# Stochastic Differential Equations

Stochastic differential equations (SDEs) form a large and very important part of the theory of stochastic calculus. Much like ordinary differential equations (ODEs), they describe the behaviour of a dynamical system over infinitesimal time increments, and their solutions show how the system evolves over time. The difference with SDEs is that they include a source of random noise., typically given by a Brownian motion. Since Brownian motion has many pathological properties, such as being everywhere nondifferentiable, classical differential techniques are not well equipped to handle such equations. Standard results regarding the existence and uniqueness of solutions to ODEs do not apply in the stochastic case, and cannot readily describe what it even means to solve such as system. I will make some posts explaining how the theory of stochastic calculus applies to systems described by an SDE.

Consider a stochastic differential equation describing the evolution of a real-valued process {Xt}t≥0,

 $\displaystyle dX_t = \sigma(X_t)\,dW_t + b(X_t)\,dt$ (1)

which can be specified along with an initial condition X0 = x0. Here, b is the drift specifying how X moves on average across the dt time, σ is a volatility term giving the amplitude of the random noise and W is a driving Brownian motion providing the source of the randomness. There are numerous situations where equations such as (1) are used, with applications in physics, finance, filtering theory, and many other areas.

In the case where σ is zero, (1) is just an ordinary differential equation dX/dt = b(X). In the general case, we can informally think of dividing through by dt to give an ODE plus an additional noise term

 $\displaystyle \frac{dX_t}{dt}=b(X_t)+\sigma(X_t)\xi_t.$ (2)

I have set ξt = dWt/dt which can be thought of as a process whose values at each time are independent zero-mean random variables. As mentioned above, though, Brownian motion is not differentiable so this does not exist in the usual sense. While it can be described by a kind of random distribution, even distribution theory is not well-equipped to handle such equations involving multiplying by the nondifferentiable process σ(Xt). Instead, (1) can be integrated to obtain

 $\displaystyle X_t=X_0+\int_0^t\sigma(X_s)\,dW_s+\int_0^tb(X_s)\,ds,$ (3)

where the right-hand-side is interpreted using stochastic integration with respect to the semimartingale W. Likewise, X will be a semimartingale, and such solutions are often referred to as diffusions.

The differential form (1) can be interpreted as a shorthand for the integral expression (3), which I will do in these notes. It can be generalized to n-dimensional processes by allowing b to take values in n, a(x) to be an n × m matrix, and W to be an m-dimensional Brownian motion. That is, W = (W1, …, Wm) where Wi are independent Brownian motions. I will sometimes write this as

 $\displaystyle dX^t_i=\sigma_{ij}(X_t)dW^j_t+b_i(X_t)dt$

where the summation convention is being applied, with subscripts or superscripts occuring more than once in a single term being summed from 1 to n.

Unlike ODEs, when dealing with SDEs we need to consider what underlying probability space the solution is defined with respect to. This leads to the existence of different classes of solutions.

• Strong solutions where X can be expressed as a measurable function of the Brownian motion W or, equivalently, X is adapted to its natural filtration.
• Weak solutions where X need not be a function of W. Such cases may require additional randomness so may not exist on the probability space with respect to which the Brownian motion W is defined. It can be necessary to extend the filtered probability space to construct these solutions.

Likewise, when considering uniqueness of solutions, there are different ways this occurs.

• Pathwise uniqueness where, up to indistinguishability, there is only one solution X. This should hold not just on one specific space containing a Brownian motion W, but on all such spaces. That is, weak solutions should be unique.
• Uniqueness in law where there may be multiple pathwise solutions, but their distribution is uniquely determined by the SDE.

There are various general conditions under which strong solutions and pathwise uniqueness are guaranteed for SDE (1) , such as the Itô result for Lipschitz continuous coefficients. I covered this situation in a previous post.

Other than using the SDE (1), such systems can also be described by an associated differential operator. For the n-dimensional case set a(x) = σ(x)σ(x)T, which is an n × n positive semidefinite matrix. Then, the second order operator L can be defined

 $\displaystyle Lf(x)=\frac12a_{ij}(x)f_{,ij}(x)+b_{i}(x)f_{,i}(x)$

operating on twice continuously differentiable functions f: ℝn → ℝ. Being able to effortlessly switch between descriptions using the SDE (1) and the operator L is a huge benefit when working with such systems. There are several different ways in which the operator can be used to describe a stochastic process, all of which relate to weak solutions and uniqueness in law of the SDE.

Markov Generator: A Markov process is a weak solution to the SDE (1) if its infinitesimal generator is L. That is, if the transition function is Pt then,

 $\displaystyle \lim_{t\rightarrow0}t^{-1}(P_tf-f)=Lf$

for suitably regular functions f.

