Brownian Motion and the Riemann Zeta Function

Intriguingly, various constructions related to Brownian motion result in quantities with moments described by the Riemann zeta function. These distributions appear in integral representations used to extend the zeta function to the entire complex plane, as described in an earlier post. Now, I look at how they also arise from processes constructed from Brownian motion such as Brownian bridges, excursions and meanders.

Recall the definition of the Riemann zeta function as an infinite series

$\displaystyle \zeta(s)=1+2^{-s}+3^{-s}+4^{-s}+\cdots$

which converges for complex argument s with real part greater than one. This has a unique extension to an analytic function on the complex plane outside of a simple pole at s = 1.

Often, it is more convenient to use the Riemann xi function which can be defined as zeta multiplied by a prefactor involving the gamma function,

$\displaystyle \xi(s)=\frac12s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s).$

This is an entire function on the complex plane satisfying the functional equation ξ(1 - s) = ξ(s).

It turns out that ξ describes the moments of a probability distribution, according to which a random variable X is positive with moments

$\displaystyle {\mathbb E}[X^s]=2\xi(s),$

(1)

which is well-defined for all complex s. In the post titled The Riemann Zeta Function and Probability Distributions, I denoted this distribution by Ψ, which is a little arbitrary but was the symbol used for its probability density. A related distribution on the positive reals, which we will denote by Φ, is given by the moments

$\displaystyle {\mathbb E}[X^s]=\frac{1-2^{1-s}}{s-1}2\xi(s)$

(2)

which, again, is defined for all complex s.

As standard, complex powers of a positive real x are defined by x^s = e^slogx, so (1,2) are equivalent to the moment generating functions of logX, which uniquely determines the distributions. The probability densities and cumulative distribution functions can be given, although I will not do that here since they are already explicitly written out in the earlier post. I will write X ∼ Φ or X ∼ Ψ to mean that random variable X has the respective distribution. As we previously explained, these are closely connected:

If X ∼ Ψ and, independently, Y is uniform on [1, 2], then X/Y ∼ Φ.
If X, Y ∼ Φ are independent then √X² + Y² ∼ Ψ.

The purpose of this post is to describe some constructions involving Brownian bridges, excursions and meanders which naturally involve the Φ and Ψ distributions.

Theorem 1 The following have distribution Φ:

√2/πZ where Z = sup_t|B_t| is the absolute maximum of a standard Brownian bridge B.

Z/√2π where Z = sup_tB_t is the maximum of a Brownian meander B.

√2πZ where Z is the sample standard deviation of a Brownian bridge B,

$\displaystyle Z=\left(\int_0^1(B_t-\bar B)^2\,dt\right)^{\frac12}$

with sample mean B̅ = ∫₀¹B_t dt.

√π/2Z where Z is the pathwise Euclidean norm of a 2-dimensional Brownian bridge B = (B¹, B²),

$\displaystyle Z=\left(\int_0^1\lVert B_t\rVert^2\,dt\right)^{\frac12}$

√τπ/2 where τ = inf{t ≥ 0: ‖B_t‖= 1} is the first time at which the norm of a 3-dimensional standard Brownian motion B = (B¹, B², B³) hits 1.

The Kolmogorov distribution is, by definition, the absolute maximum of a Brownian bridge. So, the first statement of theorem 1 is saying that Φ is just the Kolmogorov distribution scaled by the constant factor √2/π. Moving on to Ψ;

Theorem 2 The following have distribution Ψ:

√2/πZ where Z = sup_tB_t – inf_tB_t is the range of a standard Brownian bridge B.

√2/πZ where Z = sup_tB_t is the maximum of a (normalized) Brownian excursion B.

