t,0)$, but this sum would then spend macroscopic intervals of time at 0, and these intervals would have the Raleigh distribution (for Brownian excursions) rather than the exponential distribution, hence the process given by the sum would not be memoryless and Markov.

The natural alternative is to look for an increasing process $A_t$, and then it makes sense to talk about the minimal increasing process that has the desired property. A moment’s thought suggests that $A_t=-min_{s\le t}B_t$ satisfies this. So we have the decomposition

$B_t=-A_t+S_t,$

where $S_t$ is the height of B above its running minimum. So S is an ideal alternative definition of reflecting BM. In particular, when B is away from its minimum, $dB_t=dS_t$, so this has the property that it evolves exactly as the driving Brownian motion.

What we have done is to decompose a general continuous process into the sum of a decreasing continuous process and a non-negative process. This is known as the Skorohod problem, and was the subject of much interest, even in the deterministic case. Note that process A has the property that it is locally constant almost everywhere, and is continuous, yet non-constant. Unsurprisingly, since A only changes when the underlying BM is 0, A is continuous with respect to the local time process at 0. In fact, A is the local time process of the underlying Brownian motion, by comparison with the construction by direct reflection.

One alternative approach is to look instead at the generator of the process. Recall that the generator of a process is an operator on some space of functions, with $\mathcal{L}f$ giving the infinitissimal drift of $f(X_t)$. In the case of Brownian motion, the generator $(\mathcal{L}f)(x)=\frac12 f''(x)$ for bounded smooth functions f. This is equivalent to saying that

$f(X_t)-f(X_0)-\int_0^t \frac12 f''(X_s)ds$ (*)

is a martingale. This must hold also for reflected Brownian motion, whenever x is greater than 0. Alternatively, if the function f is zero in a small neighbourhood of 0, it should have the same generator with respect to reflected BM. Indeed, for a general smooth bounded function f, we can still consider the expression (*) with respect to reflected BM. We know this expression behaves as a martingale except when X is zero. If f'(0)>0, and T is some hitting time of 0, then $f(X_{T+\delta T})-f(X_T)\ge 0$, hence the expression (*) is a submartingale. So if we restrict attention to functions with f'(0)=0, the generator remains the same. Indeed, by patching together all such intervals, it can be argued that even if f'(0) is not zero,

$f(X_t)-f(X_0)-\int_0^t \frac12 f''(X_s)ds - f'(0)A_t$

is a martingale, where A is the local time process at zero.

I was aware when I started reading about this that there was another family of processes called ‘Sticky Brownian Motion’ that shared properties with Reflected BM, in that it behaves like standard BM away from zero, but is also constrained to the non-negative reals. I think this will get too long if I also talk about that here, so that can be postponed, and for now we consider reflected BM as a limit of reflected (or other) random walks, bearing in mind that there is at least one other candidate to be the limit.

Unsurprisingly, if we have a family of random walks constrained to the non-negative reals, that are zero-drift unit-variance away from 0, then if they converge as processes, the limit is Brownian away from zero, and non-negative. Note that “away from 0” means after rescaling. So the key aspect is behaviour near zero.

What is the drift of reflected BM at 0? We might suspect it is infinite because of the form of the generator, but we can calculate it directly. Given $X_0=0$, we have:

$\frac{\mathbb{E}X_t}{t}=\frac{\mathbb{E}|B_t|}{t}=\frac{\sqrt{t}\mathbb{E}|B_1|}{t},$

so letting $t\rightarrow 0$, we see indeed that the drift is infinite at 0.

For convergence of discrete processes, we really need the generators to converge. Typically we index the discrete-time processes by the time unit h, which tends to 0, and $b_h(x),a_h(x)$ are the rescaled drift and square-drift from x. We assume that we don’t see macroscopic jumps in the limit. For the case of simple random walk reflected at 0, it doesn’t matter exactly how we construct the joint limit in h and x, as the drift is uniform on x>0, but in general this does matter. I don’t want to discuss sticky BM right now, so it’s probably easiest to be vague and say that the discrete Markov processes converge to reflected BM so long they don’t spend more time than expected near 0 in the limit, as the title ‘sticky’ might suggest.