Backwards Equation: For a function f: ℝn × ℝ+ → ℝ, f(t, Xt) is a local martingale if and only if it solves the partial differential equation (PDE)

 $\displaystyle \frac{\partial f}{\partial t}+Lf=0.$

Consequently, for any time t > 0 and function g: ℝd → ℝ, if we let f be a solution to the PDE above with boundary condition f(x, t) = g(x) then, assuming integrability conditions, the conditional expectations at times s < t are

 $\displaystyle {\mathbb E}[g(X_t)\;\vert\mathcal F_s]=f(X_s,s).$

If the conditions are satisfied, this describes a Markov process and gives its transition probabilities, describing the distribution of X and implying uniqueness in law.

Forward Equation: Assuming that it is sufficiently smooth, the probability density p(t, x) of Xt satisfies the PDE

 $\displaystyle \frac{\partial p}{\partial t}=L^Tf.$

where LT is the transpose of operator L

 $\displaystyle L^Tp=\frac12(a_{ij}p)_{,ij}+(b_ip)_{,i}.$

If this PDE has a unique solution for given initial distribution, then this uniquely determines the distribution of Xt. So, if unique solutions to the forward equation exist starting at every future time, it gives uniqueness in law for X.

Martingale problem: Any weak solution to SDE (1) satisfies the property that

 $\displaystyle f(X_t)-\int_0^t Lf(X_s)\,ds$

is a local martingale for twice continuously differentiable functions f: ℝn → ℝ. This approach, which was pioneered by Stroock and Varadhan, has many benefits over the other applications of operator L described above, since it applies much more generally. We do not need to a-priori impose any properties on X such as being Markov, and as the test functions f are chosen at will, they automatically satisfy the necessary regularity properties. As well as being a very general way to describe solutions to a stochastic dynamical system, it turns out to be very fruitful. The striking and far-reaching Stroock–Varadhan uniqueness theorem, in particular, guarantees existence and uniqueness in law so long as a is continuous and positive definite and b is locally bounded.

# 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-1/2Xct 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.

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 , and use the construction above with , σ̃,  in place of T, σ, B respectively. We need to show that and B have the same distribution. Using the scaling factor S = /T, then Xt = S-1/2XtS 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 {Bt}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.

# 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 XT 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 ↦ Xσ + 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-1/2XtS 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 {Bt}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}.

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 , and use the construction above with , σ̃, τ̃,  in place of T, σ, τ, B respectively. We need to show that and B have the same distribution. Using the scaling factor S = /T, then Xt = S-1/2XtS 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 {Bt}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.

# Brownian Bridge Fourier Expansions

Brownian bridges were described in a previous post, along with various different methods by which they can be constructed. Since a Brownian bridge on an interval ${[0,T]}$ is continuous and equal to zero at both endpoints, we can consider extending to the entire real line by partitioning the real numbers into intervals of length T and replicating the path of the process across each of these. This will result in continuous and periodic sample paths, suggesting another method of representing Brownian bridges. That is, by Fourier expansion. As we will see, the Fourier coefficients turn out to be independent normal random variables, giving a useful alternative method of constructing a Brownian bridge.

There are actually a couple of distinct Fourier expansions that can be used, which depends on precisely how we consider extending the sample paths to the real line. A particularly simple result is given by the sine series, which I describe first. This is shown for an example Brownian bridge sample path in figure 1 above, which plots the sequence of approximations formed by truncating the series after a small number of terms. This tends uniformly to the sample path, although it is quite slow to converge as should be expected when approximating such a rough path by smooth functions. Also plotted, is the series after the first 100 terms, by which time the approximation is quite close to the target. For simplicity, I only consider standard Brownian bridges, which are defined on the unit interval ${[0,1]}$. This does not reduce the generality, since bridges on an interval ${[0,T]}$ can be expressed as scaled versions of standard Brownian bridges.

Theorem 1 A standard Brownian bridge B can be decomposed as

 $\displaystyle B_t=\sum_{n=1}^\infty\frac{\sqrt2Z_n}{\pi n}\sin(\pi nt)$ (1)

over ${0\le t\le1}$, where ${Z_1,Z_2,\ldots}$ is an IID sequence of standard normals. This series converges uniformly in t, both with probability one and in the ${L^p}$ norm for all ${1\le p < \infty}$.

# 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”

# Brownian Drawdowns

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.

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.