√π/2Z where Z is the pathwise Euclidean norm of a 4-dimensional Brownian bridge B = (B¹, B², B³, B⁴),

$\displaystyle Z=\left(\int_0^1\lVert B_t\rVert^2\,dt\right)^{\frac12}.$

Continue reading “Brownian Motion and the Riemann Zeta Function” →

The Minimum and Maximum of Brownian motion

If X is standard Brownian motion, what is the distribution of its absolute maximum |X|_t^∗ = sup_s ≤ t|X_s| over a time interval [0, t]? Previously, I looked at how the reflection principle can be used to determine that the maximum X_t^∗ = sup_s ≤ tX_s has the same distribution as |X_t|. This is not the same thing as the maximum of the absolute value though, which is a more difficult quantity to describe. As a first step, |X|_t^∗ is clearly at least as large as X_t^∗ from which it follows that it stochastically dominates |X_t|.

I would like to go further and precisely describe the distribution of |X|_t^∗. What is the probability that it exceeds a fixed positive level a? For this to occur, the suprema of both X and –X must exceed a. Denoting the minimum and maximum by

$\displaystyle \begin{aligned} &X_t^m=\inf_{s\le t}X_s,\\ &X_t^M=\sup_{s\le t}X_s, \end{aligned}$

then |X|_t^∗ is the maximum of X_t^M and –X_t^m. I have switched notation a little here, and am using X^M to denote what was previously written as X^∗. This is just to use similar notation for both the minimum and maximum. Using inclusion-exclusion, the probability that the absolute maximum is greater than a level a is,

$\displaystyle \begin{aligned} {\mathbb P}(\lvert X\rvert_t^* > a)={} & {\mathbb P}(X_t^M > a)+{\mathbb P}(X_t^m < -a)\\ & -{\mathbb P}(X_t^M > a{\rm\ and\ }X_t^m < -a). \end{aligned}$

As X_t^M has the same distribution as |X_t| and, by symmetry, so does –X^m, we obtain

$\displaystyle {\mathbb P}(\lvert X\rvert_t^* > a)=4{\mathbb P}(X_t > a)-{\mathbb P}(X_t^M > a{\rm\ and\ }X_t^m < -a).$

This hasn’t really answered the question. All we have done is to re-express the probability in terms of both the minimum and maximum being beyond a level. For large values of a it does, however, give a good approximation. The probability of the Brownian motion reaching a large positive value a and then dropping to the large negative value –a will be vanishingly small, so the final term in the identity above can be neglected. This gives an asymptotic approximation as a tends to infinity,

$\displaystyle \begin{aligned} {\mathbb P}(\lvert X\rvert_t^* > a) &\sim 4{\mathbb P}(X_t > a)\\ &\sim\sqrt{\frac{8t}{\pi a^2}}e^{-\frac{a^2}{2t}}. \end{aligned}$

(1)

The last expression here is just using the fact that X_t is centered Gaussian with variance t and applying a standard approximation for the cumulative normal distribution function.

For small values of a, approximation (1) does not work well at all. We know that the left-hand-side should tend to 1, whereas 4ℙ(X_t > a) will tend to 2, and the final expression diverges. In fact, it can be shown that

$\displaystyle {\mathbb P}(\lvert X\rvert_t^* < a)\sim\frac{4}{\pi}e^{-\frac{t\pi^2}{8a^2}}$

(2)

as a → 0. I gave a direct proof in this math.stackexchange answer. In this post, I will look at how we can compute joint distributions of the minimum, maximum and terminal value of Brownian motion, from which limits such as (2) will follow. Continue reading “The Minimum and Maximum of Brownian motion” →

The Brownian Drawdown Process

The drawdown of a stochastic process is the amount that it has dropped since it last hit its maximum value so far. For process X with running maximum X^∗_t = sup_s ≤ tX_s, the drawdown is thus X^∗_t – X_t, which is a nonnegative process. This is as in figure 1 below.

Brownian motion drawdown — Figure 1: Brownian motion and its drawdown process

The previous post used the reflection principle to show that the maximum of a Brownian motion has the same distribution as its terminal absolute value. That is, X^∗_t and |X_t| are identically distributed.

