Bounded noises: new theoretical developments and applications in Pharmacokinetics

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University of Pisa

Department of Mathematics

Master’s Thesis

Bounded Noises: new theoretical

developments, and applications

in Pharmacokinetics

September 19, 2014

Author

Dario Domingo

Supervisors

Dr. Alberto d’Onofrio

Prof. Franco Flandoli

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Introduction

The different areas of Mathematics find their application in a variety of largely different fields, which include for example technology (digital music, search engines, robotics), finance (insurances, pricing strategies), environmental sciences (climatology, vulcanology, weather forecasts), and medical areas such as medicine and biology (spread of epidemics, population dynamics, pharmacology). Especially in medicine and biology, one of the major contributes which mathematics can give consists in the formulation of quite good and reliable models for different observed processes.

As it often happens, practical applications also motivate further mathematical research, which then becomes object of study and investigation. In the last decades, both pure and applied reasons have led to the development of stochastic calculus. Stochastic Differential Equations (SDEs) represent a very important mathematical tool, which is used in a variety of different contexts. Examples of application of SDEs can be found for instance in physics, engineering, economics, biochemistry, and, last but not least, biology. In this last field, a wealth of research has been carried out during the last decades. Models concerning cellular differentiation, gene expression, tumour growth, as well as resistance to chemotherapy have been object of thorough study in recent years. A spontaneously arising question probably concerns the reasons that should make stochastic models more advisable than deterministic ones. Actually, throughout the nineteenth century, the only accepted idea was that any natural process obeys laws where no place for randomness is allowed. However, the real world works in a fairly different, and often more complicated, way: samples of real biological signals show the presence of fluctuations in experimental data.

At the beginning, these fluctuations were interpreted as a disturbance of a real smooth process, something that possibly had to be filtered out. During the second half of the last century, however, it became clearer to scientists that those fluctuations reflected an inherent randomness of the observed phenomenon: both an intrinsic and an extrinsic stochasticity were found to play a prominent role in a number of biological processes. The intrinsic stochasticity of a process is the one that concerns the very nature of the process and arises from its stochastic internal dynamics, often at a molecular level. The extrinsic stochasticity

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Introduction

is instead due to the interaction between the main system and the fluctuating environment around it, where the process takes place. These sources of randomness have been tradition-ally modelled by making use of Gaussian noises, either white or coloured.

Most recently, however, a vast body of research has focused on another important class of non-Gaussian stochastic processes, the one of bounded noises. The rise of scientific interest on this particular class of noises is mainly motivated by two reasons. First, in many applications, Gaussian noises are an inadequate model of the real world because of their unboundedness, which is in contrast with the nature of any real source of randomness. Second, in many relevant cases, especially from the biological area, some parameters must be positive. If the fluctuations affecting these parameters are modelled by means of a Gaussian noise, its unboundedness unavoidably makes the fluctuating parameter also assume negative values. Therefore, the model itself loses most of its reliability. We shall see an example of this last phenomenon in the first chapter of this work, where a tumor-growth model is presented. In that case, too large negative values of the Gaussian noise correspond to the fact that the immune system generates tumor cells instead of killing them, which is obviously nonsense.

The solution to these problems consists in replacing Gaussian noises with bounded noises, whose state space is then limited. In this work, examples of the use of bounded noises will be supplied within the framework of pharmacokinetic models. Pharmacokinetics and Pharmacodynamics are two important branches of Pharmacology, concerning the interplay between a drug and the living body where the latter has been administered. In particular, Pharmacokinetics deals with the processes of Absorption, Distribution, Metabolization and Excretion (ADME scheme) of the drug from the body. This is why Pharmacokinetics is in simple terms often summarized as what the body does to the drug. On the contrary, Pharmacodynamics is devoted to the analysis of both the biochemical and the physiological effects of the drug on the body, and is therefore known as the study of what a drug does to the body.

In the following chapters, we present some classes of bounded noises, make a detailed investigation on their properties, supply an efficient algorithm for their numerical simulation, and finally make use of these noises in two different pharmacokinetic models. More precisely, this work is organized as follows.

