# Energy of Taut Strings Accompanying Wiener Process

###### Abstract

Let be a Wiener process. For and let denote the minimal value of the energy taken among all absolutely continuous functions defined on , starting at zero and satisfying

The function minimizing energy is a taut string, a classical object well known in Variational Calculus, in Mathematical Statistics, and in a broad range of applications. We show that there exists a constant such that for any

and for any fixed ,

Although precise value of remains unknown, we give various theoretical bounds for it, as well as rather precise results of computer simulation.

While the taut string clearly depends on entire trajectory of , we also consider an adaptive version of the problem by giving a construction (called Markovian pursuit) of a random function based only on the values , and having minimal asymptotic energy. The solution, i.e. an optimal pursuit strategy, turns out to be related with a classical minimization problem for Fisher information on the bounded interval.

2010 AMS Mathematics Subject Classification: Primary: 60G15; Secondary: 60G17, 60F15.

Key words and phrases: Gaussian processes, Markovian pursuit, taut string, Wiener process.

## Introduction

Given a time interval and two functional boundaries , , the taut string is a function that for any (!) convex function provides minimum for the functional

among all absolutely continuous functions with given starting and final values and satisfying

The list of simultaneously optimized functionals includes energy , variation , graph length , etc.

The first instance of taut strings that we have found in the literature is in G. Dantzig’s paper [7]. Dantzig notes there that the problem under study and its solution was discussed in R. Bellman’s seminar at RAND Corporation in 1952. The taut strings were later used in Statistics, see [17] and [8]. In the book [21, Chapter 4, Subsection 4.4], taut strings are considered in connection with problems in image processing. Quite recently, taut strings were applied to a buffer management problem in communication theory, see [22].

In this article, we study the energy of the taut string going through the tube of constant width constructed around sample path of a Wiener process , i.e. for some we let , , see Fig. 1.

We focus attention on the behavior in a long run: we show that when , the taut string spends asymptotically constant amount of energy per unit of time. Precise assertions are given in Theorems 1.1 and 1.2 below. The constant shows how much energy an absolutely continuous function must spend if it is bounded to stay within a certain distance from the non-differentiable trajectory of .

Although precise value of remains unknown, we give various theoretical bounds for it in Section 4, as well as the results of computer simulation in Section 6. The latter suggest .

If we take the pursuit point of view, considering as a trajectory of a particle moving with finite speed and trying to stay close to a Brownian particle, then it is much more natural to consider constructions that define in adaptive way, i.e. on the base of the known . Recall that the taut string depends on the entire trajectory , hence it does not fit the adaptive setting. In view of Markov property of , the reasonable pursuit strategy for is to move towards with the speed depending on the distance . In this class of algorithms we find an optimal one in Section 5. The corresponding function spends in average units of energy per unit of time. Comparing of two constants shows that we have to pay more than double price for not knowing the future of the trajectory of . To our great surprise, the search of optimal pursuit strategy boils down to the well known variational problem: minimize Fisher information on the class of distributions supported on a fixed bounded interval.

We conjecture that the provided algorithm is the optimal one in the entire class of adaptive algorithms.

In Section 7 we establish some connections with other well known settings and problems.

First, we recall that the famous Strassen’s functional law of the iterated logarithm (FLIL) and its extensions handling convergence rates in FLIL actually deal exactly with the energy of taut strings. Not surprisingly, we borrowed some techniques for evaluation of this energy from FLIL research. Yet it should be noticed that FLIL requires very different range of parameters and than those emerging in our case. The FLIL tubes are much wider, hence the taut string energy is much lower than ours. This is why Strassen law with its super-slow loglog rates is so hard to reproduce in simulations, while our results handling the same type of quantities are easily observable in computer experiment.

Second, we briefly look at the taut string as a minimizer of of variation

Since is not a strictly convex function, the corresponding variational problem typically has many solutions. In [15, 18] another minimizer of is described in detail, a so called lazy function. As E. Schertzer pointed to us, the relations between the taut strings and lazy functions are yet to be clarified.

Finally, we briefly describe a discrete analogue of our problem thus giving flavor of eventual applications.

As a conclusion, Section 8 traces some forthcoming or possible developments of the treated subject.

## 1 Notation and main results

Throughout the paper, we consider uniform norms

and Sobolev-type norms

where denotes the space of absolutely continuous functions on . It is natural to call energy.

Let be a Wiener process. We are mostly interested in its approximation characteristics

and

The unique functions at which the infima are attained are called taut string, resp. taut string with fixed end.

Our main results are as follows.

###### Theorem 1.1

There exists a constant such that if , then

(1.1) | |||

(1.2) |

for any .

We may complete the mean convergence with almost sure convergence to .