For a process X started from zero, its maximum and drawdown can be written as X^∗_t – X₀ and X^∗_t – X_t. Reversing the process in time across the interval [0, t] will exchange these values. So, reversing in time and translating so that it still starts from zero will exchange the maximum value and the drawdown. Specifically, write

$\displaystyle Y_s = X_{t-s} - X_t$

for time index 0 ≤ s ≤ t. The maximum of Y is equal to the drawdown of X,

$\displaystyle Y^*_t = X^*_t-X_t.$

If X is standard Brownian motion then so is Y, since the independent normal increments property for Y follows from that of X. As already stated, the maximum Y^∗_t = X^∗_t – X_t has the same distribution as the absolute value |Y_t|= |X_t|. So, the drawdown has the same distribution as the absolute value at each time.

Lemma 1 If X is standard Brownian motion, then X^∗_t – X_t has the same distribution as |X_t| at each time t ≥ 0.

Continue reading “The Brownian Drawdown Process” →

The Maximum of Brownian Motion and the Reflection Principle

The distribution of a standard Brownian motion X at a positive time t is, by definition, centered normal with variance t. What can we say about its maximum value up until the time? This is X^∗_t = sup_s ≤ tX_s, and is clearly nonnegative and at least as big as X_t. To be more precise, consider the probability that the maximum is greater than a fixed positive value a. Such problems will be familiar to anyone who has looked at pricing of financial derivatives such as barrier options, where the payoff of a trade depends on whether the maximum or minimum of an asset price has crossed a specified barrier level.

This can be computed with the aid of a symmetry argument commonly referred to as the reflection principle. The idea is that, if we reflect the Brownian motion when it first hits a level, then the resulting process is also a Brownian motion. The first time at which X hits level a is τ = inf{t ≥ 0: X_t ≥ a}, which is a stopping time. Reflecting the process about this level at all times after τ gives a new process

Reflected Brownian motion — Figure 1: Reflecting Brownian motion when it hits level a.

Continue reading “The Maximum of Brownian Motion and the Reflection Principle” →

Brownian Meanders

Having previously looked at Brownian bridges and excursions, I now turn to a third kind of process which can be constructed either as a conditioned Brownian motion or by extracting a segment from Brownian motion sample paths. Specifically, the Brownian meander, which is a Brownian motion conditioned to be positive over a unit time interval. Since this requires conditioning on a zero probability event, care must be taken. Instead, it is cleaner to start with an alternative definition by appropriately scaling a segment of a Brownian motion.

For a fixed positive times T, consider the last time σ before T at which a Brownian motion X is equal to zero,

$\displaystyle \sigma=\sup\left\{t\le T\colon X_t=0\right\}.$

(1)

On interval [σ, T], the path of X will start from 0 and then be either strictly positive or strictly negative, and we may as well restrict to the positive case by taking absolute values. Scaling invariance says that c^-1/2X_ct is itself a standard Brownian motion for any positive constant c. So, scaling the path of X on [σ, 1] to the unit interval defines a process

$\displaystyle B_t=(T-\sigma)^{-1/2}\lvert X_{\sigma+t(T-\sigma)}\rvert.$

(2)

over 0 ≤ t ≤ 1; This starts from zero and is strictly positive at all other times.

Brownian meander construction — Figure 2: Constructing a Brownian meander

Scaling invariance shows that the law of the process B does not depend on the choice of fixed time T The only remaining ambiguity is in the choice of the fixed time T.

Lemma 1 The distribution of B defined by (2) does not depend on the choice of the time T > 0.

Proof: Consider any other fixed positive time T̃, and use the construction above with T̃, σ̃, B̃ in place of T, σ, B respectively. We need to show that B̃ and B have the same distribution. Using the scaling factor S = T̃/T, then X′_t = S^-1/2X_tS is a standard Brownian motion. Also, σ′= σ̃/S is the last time before T at which X′ is zero. So,

$\displaystyle \tilde B_t=(T-\sigma')^{-1/2}\lvert X'_{\sigma'+t(T-\sigma')}\rvert$

has the same distribution as B. ⬜

This leads to the definition used here for Brownian meanders.

Definition 2 A continuous process {B_t}_{t ∈ [0, 1]} is a Brownian meander if and only it has the same distribution as (2) for a standard Brownian motion X and fixed time T > 0.

In fact, there are various alternative — but equivalent — ways in which Brownian excursions can be defined and constructed.