# Drawdown Point Processes

For a continuous real-valued stochastic process ${\{X_t\}_{t\ge0}}$ with running maximum ${M_t=\sup_{s\le t}X_s}$, consider its drawdown. This is just the amount that it has dropped since its maximum so far,

 $\displaystyle D_t=M_t-X_t,$

which is a nonnegative process hitting zero whenever the original process visits its running maximum. By looking at each of the individual intervals over which the drawdown is positive, we can break it down into a collection of finite excursions above zero. Furthermore, the running maximum is constant across each of these intervals, so it is natural to index the excursions by this maximum process. By doing so, we obtain a point process. In many cases, it is even a Poisson point process. I look at the drawdown in this post as an example of a point process which is a bit more interesting than the previous example given of the jumps of a cadlag process. By piecing the drawdown excursions back together, it is possible to reconstruct ${D_t}$ from the point process. At least, this can be done so long as the original process does not monotonically increase over any nontrivial intervals, so that there are no intervals with zero drawdown. As the point process indexes the drawdown by the running maximum, we can also reconstruct X as ${X_t=M_t-D_t}$. The drawdown point process therefore gives an alternative description of our original process.

See figure 1 for the drawdown of the bitcoin price valued in US dollars between April and December 2020. As it makes more sense for this example, the drawdown is shown as a percent of the running maximum, rather than in dollars. This is equivalent to the approach taken in this post applied to the logarithm of the price return over the period, so that ${X_t=\log(B_t/B_0)}$. It can be noted that, as the price was mostly increasing, the drawdown consists of a relatively large number of small excursions. If, on the other hand, it had declined, then it would have been dominated by a single large drawdown excursion covering most of the time period.

For simplicity, I will suppose that ${X_0=0}$ and that ${M_t}$ tends to infinity as t goes to infinity. Then, for each ${a\ge0}$, define the random time at which the process first hits level ${a}$,

 $\displaystyle \tau_a=\inf\left\{t\ge 0\colon X_t\ge a\right\}.$

By construction, this is finite, increasing, and left-continuous in ${a}$. Consider, also, the right limits ${\tau_{a+}=\lim_{b\downarrow0}\tau_b}$. Each of the excursions on which the drawdown is positive is equal to one of the intervals ${(\tau_a,\tau_{a+})}$. The excursion is defined as a continuous stochastic process ${\{D^a_t\}_{t\ge0}}$ equal to the drawdown starting at time ${\tau_a}$ and stopped at time ${\tau_{a+}}$,

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

This is a continuous nonnegative real-valued process, which starts at zero and is equal to zero at all times after ${\tau_{a+}-\tau_a}$. Note that there uncountably many values for ${a}$ but, the associated excursion will be identically zero other than for the countably many times at which ${\tau_{a+} > \tau_a}$. We will only be interested in these nonzero excursions.

As usual, we work with respect to an underlying probability space ${(\Omega,\mathcal F,{\mathbb P})}$, so that we have one path of the stochastic process X defined for each ${\omega\in\Omega}$. Associated to this is the collection of drawdown excursions indexed by the running maximum.

 $\displaystyle S=\left\{(a,D^a)\colon a\ge0,\ D^a\not=0\right\}.$

As S is defined for each given sample path, it depends on the choice of ${\omega\in\Omega}$, so is a countable random set. The sample paths of the excursions ${D^a}$ lie in the space of continuous functions ${{\mathbb R}_+\rightarrow{\mathbb R}}$, which I denote by E. For each time ${t\ge0}$, I use ${Z_t}$ to denote the value of the path sampled at time t,

 \displaystyle \begin{aligned} &E=\left\{z\colon {\mathbb R}_+\rightarrow{\mathbb R}{\rm\ is\ continuous}\right\}.\\ &Z_t\colon E\rightarrow{\mathbb R},\\ & Z_t(z)=z_t. \end{aligned}

Use ${\mathcal E}$ to denote the sigma-algebra on E generated by the collection of maps ${\{Z_t\colon t\ge0\}}$, so that ${(E,\mathcal E)}$ is the measurable space in which the excursion paths lie. It can be seen that ${\mathcal E}$ is the Borel sigma-algebra generated by the open subsets of E, with respect to the topology of compact convergence. That is, the topology of uniform convergence on finite time intervals. As this is a complete separable metric space, it makes ${(E,\mathcal E)}$ into a standard Borel space.

Lemma 1 The set S defines a simple point process ${\xi}$ on ${{\mathbb R}_+\times E}$,

 $\displaystyle \xi(A)=\#(S\cap A)$

for all ${A\in\mathcal B({\mathbb R}_+)\otimes\mathcal E}$.

From the definition of point processes, this simply means that ${\xi(A)}$ is a measurable random variable for each ${A\in \mathcal B({\mathbb R}_+)\otimes\mathcal E}$ and that there exists a sequence ${A_n\in \mathcal B({\mathbb R}_+)\otimes\mathcal E}$ covering E such that ${\xi(A_n)}$ are almost surely finite. The set of drawdowns for the point process corresponding to the bitcoin prices in figure 1 are shown in figure 2 below.