The two ways in which this can happen is if the volatility term $a_h(x)$ is too small, in which case the process looks almost deterministic near 0, or if the drift doesn’t increase fast enough. And indeed, this leads to two conditions. The first is straightforward, if $a_h(x)$ is bounded below, in the sense that $\liminf_{h,x\rightarrow 0} a_h(x)\ge C>0$, then we have convergence to reflected BM. Alternatively, the only danger can arise down those subsequences where $a_h(x)\rightarrow 0$, so if we have that $b_h(x)\rightarrow +\infty$ whenever $h,x,a_h(x)\rightarrow 0$, then this convergence also holds.

Next time I’ll discuss what sticky BM means, what it doesn’t mean, why it isn’t easy to double the local time, and how to obtain sticky BM as a limit of discrete random walks in a similar way to the above.

REFERENCES

S. Varadhan – Chapter 16 from a Lecture Course at NYU can be found here.

# Critical Components in Erdos-Renyi

In various previous posts, I’ve talked about the phase transition in the Erdos-Renyi random graph process. Recall the definition of the process. Here we will use the Gilbert model G(n,p), where we have n vertices, and between any pair of vertices we add an edge, independently of other pairs with probability p. We are interested in the sparse scaling, where the typical vertex has degree O(1) in n, and so p=c/n for constant c>0, and we assume throughout that n is large. We could alternatively have considered the alternative Erdos-Renyi model where we choose uniformly at random from the set of graphs with n vertices and some fixed number of edges. Almost all the results present work equally well in this setting.

As proved by Erdos and Renyi, the typical component structure of such a graph changes noticeably around the threshold c=1. Below this, in the subcritical regime, all the components are small, meaning of size at most order O(log n). Above this, in the supercritical regime, there is a single giant component on some non-zero proportion of the vertices. The rest of the graph looks subcritical. The case c=1 exhibits a phase transition between these qualitatively different behaviours. They proved that here, the largest component is with high probability O(n^2/3). It seems that they thought this result held whenever c=1-o(1), but it turns out that this is not the case. In this post, I will discuss some aspects of behaviour around criticality, and the tools needed to treat them.

The first question to address is this: how many components of size n^{2/3} are there? It might be plausible that there is a single such component, like for the subsequent giant component. It might also be plausible that there are n^1/3 such components, so O(n) vertices are on such critical components. As then it is clear how we transition out of criticality into supercriticality – all the vertices on critical components coalesce to form the new giant component.

In fact neither of these are correct. The answer is that for all integers k>0, with high probability the k-th largest component is on a size scale of n^2/3. This is potentially a confusing statement. It looks like there are infinitely many such components, but of course for any particular value of n, this cannot be the case. We should think of there being w(1) components, but o(n^b) for any b>0.

The easiest way to see this is by a duality argument, as we have discussed previously for the supercritical phase. If we remove a component of size O(n^2/3), then what remains is a random graph with n-O(n^2/3) vertices, and edge probability the same as originally. It might make sense to rewrite this probability 1/n as

$\frac{1}{n-O(n^{2/3})}\cdot \frac{n-O(n^{2/3})}{n}=\frac{1-O(n^{-1/3})}{n-O(n^{2/3})}.$

The approximation in the final numerator is basically the same as

$1-o\left(n-O(n^{2/3})\right).$

Although we have no concrete reasoning, it seems at least plausible that this should look similar in structure to G(n,1/n). In particular, there should be another component of size

$O\left([n-O(n^{2/3})]^{2/3}\right)=O(n^{2/3})$.