As a scaled segment of a Brownian motion before a time T and after it last hits 0. This is definition 2.
As a Brownian motion conditioned on being positive. See theorem 4 below.
As a segment of a Brownian excursion. See lemma 5.
As the path of a standard Brownian motion starting from its minimum, in either the forwards or backwards direction. See theorem 6.
As a Markov process with specified transition probabilities. See theorem 9 below.
As a solution to an SDE. See theorem 12 below.

Continue reading “Brownian Meanders” →

Brownian Excursions

A normalized Brownian excursion is a nonnegative real-valued process with time ranging over the unit interval, and is equal to zero at the start and end time points. It can be constructed from a standard Brownian motion by conditioning on being nonnegative and equal to zero at the end time. We do have to be careful with this definition, since it involves conditioning on a zero probability event. Alternatively, as the name suggests, Brownian excursions can be understood as the excursions of a Brownian motion X away from zero. By continuity, the set of times at which X is nonzero will be open and, hence, can be written as the union of a collection of disjoint (and stochastic) intervals (σ, τ).

In fact, Brownian motion can be reconstructed by simply joining all of its excursions back together. These are independent processes and identically distributed up to scaling. Because of this, understanding the Brownian excursion process can be very useful in the study of Brownian motion. However, there will by infinitely many excursions over finite time periods, so the procedure of joining them together requires some work. This falls under the umbrella of ‘excursion theory’, which is outside the scope of the current post. Here, I will concentrate on the properties of individual excursions.

In order to select a single interval, start by fixing a time T > 0. As X_T is almost surely nonzero, T will be contained inside one such interval (σ, τ). Explicitly,

$\displaystyle \begin{aligned} &\sigma=\sup\left\{t\le T\colon X_t=0\right\},\\ &\tau=\inf\left\{t\ge T\colon X_t=0\right\}, \end{aligned}$

(1)

so that σ < T < τ < ∞ almost surely. The path of X across such an interval is t ↦ X_σ + t for time t in the range [0, τ - σ]. As it can be either nonnegative or nonpositive, we restrict to the nonnegative case by taking the absolute value. By invariance, S^-1/2X_tS is also a standard Brownian motion, for each fixed S > 0. Using a stochastic factor S = τ – σ, the width of the excursion is normalised to obtain a continuous process {B_t}_{t ∈ [0, 1]} given by

$\displaystyle B_t=(\tau-\sigma)^{-1/2}\lvert X_{\sigma+t(\tau-\sigma)}\rvert.$

(2)

By construction, this is strictly positive over 0 < t < 1 and equal to zero at the endpoints t ∈ {0, 1}.

Figure 2: Constructing a Brownian excursion

The only remaining ambiguity is in the choice of the fixed time T.

Lemma 1 The distribution of B defined by (2) does not depend on the choice of the time T > 0.

Proof: This follows from scaling invariance of Brownian motion. Consider any other fixed positive time T̃, and use the construction above with T̃, σ̃, τ̃, B̃ in place of T, σ, τ, B respectively. We need to show that B̃ and B have the same distribution. Using the scaling factor S = T̃/T, then X′_t = S^-1/2X_tS is a standard Brownian motion. Also, σ′= σ̃/S and τ′= τ̃/S are random times given in the same way as σ and τ, but with the Brownian motion X′ in place of X in (1). So,

$\displaystyle \tilde B_t=(\tau^\prime-\sigma^\prime)^{-1/2}\lvert X^\prime_{\sigma^\prime+t(\tau^\prime-\sigma^\prime)}\rvert$

has the same distribution as B. ⬜

This leads to the definition used here for Brownian excursions.

Definition 2 A continuous process {B_t}_{t ∈ [0, 1]} is a Brownian excursion if and only it has the same distribution as (2) for a standard Brownian motion X and time T > 0.

In fact, there are various alternative — but equivalent — ways in which Brownian excursions can be defined and constructed.