# Criteria for Poisson Point Processes

If S is a finite random set in a standard Borel measurable space ${(E,\mathcal E)}$ satisfying the following two properties,

• if ${A,B\in\mathcal E}$ are disjoint, then the sizes of ${S\cap A}$ and ${S\cap B}$ are independent random variables,
• ${{\mathbb P}(x\in S)=0}$ for each ${x\in E}$,

then it is a Poisson point process. That is, the size of ${S\cap A}$ is a Poisson random variable for each ${A\in\mathcal E}$. This justifies the use of Poisson point processes in many different areas of probability and stochastic calculus, and provides a convenient method of showing that point processes are indeed Poisson. If the theorem applies, so that we have a Poisson point process, then we just need to compute the intensity measure to fully determine its distribution. The result above was mentioned in the previous post, but I give a precise statement and proof here. Continue reading “Criteria for Poisson Point Processes”

# Poisson Point Processes

The Poisson distribution models numbers of events that occur in a specific period of time given that, at each instant, whether an event occurs or not is independent of what happens at all other times. Examples which are sometimes cited as candidates for the Poisson distribution include the number of phone calls handled by a telephone exchange on a given day, the number of decays of a radio-active material, and the number of bombs landing in a given area during the London Blitz of 1940-41. The Poisson process counts events which occur according to such distributions.

More generally, the events under consideration need not just happen at specific times, but also at specific locations in a space E. Here, E can represent an actual geometric space in which the events occur, such as the spacial distribution of bombs dropped during the Blitz shown in figure 1, but can also represent other quantities associated with the events. In this example, E could represent the 2-dimensional map of London, or could include both space and time so that ${E=F\times{\mathbb R}}$ where, now, F represents the 2-dimensional map and E is used to record both time and location of the bombs. A Poisson point process is a random set of points in E, such that the number that lie within any measurable subset is Poisson distributed. The aim of this post is to introduce Poisson point processes together with the mathematical machinery to handle such random sets.

The choice of distribution is not arbitrary. Rather, it is a result of the independence of the number of events in each region of the space which leads to the Poisson measure, much like the central limit theorem leads to the ubiquity of the normal distribution for continuous random variables and of Brownian motion for continuous stochastic processes. A random finite subset S of a reasonably ‘nice’ (standard Borel) space E is a Poisson point process so long as it satisfies the properties,

• If ${A_1,\ldots,A_n}$ are pairwise-disjoint measurable subsets of E, then the sizes of ${S\cap A_1,\ldots,S\cap A_n}$ are independent.
• Individual points of the space each have zero probability of being in S. That is, ${{\mathbb P}(x\in S)=0}$ for each ${x\in E}$.

The proof of this important result will be given in a later post.

We have come across Poisson point processes previously in my stochastic calculus notes. Specifically, suppose that X is a cadlag ${{\mathbb R}^d}$-valued stochastic process with independent increments, and which is continuous in probability. Then, the set of points ${(t,\Delta X_t)}$ over times t for which the jump ${\Delta X}$ is nonzero gives a Poisson point process on ${{\mathbb R}_+\times{\mathbb R}^d}$. See lemma 4 of the post on processes with independent increments, which corresponds precisely to definition 5 given below. Continue reading “Poisson Point Processes”

# Local Time Continuity

The local time of a semimartingale at a level x is a continuous increasing process, giving a measure of the amount of time that the process spends at the given level. As the definition involves stochastic integrals, it was only defined up to probability one. This can cause issues if we want to simultaneously consider local times at all levels. As x can be any real number, it can take uncountably many values and, as a union of uncountably many zero probability sets can have positive measure or, even, be unmeasurable, this is not sufficient to determine the entire local time ‘surface’

 $\displaystyle (t,x)\mapsto L^x_t(\omega)$

for almost all ${\omega\in\Omega}$. This is the common issue of choosing good versions of processes. In this case, we already have a continuous version in the time index but, as yet, have not constructed a good version jointly in the time and level. This issue arose in the post on the Ito–Tanaka–Meyer formula, for which we needed to choose a version which is jointly measurable. Although that was sufficient there, joint measurability is still not enough to uniquely determine the full set of local times, up to probability one. The ideal situation is when a version exists which is jointly continuous in both time and level, in which case we should work with this choice. This is always possible for continuous local martingales.

Theorem 1 Let X be a continuous local martingale. Then, the local times

 $\displaystyle (t,x)\mapsto L^x_t$

have a modification which is jointly continuous in x and t. Furthermore, this is almost surely ${\gamma}$-Hölder continuous w.r.t. x, for all ${\gamma < 1/2}$ and over all bounded regions for t.