In fact, the formal proof of this proceeds by an identical argument, only using the exploration process. Because I’ve described this several times before, I’ll be brief. We track how far we have gone through each component in a depth-first walk. In both the supercritical and subcritical cases, when we scale correctly we get a random path which is basically deterministic in the limit (in n). For exactly the same reasons as visible CLT fluctuations for partial sums of RVs with expectation zero, we start seeing interesting effects at criticality.

The important question is the order of rescaling to choose. At each stage of the exploration process, the number of vertices added to the stack is binomial. We want to distinguish between components of size $O(n^{2/3})$ so we should look at the exploration process at time $sn^{2/3}$. The drift of the exploration process is given by the expectation of a binomial random variable minus one (since we remove the current vertex from the stack as we finish exploring it). This is given by

$\mathbb{E}=\left[n-sn^{2/3}\right]\cdot \frac{1}{n}-1=-sn^{-1/3}.$

Note that this is the drift in one time-step. The drift in $n^{2/3}$ time-steps will accordingly by $sn^{1/3}$. So, if we rescale time by $n^{2/3}$ and space by $n^{1/3}$, we should get a nice stochastic process. Specifically, if Z is the exploration process, then we obtain:

$\frac{1}{n^{1/3}}Z^{(n)}_{sn^{2/3}} \rightarrow_d W_s,$

where W is a Brownian motion with inhomogeneous drift -s at time s. The net effect of such a drift at a fixed positive time is given by integrating up to that time, and hence we might say the process has quadratic drift, or is parabolic.

We should remark that our binomial expectation is not entirely correct. We have discounted those $sn^{2/3}$ vertices that have already been explored, but we have not accounted for the vertices currently in the stack. We should also be avoiding considering these. However, we now have a heuristic for the approximate number of these. The number of vertices in the stack should be $O(n^{1/3})$ at all times, and so in particular will always be an order of magnitude smaller than the number of vertices already considered. Therefore, they won’t affect this drift term, though this must be accounted for in any formal proof of convergence. On the subject of which, the mode of convergence is, unsurprisingly, weak convergence uniformly on compact sets. That is, for any fixed S, the convergence holds weakly on the random functions up to time $sn^{2/3}$.

Note that this process will tend to minus infinity almost surely. Component sizes are given by excursions above the running minimum. The process given by the height of the original process above the running minimum is called reflected. Essentially, we construct the reflected process by having the same generator when the current value is positive, and forcing the process up when it is at zero. There are various ways to construct this more formally, including as the scaling limit of some simple random walks conditioned never to stay non-negative.

The cute part of the result is that it holds equally well in a so-called critical window either side of the critical probability 1/n. When the probability is $\frac{1+tn^{-1/3}}{n}$, for any $t\in \mathbb{R}$, the same argument holds. Now the drift at time s is t-s, though everything else still holds.

This result was established by Aldous in [1], and gives a mechanism for calculating distributions of component sizes and so on through this critical window.

In particular, we are now in a position to answer the original question regarding how many such components there were. The key idea is that because whenever we exhaust a component in the exploration process, we choose a new vertex uniformly at random, we are effectively choosing a component according to the size-biased distribution. Roughly speaking, the largest components will show up near the beginning. Note that a critical $O(n^{2/3})$ component will not necessarily be exactly the first component in the exploration process, but the components that are explored before this will take up sufficiently few vertices that they won’t show up in the scaling of the limit.

In any case, the reflected Brownian motion ‘goes on forever’, and the drift is eventually very negative, so there cannot be infinitely wide excursions, hence there are infinitely many such critical components.

If we care about the number of cycles, we can treat this also via the exploration process. Note that in any depth-first search we are necessarily only interested in a spanning tree of the host graph. Anyway, when we are exploring a vertex, there could be extra edges to other vertices in the stack, but not to vertices we’ve already finished exploring (otherwise the edge would have been exposed then). So the expected number of excess edges into a vertex is proportional to the height of the exploration process at that vertex. So the overall expected number of excess edges, conditional on the exploration process is the area under the curve. This carries over perfectly well into the stochastic process limit. It is then a calculation to verify that the area under the curve is almost surely infinite, and thus that we expect there to be infinitely many cycles in a critical random graph.