As a normalized excursion away from zero of a Brownian motion. This is definition 2.
As a normalized excursion away from zero of a Brownian bridge. This is theorem 6.
As a Brownian bridge conditioned on being nonnegative. See theorem 9 below.
As the sample path of a Brownian bridge, translated so that it has minimum value zero at time 0. This is a very interesting and useful method of directly computing excursion sample paths from those of a Brownian bridge. See theorem 12 below, sometimes known as the Vervaat transform.
As a Markov process with specified transition probabilities. See theorem 15 below.
As a transformation of Bessel process paths, see theorem 16 below.
As a Bessel bridge of order 3. This can be represented either as a Bessel process conditioned on hitting zero at time 1., or as the vector norm of a 3-dimensional Brownian bridge. See lemma 17 below.
As a solution to a stochastic differential equation. See theorem 18 below.

Continue reading “Brownian Excursions” →

Brownian Bridges

A Brownian bridge can be defined as standard Brownian motion conditioned on hitting zero at a fixed future time T, or as any continuous process with the same distribution as this. Rather than conditioning, a slightly easier approach is to subtract a linear term from the Brownian motion, chosen such that the resulting process hits zero at the time T. This is equivalent, but has the added benefit of being independent of the original Brownian motion at all later times.

Lemma 1 Let X be a standard Brownian motion and ${T > 0}$ be a fixed time. Then, the process

$\displaystyle B_t = X_t - \frac tTX_T$ (1)

over ${0\le t\le T}$ is independent from ${\{X_t\}_{t\ge T}}$ .

Proof: As the processes are joint normal, it is sufficient that there is zero covariance between them. So, for times ${s\le T\le t}$ , we just need to show that ${{\mathbb E}[B_sX_t]}$ is zero. Using the covariance structure ${{\mathbb E}[X_sX_t]=s\wedge t}$ we obtain,

$\displaystyle {\mathbb E}[B_sX_t]={\mathbb E}[X_sX_t]-\frac sT{\mathbb E}[X_TX_t]=s-\frac sTT=0$

as required. ⬜

This leads us to the definition of a Brownian bridge.

Definition 2 A continuous process ${\{B_t\}_{t\in[0,T]}}$ is a Brownian bridge on the interval ${[0,T]}$ if and only it has the same distribution as ${X_t-\frac tTX_T}$ for a standard Brownian motion X.

In case that ${T=1}$ , then B is called a standard Brownian bridge.

There are actually many different ways in which Brownian bridges can be defined, which all lead to the same result.

As a Brownian motion minus a linear term so that it hits zero at T. This is definition 2.
As a Brownian motion X scaled as ${tT^{-1/2}X_{T/t-1}}$ . See lemma 9 below.
As a joint normal process with prescribed covariances. See lemma 7 below.
As a Brownian motion conditioned on hitting zero at T. See lemma 14 below.
As a Brownian motion restricted to the times before it last hits zero before a fixed positive time T, and rescaled to fit a fixed time interval. See lemma 15 below.
As a Markov process. See lemma 13 below.
As a solution to a stochastic differential equation with drift term forcing it to hit zero at T. See lemma 18 below.

There are other constructions beyond these, such as in terms of limits of random walks, although I will not cover those in this post. Continue reading “Brownian Bridges” →

The Riemann Zeta Function and Probability Distributions

Phi and Psi densities — Figure 1: Probability densities used to extend the zeta function

The famous Riemann zeta function was first introduced by Riemann in order to describe the distribution of the prime numbers. It is defined by the infinite sum

$\displaystyle \begin{aligned} \zeta(s) &=1+2^{-s}+3^{-s}+4^{-s}+\cdots\\ &=\sum_{n=1}^\infty n^{-s}, \end{aligned}$

(1)

which is absolutely convergent for all complex s with real part greater than one. One of the first properties of this is that, as shown by Riemann, it extends to an analytic function on the entire complex plane, other than a simple pole at ${s=1}$ . By the theory of analytic continuation this extension is necessarily unique, so the importance of the result lies in showing that an extension exists. One way of doing this is to find an alternative expression for the zeta function which is well defined everywhere. For example, it can be expressed as an absolutely convergent integral, as performed by Riemann himself in his original 1859 paper on the subject. This leads to an explicit expression for the zeta function, scaled by an analytic prefactor, as the integral of ${x^s}$ multiplied by a function of x over the range ${ x > 0}$ . In fact, this can be done in a way such that the function of x is a probability density function, and hence expresses the Riemann zeta function over the entire complex plane in terms of the generating function ${{\mathbb E}[X^s]}$ of a positive random variable X. The probability distributions involved here are not the standard ones taught to students of probability theory, so may be new to many people. Although these distributions are intimately related to the Riemann zeta function they also, intriguingly, turn up in seemingly unrelated contexts involving Brownian motion.

