Perturbative Expansion for the Maximum of Fractional Brownian Motion

Perturbative Expansion for the Maximum of Fractional Brownian Motion

Mathieu Delorme and Kay Jörg Wiese CNRS-Laboratoire de Physique Théorique de l’Ecole Normale Supérieure, 24 rue Lhomond, 75005 Paris, France

Brownian motion is the only random process which is Gaussian, stationary and Markovian. Dropping the Markovian property, i.e. allowing for memory, one obtains a class of processes called Fractional Brownian motion, indexed by the Hurst exponent . For , Brownian motion is recovered. We develop a perturbative approach to treat the non-locality in time in an expansion in . This allows us to derive analytic results beyond scaling exponents for various observables related to extreme value statistics: The maximum of the process and the time at which this maximum is reached, as well as their joint distribution. We test our analytical predictions with extensive numerical simulations for different values of . They show excellent agreement, even for far from .

I Introduction

Random processes are ubiquitous in nature. Though many processes can successfully be mExtensive numerical simulations for different values of test these analytical predictions and show excellent agreement, even for large .odeled by Markov chains and are well analyzed by tools of statistical mechanics, there are also interesting and realistic systems which do not evolve with independent increments, and thus are non-Markovian, i.e. history dependent. Dropping the Markov property, but demanding that a continuous process be scale-invariant and Gaussian with stationary increments defines an enlarged class of random processes, known as fractional Brownian motion (fBm). Such processes appear in a broad range of contexts: Anomalous diffusion BouchaudGeorges1990 (), diffusion of a marked monomer inside a polymer WalterFerrantiniCarlonVanderzande2012 (); AmitaiKantorKardar2010 (), polymer translocation through a pore AmitaiKantorKardar2010 (); ZoiaRossoMajumdar2009 (); DubbeldamRostiashvili2011 (); PalyulinAlaNissilaMetzler2014 (), single-file diffusion in ion channels KuklaKornatowskiDemuthGirnusal1996 (); WeiBechingerLeiderer2000 (), the dynamics of a tagged monomer GuptaRossoTexier2013 (); Panja2011 (), finance (fractional Black-Scholes, fractional stochastic volatility models, and their limitations) CutlandKoppWillinger1995 (); Rogersothers1997 (); RostekSchobel2013 (), hydrology MandelbrotWallis1968 (); MolzLiuSzulga1997 (), and many more.

While averaged quantities have been studied extensively and are well characterized, it is often more important to understand the extremal behavior of these processes, or the time the process satisfies a given criterion GumbelBook (). These quantities are associated with failure in fracture or earthquakes, a crash in the stock market, the breakage of dams, the time one has to heat, etc. For Brownian motion the three arcsine laws are well studied examples. They state that for a Brownian process , with , and , three observables have the same probability distribution, namely


The observables in question are (see Fig. 1)

  1. First (Lévy’s) arcsine law: The time the process is positive, (red in Fig. 1),

  2. Second arcsine law: The last time the process is at its initial position, (blue in Fig. 1),

  3. Third arcsine law: The time at which the process achieves its maximum (which is almost surely unique), (green in Fig. 1)

Figure 1: The three arcsine laws discussed in the main text. , in green, is the time where the process achieves its maximum. , in blue, is the last time the process is at its starting value . Finally, , in red, is the time spend in the positive half space, which is the sum of the red intervals .

While these laws are well-studied for Brownian motion, little is known about their generalization to other random processes. In this article, we will generalize the third arcsine law to fractional Brownian motion, and obtain the distribution of the achieved maximum.

Fractional Brownian motion (fBm) is a random process characterized by the Hurst exponent which quantifies the growth of the 2-point function in time,


Up to now, analytical tools to study its extreme value statistics were available only for Brownian motion, i.e. . In this article, we aim to extend this to . This is achieved by constructing a path integral, and evaluating it perturbatively around a Brownian, setting . This technique has been introduced in Ref. WieseMajumdarRosso2010 (). We will calculate the probability distribution of the maximum of the process and the time at which the maximum is reached, as well as their joint distribution. A short account of this work was published in Ref. DelormeWiese2015 ().

The article is structured as follows: Section II defines the fBm, discusses its relation to anomalous diffusion, and defines the observables related to extremal value statistics we wish to study.