REFERENCES

[1] Aldous D. – Brownian excursions, critical random graphs and the multiplicative coalescent

# Supremum of Brownian Motion

We define the supremum process of Brownian Motion by:

$S_t:=\sup_{0\leq s\leq t}B_s.$

Here are two facts about Brownian Motion. Firstly, the Reflection Principle:

$\mathbb{P}(S_t\geq b,B_t\leq a)=\mathbb{P}(B_t\geq 2b-a),$

which we motivate by ‘stopping’ at time $S_t$, and using the SMP for Brownian Motion, even though it isn’t a stopping time. By setting a=b, we get:

$\mathbb{P}(S_t\geq b)=\mathbb{P}(S_t\geq b,B_t\leq b)+\mathbb{P}(B_t\geq b)=2\mathbb{P}(B_t\geq b)=\mathbb{P}(|B|\geq b),$

and conclude that

$S_t\stackrel{d}{=}|B_t|\quad\text{for each }t\geq 0.$

The second fact comes from the decomposition of BM into local times and excursions:

$(S_t,S_t-B_t)_{t\geq 0}\stackrel{d}{=}(L_t,|B_t|)_{t\geq 0},$

where L is the local time process at 0, and this equality in distribution holds for the processes. See the previous post on excursion theory for explanation of what local times mean.

In particular, combining these two facts gives:

$S_t\stackrel{d}{=}S_t-B_t\quad\text{for every }t\geq 0.$

I thought that was rather surprising, and wanted to think of a straightforward reason why this should be true. I think the following works:

Brownian motion is time-reversible. In particular, as processes, we have

$(B_s)_{s\geq 0}\stackrel{d}{=}(B_{t-s}-B_t)_{s\geq 0}$

$\Rightarrow \sup_{0\leq r\leq t}B_r\stackrel{d}{=}\sup_{0\leq r\leq t}(B_{t-r}-B_t)$

$\Rightarrow S_t\stackrel{d}{=}S_t-B_t.$

# Subordinators and the Arcsine rule

After the general discussion of Levy processes in the previous post, we now discuss a particular class of such processes. The majority of content and notation below is taken from chapters 1-3 of Jean Bertoin’s Saint-Flour notes.

We say $X_t$ is a subordinator if:

• It is a right-continuous adapted stochastic process, started from 0.
• It has stationary, independent increments.
• It is increasing.

Note that the first two conditions are precisely those required for a Levy process. We could also allow the process to take the value $\infty$, where the hitting time of infinity represents ‘killing’ the subordinator in some sense. If this hitting time is almost surely infinite, we say it is a strict subordinator. There is little to be gained right now from considering anything other than strict subordinators.

Examples

• A compound Poisson process, with finite jump measure supported on $[0,\infty)$. Hereafter we exclude this case, as it is better dealt with in other languages.
• A so-called stable Levy process, where $\Phi(\lambda)=\lambda^\alpha$, for some $\alpha\in(0,1)$. (I’ll define $\Phi$ very soon.) Note that checking that the sample paths are increasing requires only that $X_1\geq 0$ almost surely.
• The hitting time process for Brownian Motion. Note that this does indeed have jumps as we would need. (This has $\Phi(\lambda)=\sqrt{2\lambda}$.)