In this post, I derive two probability distributions related to the extension of the Riemann zeta function, and describe some of their properties. I also show how they can be constructed as the sum of a sequence of gamma distributed random variables. For motivation, some examples are given of where they show up in apparently unrelated areas of probability theory, although I do not give proofs of these statements here. For more information, see the 2001 paper Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions by Biane, Pitman, and Yor. Continue reading “The Riemann Zeta Function and Probability Distributions” →

Brownian Drawdowns

BM drawdown — Figure 1: Brownian motion and drawdown

Here, I apply the theory outlined in the previous post to fully describe the drawdown point process of a standard Brownian motion. In fact, as I will show, the drawdowns can all be constructed from independent copies of a single ‘Brownian excursion’ stochastic process. Recall that we start with a continuous stochastic process X, assumed here to be Brownian motion, and define its running maximum as ${M_t=\sup_{s\le t}X_s}$ and drawdown process ${D_t=M_t-X_t}$ . This is as in figure 1 above.

Next, ${D^a}$ was defined to be the drawdown ‘excursion’ over the interval at which the maximum process is equal to the value ${a \ge 0}$ . Precisely, if we let ${\tau_a}$ be the first time at which X hits level ${a}$ and ${\tau_{a+}}$ be its right limit ${\tau_{a+}=\lim_{b\downarrow a}\tau_b}$ then,

$\displaystyle D^a_t=D_{({\tau_a+t})\wedge\tau_{a+}}=a-X_{({\tau_a+t)}\wedge\tau_{a+}}.$

Next, a random set S is defined as the collection of all nonzero drawdown excursions indexed the running maximum,

$\displaystyle S=\left\{(a,D^a)\colon D^a\not=0\right\}.$

The set of drawdown excursions corresponding to the sample path from figure 1 are shown in figure 2 below.

BM drawdown excursions — Figure 2: Brownian drawdown excursions

As described in the post on semimartingale local times, the joint distribution of the drawdown and running maximum ${(D,M)}$ , of a Brownian motion, is identical to the distribution of its absolute value and local time at zero, ${(\lvert X\rvert,L^0)}$ . Hence, the point process consisting of the drawdown excursions indexed by the running maximum, and the absolute value of the excursions from zero indexed by the local time, both have the same distribution. So, the theory described in this post applies equally to the excursions away from zero of a Brownian motion.

Before going further, let’s recap some of the technical details. The excursions lie in the space E of continuous paths ${z\colon{\mathbb R}_+\rightarrow{\mathbb R}}$ , on which we define a canonical process Z by sampling the path at each time t, ${Z_t(z)=z_t}$ . This space is given the topology of uniform convergence over finite time intervals (compact open topology), which makes it into a Polish space, and whose Borel sigma-algebra ${\mathcal E}$ is equal to the sigma-algebra generated by ${\{Z_t\}_{t\ge0}}$ . As shown in the previous post, the counting measure ${\xi(A)=\#(S\cap A)}$ is a random point process on ${({\mathbb R}_+\times E,\mathcal B({\mathbb R}_+)\otimes\mathcal E)}$ . In fact, it is a Poisson point process, so its distribution is fully determined by its intensity measure ${\mu={\mathbb E}\xi}$ .

Theorem 1 If X is a standard Brownian motion, then the drawdown point process ${\xi}$ is Poisson with intensity measure ${\mu=\lambda\otimes\nu}$ where,

${\lambda}$ is the standard Lebesgue measure on ${{\mathbb R}_+}$ .