Section III introduces the path integral we need to calculate, followed by its perturbative expansion in . This defines the main integrals to be calculated, for which we also give a diagrammatic representation. As the calculations are rather tedious, they are relegated to appendix C.

Section IV presents our results: We start by recalling scaling relations in section IV.1, before introducing our most general formula, the probability to start at , to reach the minimum at time time , and to finish at time in . This allows us to derive several simpler results: First the distribution of times when the maximum is achieved, for a Brownian known as the third arcsine law (section IV.3). Second, the distribution of the value of this maximum. And third, the joint distribution of maximum, and the time when this maximum is taken.

Extensive numerical simulations for different values of test these analytical predictions in section V.

Conclusions are given in Section VI, followed by several appendices: Appendix A gives details on the perturbation expansion. Appendix B reviews results from WieseMajumdarRosso2010 (), including a new derivation of the latter. Appendix C calculates the main new, and most difficult, contribution. Appendix D gives details on the corrections to the third Arcsine Law, while for the attained maximum and its cumulative distribution this is done in appendices E and F. Appendix G gives a list of used inverse Laplace transforms. Finally, in appendix H is verified that the second cumulant is correctly reproduced.

Ii Fractional Brownian motion and Observables

ii.1 Definition of the fBm

FBm is a generalization of standard Brownian motion to other fractal dimensions, introduced in its final form by Mandelbrot and Van Ness MandelbrotVanNess1968 (). It is a Gaussian process , starting at zero, , with mean and covariance function (variance)


A fBm starting at a non-zero value is defined as , with as above. The parameter appearing in (7) is the Hurst exponent. Standard Brownian motion corresponds to ; there the covariance function (7) reduces to . Unless , the process is non-Markovian , i.e. its increments are not independent: For they are correlated, whereas for they are anti-correlated:


It is important to note that the process is stationary, as the second moment (and thus the whole distribution) of the increments is a function of the time difference only,


The fact that a fBm process is non-Markovian makes its study difficult, as most of the standard stochastic-process tools (decomposing transition probabilities into products of propagators, or writing the evolution of a density using a Fokker-Plank equation) rely on the Markov property.

ii.2 Anomalous diffusion

Anomalous diffusion is another interesting property of the fBm. It is caracterized by the non-linear growth (for ) of the second moment of the process,


For , a fBm is a sub-diffusive process, while for , it is super-diffusive.

Anomalous diffusion is usually implied by a stronger property (but equivalent in the case of a Gaussian process): self-similarity of exponent . It means that rescaling time by and space by leaves every averaged observable defined on the process invariant,


This property is stronger in the sense that the growth of every moment, and not only the second one, is governed by the same exponent : .

It is well known that standard Brownian motion is the only continuous process with stationary, independent (Markovian) and Gaussian increments. As a consequence, every process in this class is -self-similar, i.e. exhibits normal diffusion. To obtain an anomalous diffusive process, one of these three hypotheses has to be removed. This gives three main classes of anomalous diffusion:

  • heavy tails of the increments (Levy-flight process) or heavy tails in the waiting time between increments (CTRW processes); these processes are non-Gaussian.

  • time dependence of the diffusive constant: the distribution of the increments is time dependent, i.e. the process is non-stationary.

  • correlations between increments: the process is non-Markovian

FBm is the only process which is Gaussian, stationary, and statistically self-similar. As the first two hypotheses are natural in a large class of processes appearing in nature, and self-similarity with exponent is equivalent to anomalous diffusion for a Gaussian process, fBm appears as an important representative for anomalous diffusion.

Interestingly, several processes commonly used in physics, mathematics, and computer science belong to the fBm class. For expample, it was recently proven that the dynamics of a tagged particle in single-file diffussion (cf. WeiBechingerLeiderer2000 (); KrapivkyMallickSadhu2014 (); KrapivkyMallickSadhu2015a (); KrapivkyMallickSadhu2015 ()) has at large times the fBm covariance function (7) with Hurst exponent .

ii.3 Extreme-value statistics (EVS)

Figure 2: Two realisations of fBm paths for different values of , generated using the same random numbers for the Fourier modes in the Davis and Harte procedure Dieker (). The observables and are given.