Properties

• In general, we describe Levy processes by their characteristic exponent. As a subordinator takes values in $[0,\infty)$, we can use the Laplace exponent instead:

$\mathbb{E}\exp(-\lambda X_t)=:\exp(-t\Phi(\lambda)).$

• We can refine the Levy-Khintchine formula;

$\Phi(\lambda)=k+d\lambda+\int_{[0,\infty)}(1-e^{-\lambda x})\Pi(dx),$

• where k is the kill rate (in the non-strict case). Because the process is increasing, it must have bounded variation, and so the quadratic part vanishes, and we have a stronger condition on the Levy measure: $\int(1\wedge x)\Pi(dx)<\infty$.
• The expression $\bar{\Pi}(x):=k+\Pi((x,\infty))$ for the tail of the Levy measure is often more useful in this setting.
• We can think of this decomposition as the sum of a drift, and a PPP with characteristic measure $\Pi+k\delta_\infty$. As we said above, we do not want to consider the case that X is a step process, so either d>0 or $\Pi((0,\infty))=\infty$ is enough to ensure this.

Analytic Methods

We give a snapshot of a couple of observations which make these nice to work with. Define the renewal measure U(dx) by:

$\int_{[0,\infty)}f(x)U(dx)=\mathbb{E}\left(\int_0^\infty f(X_t)dt\right).$

If we want to know the distribution function of this U, it will suffice to consider the indicator function $f(x)=1_{X_t\leq x}$ in the above.

The reason to exclude step processes specifically is to ensure that X has a continuous inverse:

$L_x=\sup\{t\geq 0:X_t\leq x\}$ so $U(x)=\mathbb{E}L_x$ is continuous.

In fact, this renewal measure characterises the subordinator uniquely, as we see by taking the Laplace transform:

$\mathcal{L}U(\lambda)=\int_{[0,\infty)}e^{-\lambda x}U(dx)=\mathbb{E}\int e^{-\lambda X_t}dt$

$=\int \mathbb{E}e^{-\lambda X_t}dt=\int\exp(-t\Phi(\lambda))dt=\frac{1}{\Phi(\lambda)}.$

The Arcsine Law

X is Markov, which induces a so-called regenerative property on the range of X, $\mathcal{R}$. Formally, given s, we do not always have $s\in\mathcal{R}$ (as the process might jump over s), but we can define $D_s=\inf\{t>s:t\in\mathcal{R}\}$. Then

$\{v\geq 0:v+D_s\in\mathcal{R}\}\stackrel{d}{=}\mathcal{R}.$

In fact, the converse holds as well. Any random set with this regenerative property is the range of some subordinator. Note that $D_s$ is some kind of dual to X, since it is increasing, and the regenerative property induces some Markovian properties.

In particular, we consider the last passage time $g_t=\sup\{s, in the case of a stable subordinator with $\Phi(\lambda)=\lambda^\alpha$. Here, $\mathcal{R}$ is self-similar with scaling exponent $\alpha$. The distribution of $\frac{g_t}{t}$ is thus independent of t. In this situation, we can derive the generalised arcsine rule for the distribution of $g_1$:

$\mathbb{R}(g_1\in ds)=\frac{\sin \alpha\pi}{\pi}s^{\alpha-1}(1-s)^{-\alpha}ds.$

The most natural application of this is to the hitting time process of Brownian Motion, which is stable with $\alpha=\frac12$. Then $g_1=S_1-B_1$, in the usual notation for the supremum process. Furthermore, we have equality in distribution of the processes (see previous posts on excursion theory and the short aside which follows):

$(S_t-B_t)_{t\geq 0}\stackrel{d}{=}(|B_t|)_{t\geq 0}.$

So $g_1$ gives the time of the last zero of BM before time 1, and the arcsine law shows that its distribution is given by:

$\mathbb{P}(g_1\leq t)=\frac{2}{\pi}\text{arcsin}\sqrt{t}.$

# Brownian Excursions and Local Time

I’ve been spending a fair bit of time this week reading and thinking about the limits of various combinatorial objects, in particular letting the number of vertices tend to $\infty$ in models of random graphs with various constraints. Perhaps predictably, like so many continuous stochastic objects, yet again the limiting ‘things’ turn out to be closely linked to Brownian Motion. As a result, I’ve ended up reading a bit about the notion of local time, and thought it was sufficiently elegant even by itself to justify a quick post.