${\nu}$ is a sigma-finite measure on E given by

$\displaystyle \nu(f) = \lim_{\epsilon\rightarrow0}\epsilon^{-1}{\mathbb E}_\epsilon[f(Z^{\sigma})]$ (1)

for all bounded continuous continuous maps ${f\colon E\rightarrow{\mathbb R}}$ which vanish on paths of length less than L (some ${L > 0}$ ). The limit is taken over ${\epsilon > 0}$ , ${{\mathbb E}_\epsilon}$ denotes expectation under the measure with respect to which Z is a Brownian motion started at ${\epsilon}$ , and ${\sigma}$ is the first time at which Z hits 0. This measure satisfies the following properties,

${\nu}$ -almost everywhere, there exists a time ${T > 0}$ such that ${Z > 0}$ on ${(0,T)}$ and ${Z=0}$ everywhere else.

for each ${t > 0}$ , the distribution of ${Z_t}$ has density

$\displaystyle p_t(z)=z\sqrt{\frac 2{\pi t^3}}e^{-\frac{z^2}{2t}}$ (2)

over the range ${z > 0}$ .

over ${t > 0}$ , ${Z_t}$ is Markov, with transition function of a Brownian motion stopped at zero.

Continue reading “Brownian Drawdowns” →

Semimartingale Local Times

Figure 1: Brownian motion B with local time L and auxiliary Brownian motion W

For a stochastic process X taking values in a state space E, its local time at a point ${x\in E}$ is a measure of the time spent at x. For a continuous time stochastic process, we could try and simply compute the Lebesgue measure of the time at the level,

$\displaystyle L^x_t=\int_0^t1_{\{X_s=x\}}ds.$

(1)

For processes which hit the level ${x}$ and stick there for some time, this makes some sense. However, if X is a standard Brownian motion, it will always give zero, so is not helpful. Even though X will hit every real value infinitely often, continuity of the normal distribution gives ${{\mathbb P}(X_s=x)=0}$ at each positive time, so that that ${L^x_t}$ defined by (1) will have zero expectation.

Rather than the indicator function of ${\{X=x\}}$ as in (1), an alternative is to use the Dirac delta function,

$\displaystyle L^x_t=\int_0^t\delta(X_s-x)\,ds.$

(2)

Unfortunately, the Dirac delta is not a true function, it is a distribution, so (2) is not a well-defined expression. However, if it can be made rigorous, then it does seem to have some of the properties we would want. For example, the expectation ${{\mathbb E}[\delta(X_s-x)]}$ can be interpreted as the probability density of ${X_s}$ evaluated at ${x}$ , which has a positive and finite value, so it should lead to positive and finite local times. Equation (2) still relies on the Lebesgue measure over the time index, so will not behave as we may expect under time changes, and will not make sense for processes without a continuous probability density. A better approach is to integrate with respect to the quadratic variation,

$\displaystyle L^x_t=\int_0^t\delta(X_s-x)d[X]_s$

(3)

which, for Brownian motion, amounts to the same thing. Although (3) is still not a well-defined expression, since it still involves the Dirac delta, the idea is to come up with a definition which amounts to the same thing in spirit. Important properties that it should satisfy are that it is an adapted, continuous and increasing process with increments supported on the set ${\{X=x\}}$ ,

$\displaystyle L^x_t=\int_0^t1_{\{X_s=x\}}dL^x_s.$

Local times are a very useful and interesting part of stochastic calculus, and finds important applications to excursion theory, stochastic integration and stochastic differential equations. However, I have not covered this subject in my notes, so do this now. Recalling Ito’s lemma for a function ${f(X)}$ of a semimartingale X, this involves a term of the form ${\int f^{\prime\prime}(X)d[X]}$ and, hence, requires ${f}$ to be twice differentiable. If we were to try to apply the Ito formula for functions which are not twice differentiable, then ${f^{\prime\prime}}$ can be understood in terms of distributions, and delta functions can appear, which brings local times into the picture. In the opposite direction, which I take in this post, we can try to generalise Ito’s formula and invert this to give a meaning to (3). Continue reading “Semimartingale Local Times” →

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