The objective of this article is to study fBm in the context of what is now called extreme-value statistics. While the knowledge of averages or of the typical behavior is an important step in understanding and comparing stochastic models to experiments or data, there are situations were the interest lies in the extremes or rare events. For example, the physics of disordered systems at low temperatures is governed by the states with a (close to) minimal energy in the random energy landscape. Extreme weather conditions are of importance in the dimensioning of infrastructures such as dams and bridges. More generally, extreme value questions appear naturally in many optimization problem.

The simplest and first case studied for these extreme-value statistics was the distribution of the maximum of a large number of independent and identically distributed random variables, which is now well understood in the large- limit thanks to the classification of the Fisher-Tippett-Gnedenko theorem: Depending on the initial distribution of the variables, the rescaled maximum follows either a Weibull, Gumbel or Fréchet distribution GumbelBook (); BouchaudMezard1997 (). This is the equivalent of the central-limit theorem, which classifies the sums, or equivalently averages, of a large number of independent identically distributed (i.i.d.) variables.

The case of strongly correlated variables was a natural extension to this problem, as many physically relevant situations present significant deviations from the i.i.d. case. Many results were derived for random walks and Brownian motion SchehrLeDoussal2010 (); DeanMajumdar2001 (). The distribution of the largest eigenvalue is also a central question in random matrix theory MajumdarSchehr2014 (). Finally, some previous study in the context of non-Markovian processes can be found in Ref. DerridaHakimZeitak1996 (); Majumdar1999 (); MajumdarRossoZoia2010 ().

In this article we study the extremal properties of a fractionnal Brownian motion . The main observables are the maximum and the time when this maximum is reached. Figure 2 shows an illustration for different values of , using the same random numbers for the Fourier modes. We will denote and their respective probability distributions. Previous studies on these distributions, focusing on the small-scale behaviour, can be found in Refs. Sinai1997 (); Molchan1999 ().

These observables are closely linked to other quantities of interest, such as the first-return time, the survival probability, the persistence exponent, and the statistics of records.

Iii The perturbative approach

iii.1 Path integral formulation and the Action

Following the ideas of MajumdarSire1996 (); OerdingCornellBray1997 (); WieseMajumdarRosso2010 (), we start with the path-integral,


It sums over all paths , weighted by their probability , starting at , passing through (close to ) at time , and ending in , while staying positive for all . The latter is enforced by the product of Heaviside functions . This path integral depends on the Hurst exponent through the action. Since is a Gaussian process, the action can (at least formally) be constructed from the covariance function of ,


Here . This, however, is not enough to evaluate the path integral (12), since it is not evident how to implement the product of -functions. Following the formalism of Ref. WieseMajumdarRosso2010 (), we use standard Brownian motion as a starting point for a perturbative expansion, setting with a small parameter; then the action at first order in is (we refer to the appendix of Ref. WieseMajumdarRosso2010 () for the derivation)


The time is a regularization cutoff for coinciding times (a UV cutoff). We will see that it has no impact on the distribution of observables which can be extracted from the path integral. (One can also introduce discrete times spaced by WieseMajumdarRosso2010 ()).

The first line of Eq. (14), which we denote , is the action for standard Brownian motion, with a rescaled diffusion constant


It is a dimensionfull constant, as fBm and standard Brownian motion do not have the same time dimension. The second line, which we denote , is the first correction to the action. It is non-local in time, which implies that the process is non-Markovian (even if we neglect terms). We check this expansion of the action in appendix H, where we compute the covariance of the process from a path integral, and recover Eq. (7) at first order in .

As we will see in section IV, this path integral , in the limit of , encodes a plethora of information about the maximum of the process: both distributions and can be extracted from it, as well as the joint distribution. Further, the same distributions in the case of a fBm bridge.