###### Theorem 1.2

For any fixed , when , we have

## 2 Basic properties of and

We prepare the proofs of the main results given below in Subsections 3.2 and 3.3 by exploring scaling and concetration properties of the taut string’s energy.

### 2.1 Scaling

Given two functions and on , let us rescale them onto the time interval by letting

Then

and

The boundary conditions are also transformed properly: namely, is equivalent to , while is equivalent to . Therefore, belongs to the set iff belongs to the analogous set .

### 2.2 Finite moments

We will show now that both and have finite exponential moments. Yet in the following we only need that

(2.3) | |||

(2.4) |

Let be an even integer. Then is inverse to an integer, and we may cut the time interval into intervals of length . Let be the linear interpolation of based on the knots , . Clearly, we have either or . It follows that

(2.5) |

Notice that

(2.6) | |||||

where are independent Brownian bridges, and

(2.7) | |||||

where are i.i.d. standard normal random variables.

Now we may evaluate the probabilities in (2.5). By using (2.6), we obtain

On the other hand, by using Cramér–Chernoff theorem and (2.7),

for all and some universal constant . It follows that

Hence,

whenever . By scaling we also have

for any and sufficiently small positive . It follows from the definitions that

(2.8) |

Hence, the exponential moment of is finite, too.

### 2.3 Relations between and

We already noticed in (2.8) that . We will show now that a kind of converse estimate is also true.

###### Proposition 2.1

For all positive it is true that

(2.9) |

###### Proof.

Let us fix for a while the time interval and let us approximate the trajectory of Wiener process by functions starting from some arbitrary point . Let and let be the taut string with fixed end at which is attained. Then we have , , , . Let

Then , ,

(2.10) |

and

(2.11) | |||||

Now we pass to the lower bound for . Let us fix and produce an approximation for on with the fixed end. First, let , be the taut string at which is attained. The end point is not fixed, thus need not vanish. Nevertheless we still have

Now we approximate the auxiliary Wiener process

by the function defined above and let

At the boundary point the first definition yields the value , the second definition yields ; the two values coincide by the definition of function .

Moreover,

Therefore, the extended function provides an absolutely continuous approximation with fixed end to on . Furthermore, by (2.10) for we have

Finally, by (2.11),

We conclude that

(2.12) | |||||

and turn this relation into the desired bound

Notice that and are independent and . By taking expectations we get the desired relation (2.9). ∎

### 2.4 Concentration

We first notice an almost obvious Lipschitz property of the functionals under consideration.

###### Proposition 2.2

For any , any we have

(2.13) |

and

(2.14) |

It is remarkable that the Lipschitz constant in the right hand side does not depend on and .

###### Proof.

In the rest of the subsection parameters and are fixed, and we drop them from our notation, thus writing instead of , etc. Let be a median for the random variable . The famous concentration inequality for Lipschitz functionals of Gaussian random vectors (see [16, Section 12]) asserts that for any

where is a standard normal random variable.

It follows that

Moreover,

and

Finally, we infer

(2.15) |

We will also need that for any

(2.16) | |||||

Similarly, for the median of we obtain

and

(2.17) |

## 3 Asymptotics

### 3.1 Asymptotics of the second moments and medians

Recall that and are the second moments. We prove the following.

###### Proposition 3.1

There exists a constant such that if , then

(3.1) | |||

(3.2) |

###### Proof.

In proving (3.1), the following sub-additivity property plays the key role. For any we have

(3.3) |

where is a Wiener process. This means that we may approximate by taut strings with fixed ends separately on the intervals and by gluing them at due to the fixed end condition imposed on the first string.

Notice that does not possess such a nice subadditivity property.

By taking expectations in (3.3), we obtain

(3.4) |

Since is a decreasing random function w.r.t. argument , the function is also decreasing in . By the scaling argument (2.1) we observe that for any fixed

(3.5) |

is an increasing function w.r.t. the argument .

Remark: We will show later in Subsection 4.3 that .

We may complete the convergence of second moments with convergence of medians.

###### Corollary 3.2

Let , resp. , be a median of , resp. . If , then

(3.6) | |||

(3.7) |

### 3.2 -convergence

Proof of Theorem 1.1. Let . We have to prove that if , then

(3.8) | |||

(3.9) |

### 3.3 Almost sure convergence

Proof of Theorem 1.2. For any fixed , when , we must prove

(3.10) | |||

(3.11) |

Consider first an exponential subsequence with arbitrary fixed . By moment estimate (2.17) and Chebyshev inequality, for any we have

Borel–Cantelli lemma yields

Taking the convergence of medians (3.6) into account, we obtain

Since the function is non-decreasing, for any we have the chain

It follows that

By letting we obtain (3.10).