Local Time

In general, we might be interested in calculating a stochastic integral like

$\int_0^t f(B_s)ds$.

Note that, except in some highly non-interesting cases, this is a random variable. Our high school understanding of Riemannian integration encourages thinking of this as a ‘pathwise’ integral along the path evolving in time. But of course, that’s orthogonal to the approach we start thinking about when we are introduced to the Lebesgue integral. There we think about potential values of the integrand, and weight their contribution by the (Lebesgue) measure of the subset of the domain in which they appear.

Can we do the same for the stochastic integral? That is, can we find a measure which records how long the Brownian Motion spends at a point x? This measure will not be deterministic – effectively the stochastic behaviour of BM will be encoded through the measure rather than the argument of the function.

The answer is yes, and the measure in question is referred to as local time. More formally, we want

$\int_0^t f(B_s)ds=\int_\mathbb{R}f(x)L(t,x)dx.$ (*)

where the local time L(t,x) is a random process, increasing for fixed x. Informally, one could take

$\partial_t L(t,x) \propto 1(B_t=x)$

but clearly in practice that won’t do at all for a definition, and so instead we use (*). In the usual way, if we want (*) to hold for all reasonably nice functions f, it suffices to check it for the indicator functions of Borel sets. L(t,.) is therefore often referred to as occupation density, while L(.,A) is local time.

Local Time as natural index for Excursions

An excursion, for example of Brownian Motion, is a segment of the path that has zero value only at its endpoints. Alternatively, it is a maximal open interval of time such that the path is away from 0. We want to specify the measure on these excursions. Here are some obvious difficulties.

By Blumenthal’s 0-1 law, BM started from zero hits zero infinitely often in any time interval [0,e], so in the same way that there is no first positive rational, there is no first excursion. We could pick the excursion occurring in progress at a fixed time t, but this is little better. Firstly, the resultant measure is size-biased by the length of the excursion, and more importantly, the proximity of t to the origin may be significant unless we know of some memorylessness type of property to excursions.

Local time allows us to solve these problems. We restrict attention to $L_t:=L(t,0)$, the occupation density of 0. Let’s think about some advantages of indexing excursions by local time rather than by the start time:

• The key observation is that local time remains constant on excursions. That is, if we are avoiding 0, the local time at 0 cannot grow because the BM spends no time there!
• If we use start time, then we have a countably infinite number of small excursions accumulating close to 0, ie with very small start time. However, local time increases rapidly when there are lots of small excursions. Remember, lots of small excursions means that the BM hits 0 lots of times. So local time grows quickly through the annoying bits, and effectively provides a size-biasing for excursions that allows us to ignore the effects of the ‘Blumenthal excursions’ near time 0.
• When indexed by time, excursions might be Markovian, in the sense that subsequent excursions (and in particular their lengths) are independent of past excursions.This is certainly not the case if you index by start time! If an excursion starts at time t and has length u, then the ‘next’ excursions, in as much as that makes sense, must surely start at time t+u.

We know there are only countably many excursions, hence there are only countably many local times which pertain to an excursion. This motivates considering the set of excursions as a Poisson Point Process on local time. Once you’ve had this idea, everything follows quite nicely. Working out the distribution of the constant rate (which is a measure on the set of excursions) remains, but essentially we now have a sensible framework for tracking the process of excursions, and from this we can reconstruct the original Brownian Motion.

# SDEs and L-Diffusions

The motivation for the development of differential calculus by Newton et al. was to enable us to deduce concrete properties of, say, particle motion defined implicitly through ODEs. And we proceed similarly for the stochastic differential. Having defined all the terms through Ito’s formula, and concluded that BM is in some sense the canonical stochastic process, we seek to solve so-called stochastic differential equations of the form:

$dX_t=b(X_t)dt+\sigma(X_t)dB_t$

While there is no reason not to consider processes in $\mathbb{R}^d$, it is reasonable interesting to consider processes in one dimension. As with normal ODEs and PDEs, we have some intuitive notion if we specify some initial conditions, we should be able to set the differential equation up and ‘let it go’ like a functional clockwork mouse. Of course, we are conscious of the potential problems with uniqueness of solutions, stability of solutions, and general mathematical awkwardness that derives from the fact that we can’t treat all DEs as physical systems, with all the luxuries of definiteness that the physical world automatically affords. To establish some concreteness, we set up some definitions.