It is important to note that the limit of is non-trivial, as it forces the process to go close to an absorbing boundary which leads to non-trivial scaling involving the persistence exponent defined below in section IV.1.

iii.2 The order- term

Having expressed the perturbative expansion of the action, the main task is to compute the path integral (12), at first order in , and in the limit of small . Expanding the exponential of the action in (12),


allows us to compute the path integral perturbatively in the non-local interaction , defined as the second line of Eq. (14),


This gives


is the term with no non-local interaction, while is the term with one interaction (it is proportional to because the non-local interaction itself has an amplitude of order ). Formally, the order- term is


where is the action of a standard Brownian motion,


Since Brownian motion is a Markov process, this action is local in time. It allows us to write the path integral as a product,


In the second line, the constraint is enforced by the boundary conditions of the path integral. In the last line, we expressed each path integral in terms of the propagator of standard Brownian motion, constraint to the positive half space. It is obtained via the method of images,


for an arbitrary diffusive constant . We now use that the diffusive constant is . This allows us to express the path integral (12) at leading order in , and in the limit of small , as


To include the order- term in the diffusive constant to get the full result for at order , we use Eq. (15) expanded in ,


It is interesting to note that the order- term appearing here can also be computed from the result (23) as


iii.3 The first-order terms

To go beyond Brownian motion and include non-Markovian effects, i.e. interactions non-local in time, we need to compute the first-order correction in the expansion (III.2), which is called and reads


As before, we denote . To compute , we decompose it into three terms, distinguished by their time ordering. Denote the part where , the part where , and the term where . Then


In the first term, the interaction affects only the process in the time interval , and there is no coupling with the process on the time interval . This leads, as shown in appendix A, to a factorized expression for ,


Here is the order- correction to the propagator of fBm in the half space (i.e. constrained to remain positive). This object, which we need in the limit of , was studied and computed in Ref. WieseMajumdarRosso2010 (). The result is recalled in appendix B, and recalculated using more efficient technology developed here. The second term is similar to the first, swapping the two time intervals,


The third term, , is more complicated as the interaction couples the two time intervals and . We can still take advantage of locality in time of the action to write the path integral (26), with time integrals restricted to , as a product of simpler path integrals:


In this expression, all path integrals can be expressed in terms of the bare propagator ; we refer to appendix A for how to deal with the terms containing . We have not written the cut-off as there are no short-time divergences that need to be regularized (contrary to the terms and ). The structure of the time integrals, which are products of convolutions, suggests to use Laplace transforms (with respect to the time variables: , ). This, and the identity


give us a simple form for the double Laplace transform of , which we will denote with a tilde (for details see appendix A),






Figure 3: Left: Graphical representation of the contribution to the path-integral given in Eq. (12). The red curve represents the non-local interaction in the action, second line of Eq. (14), while blue lines are bare propagators. We also indicate the Laplace variable which appears in each time slice in Eq. (32). Right: Graphical representation of

The Laplace-transformed constrained propagator appearing in this expression is


The Laplace transformation gives another simplification: the space dependence is now exponential, as compared to the Gaussian form of , which renders the space integrations elementary. (Without the Laplace transform, already the first space integration gives an error-function, and the remaining integrations are highly non-trivial). Nevertheless, the final result for is complicated, and requires to compute the three integrals in Eq. (32), and two inverse Laplace transformations. These steps are performed in appendix C.

iii.4 Graphical representation

It is useful to give a diagrammatic representation for the terms of the perturbative expansion. We denote bare propagators (33) with a solid blue lines. The interaction between two points and is represented in red. As can be seen from Eq. (32), it acts as on the propagator starting at , on the propagator starting at ; it also translates the Laplace variable of each time slice between these two points by . The space variables , and the interaction variable (which has the inverse dimension of time) have to be integrated from to . In case of divergences, the integration has to be cut off with a large- cutoff (c.f. appendix G for the link between the short time cutoff and the large cutoff).

The contribution of , is computed in detail in Appendix C, and represented in Figure 3 (left), together with the contribution (right).

Iv Analytical Results

We present here some known scaling results about extremal properties of the fBm. We then show how our perturbative expansion, and the computation of , allows us to obtain analytical results on the distributions beyond these scaling arguments. Some of our results were already presented in a Letter DelormeWiese2015 ().

iv.1 Scaling results

Let us start with the the survival probability , and the persistence exponent , defined for any random process with as


For a review of these concepts in the context of theoretical physics, we refer to BrayMajumdarSchehr2013 (). In a large class of processes the exponent is independent of , and characterizes the power-law decay for the probability of long positive excursions. For fractional Brownian motion with Hurst exponent it was shown that Molchan1999 (); Aurzada2011 (). To understand the link of with the maximum distribution for fBm, we use self affinity of the process to write as


Here is a scaling function depending on . The survival probability is related to the maximum distribution by


This states that due to translational invariance a realisation of a fBm starting at and remaining positive is the same as a realisation starting at and having a minimum larger than . Finally, the symmetry (for a fBm starting at ) gives the correspondence between minima and maxima. These considerations allow us to predict the scaling behavior of at small from the large- behaviour of Molchan1999 (),


and finally


For the distribution of the time at which the maximum is achieved we can estimate the behavior close to the origin by assuming that small values of the maximum are reached close to the origin. Starting with


and using scaling, , we obtain


This should be valid when (or ). By time reversal symmetry , we also have


iv.2 The complete result for

We present here the final result for , defined in Eq. (12), at order . This path integral was first expanded, c.f. Eq. (III.2), by treating the non-local term in the action (14) perturbatively. The first term of this expansion is given in Eq. (24), while the second term was split into three contributions , and , see Eq. (27). The first two terms can be obtained explicitly from (99), while the third one is computed in appendix C, the result being split between (113), (129) and (145).

In order to display a compact form, we choose (which is equivalent to rescaling and by and and by ) and introduce new rescaled (dimensionless) variables,


In these new variables, the final result is


The special function appearing in this expression is


iv.3 The third Arcsine Law: Distribution of the time when the maximum is reached

To simplify the result (IV.2), we can extract from it the distribution of a single observable. We start with the probability distribution of , the time when the fBm achieves its maximum. For Brownian motion (), this distribution is well known as the third arcsine law, because the cumulative distribution involves the arcsin function c.f. Eq. (1),


Until now, only scaling properties were known for this distribution in the general case MajumdarRossoZoia2010b (), as recalled in Eq. (40).

The path integral (12), in the limit of , selects paths which go through at time while staying positive. This means that we sum over paths reaching their minimum (in the interval , and which is almost surely unique) at , starting at and ending at . This is equivalent to summing over paths starting at , reaching their minimum with value at time , and ending in . Integrating over and finally gives the sum over all paths reaching their minimum in , independent of the value of this minimum, and the end point. Up to a normalization, this is the probability distribution of . By symmetry, this is the same as the distribution of . Formally, it reads


The normalization depends on and . It ensures that is normalized; it can be expressed in terms of as


At order , starting from Eq. (23) and integrating over and allows us to recover Eq. (46) with normalisation .

Figure 4: Distribution of for and (red) or (blue) given in Eq. (51) (plain lines) compared to the scaling ansatz, i.e.  (dashed lines) and numerical simulations (dots). For realisations with are less probable (by about ) than expected from scaling. For the correction has the opposite sign.

For the order- correction, the integrations over and are lengthy. This is done in appendix D. It allows us to write an -expansion for the distribution of in the form


The result (158) reads


where and . It takes a simple form if we exponentiate this order- correction,


The term in gives the expected change, from Eq. (40) and (41), in the scaling form of the Arcsine law, . The regular part induces a non-trivial change in the shape,


The time reversal symmetry (corresponding to ) is explicit and the constant ensures normalization. The contribution of to the probability that the maximum is attained at time is quite noticeable, as shown in Fig. 4.

Figure 5: Scaling function for the distribution of the maximum, as defined in Eq. (35), for different values of : in red, in yellow, in green, and in blue. The plain lines represent the analytic prediction from our perturbative theory (at first order in ) given in Eq. (55); the symbols are results from numerical simulations, c.f. section V.
Figure 6: Left: Numerical estimation of for different values of on a discrete system of size , using realizations. Plain curves represent the theoretical prediction (52), vertically translated for better visualization. Error bars are estimates. Note that for , and the expansion parameter is positive, while for , and it is negative.
Right: Deviation for large between the theoretical prediction (52) and the numerical estimations (77), rescaled by , c.f. Eq. (78). These curves collapse for different values of , allowing for an estimate of the correction to , as written in Eq. (79).

iv.4 The distribution of the maximum

We now present results for the distribution of the maximum . For standard Brownian motion


On the other hand, the scaling results presented in IV.1 predict that for any , behaves at small scale as