• For a solution to the SDE, $E(\sigma,b)$, we require a nice filtration $\mathcal{F}$ and a BM adapted to that filtration to drive the process $X_t$, which satisfies $X_t=X_0+\int_0^t\sigma(X_s)dB_s+\int_0^tb(X_s)ds$, and we require this for each $x_0\in\mathbb{R}^d$ s.t. $X_0=x_0$ a.s.
• Uniqueness in law: all solutions to $E(\sigma,b)$ starting from each x have the same law. Obviously, this places no restriction on the underlying probability space and filtration.
• A stronger condition is Pathwise uniqueness: Given the filtration, solutions are almost surely indistinguishable (that is, paths are equal everywhere).
• We have not specified any conditions on the filtration $\mathcal{F}$. It would be natural to consider only the minimal such filtration that works. If we can take $\mathcal{F}=\mathcal{F}^B$, the natural filtration of the driving BM, we say the solutions is strong. If every solution is strong, then we have pathwise uniqueness, otherwise we would have a solution where we could choose which path to follow independently of the BM.

The key theorem here is Yamada-Watanabe: If there exist solutions and we have pathwise uniqueness, then we have uniqueness in law. Then for every $(\mathcal{F}_t)$, and $\mathcal{F}_t$-BM, the unique solution is strong.

Existence of solutions is particularly tractable when $\sigma,b$ are Lipschitz, as this opens the way for implicit constructions as the fixed points of contracting mappings. We make particular use of Gronwall’s lemma, which confirms an intuitive thought that differential inequalities have solutions bounded by solutions to the corresponding ODE. Concretely, for $latex f\geq 0,\,f(t)\leq a+b\int_0^tf(s)ds,\quad 0\leq t\leq T$, the lemma states that $f(t)\leq a\exp(bt)$. The case a=0 is obviously of particular interest for demonstrating convergence results. We deploy this method to show that when $\sigma,b$ are Lipschitz, the SDE $dX_t=\sigma(X_t)dB_t+b(X_t)dt$ has pathwise uniqueness and for any triple of filtration $(\mathcal{F}_t)$, $\mathcal{F}_t$-adapted BM, and starting point x, there is a strong solution. Uniqueness in law then follows by Yamada-Watanabe, but we knew this anyway by composing measurable maps.

Now, given L, an operator on $C^2$ functions by:

$Lf(x)=\frac12\sum_{i,j}a_{i,j}(x)\frac{\partial^2 f}{\partial x^i\partial x^j}+\sum_i b_i(x)\frac{\partial f}{\partial x^i}$

We define X to be an L-diffusion if X’s local behaviour is specified (in distribution) by L(X). The first sum in the expression for L corresponds to diffusivity, while the second corresponds to (deterministic) drift. Formally, for a, b, bounded $X_t$ a L-diffusion is $\forall f\in C_b^2$:

$M_t^f:=f(X_t)-f(X_0)-\int_0^t Lf(X_s)ds$ is a martingale.

Alternatively, can relax boundedness condition, and require $M_t^f\in\mathcal{M}_{c,loc}$. To make a link to SDEs, define $a=\sigma\sigma^T$ (so in one dimension $a=\sqrt{\sigma}$), then solutions to $dX_t=\sigma(X_t)dB_t+b(X_t)dt$ are L-diffusions if boundedness conditions are met. Remember bounded implies Lipschitz implies solutions to SDEs. The result then follows directly from Ito’s formula.