In the first chapter we very briefly recall the reasons which led us to the study of bounded noises, by also providing a practical example of the problems arising where Gaussian noises are used in modelling. Then, three classes of bounded noises are presented. Since two of them are defined through Stochastic Differential Equations, the need of detailed investigation of mathematical questions such as strong existence and uniqueness of these SDEs arises. During this investigation, carried out in Chapter 2 and 3, it turns out that, for suitable choices of some parameters, one of the noises is not bounded. This was something unexpected, that

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Introduction

has not yet been investigated in the related literature. A physical interpretation of this apparent paradox is also supplied.

Once the study of existence, uniqueness and boundedness of the noises has been carried out, further mathematical investigation on relationships between them is performed. Both similarities and differences between them are stressed. In the same chapter, we also come to the definition of another stochastic differential equation, which has already been studied in [1] and whose solution, under suitable conditions, also supplies a bounded noise.

In Chapter 5 we move forward to seeking a suitable algorithm for the numerical gener-ation of the noises. Indeed, due to both round-off errors and global discretizgener-ation errors, a simulated trajectory of the noise might overcome the bounded interval which represents the state space of the process in study. A solution which exploits the concept of Brownian Bridges is finally proposed, and the algorithm is validated on the noises of interest.

After that, a simple pharmacokinetic model concerning the dissolution profile of a drug in a living body is investigated. The system was originally introduced in [2]. There, however, a white noise was employed. We instead proceed by analyzing the main properties of the model where bounded noises are used in order to randomize the dissolution rate at which the process takes place. The difference from the deterministic case, as well as the densities of the biological half-life of the drug are analysed for different noises and for different autocor-relation times. In some cases, noise-induced transitions can be observed in these densities, especially when the stationary density of the perturbing noise is bimodal.

Finally, in Chapter 8, a two-dimensional linear model by Susanne Ditlevsen and Andrea de Gaetano [3] is studied. In the original article, however, the white noise was used. The fluctuating parameter, here perturbed by bounded noises, models the rate at which the metabolic process of a lipid (the dodecanedioic acid) occurs within liver cells. Here, according to the experimental data reported in the original article [3], we noticed that asymmetric bounded noises appeared more suitable to fit the empirical data than symmetric ones. Thus, in a previous chapter, we briefly introduce a way of generating asymmetric bounded noises, which are then exploited in the chapter concerning the two-dimensional model of Ditlevsen and de Gaetano. There, not only are the statistical variability of the trajectories and the half-life of the dodecanedioic acid explored, but also an attempt for parameter estimation via simulated annealing is carried out.

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Chapter 1 Introduction to Bounded Noises for

Modelling Biological Processes

During the Introduction, we have stressed that the dynamics of a number of processes of the physical world, especially biological processes, is affected by both an intrinsic and an extrinsic randomness. Traditionally, these two sources of randomness have been modelled by means of Gaussian noises, which are also of relatively simple mathematical tractability. However, in many applications, Gaussian noises are an inadequate model of the real world because of their unboundedness, which is in contrast with the nature of any real source of randomness, and which, in many relevant cases, can lead to evident inconsistencies.

In the next section we shall then briefly consider a model of tumor growth (see [4, 5]) and look in more detail at the consequences which the use of Gaussian noises involves. It will then become clear that the introduction of suitable bounded noises is needed in order to obtain faithful models which do not involve absurd consequences. Thus, in Section 1.2 we shall present three specific classes of bounded noises which will be used throughout this work.

1.1 White noise effects on a tumor-growth model

The model considered in this section adapts the well-known prey-predators models to the case of the interaction between tumor cells and the immune system (IS). We address the reader to the articles [4], [5] and references therein for more detailed explanations on the interpretation of the model and of the related biological parameter.

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Chapter 1. Introduction to Bounded Noises for Modelling Biological Processes the evolution of X follows a deterministic behavior. The original model reads as follows:

˙

X = (p0− δ0− m0)X − j X2−

β0X2

1 + X_c2

. (1.1)

The quantities p0X and δ0X respectively represent the proliferation rate and the apoptotic

rate of the tumor cells (let us clarify that the Apoptosis is the process of programmed cell death which may occur in some multicellular organisms). Further, the term −m0X models

the interaction between the tumor cells and the innate immune defences, and the term jX2

accounts instead for intercellular competition. Finally, the last term β0X2/[1 + (X/c)2] is

the one which models the effects of the immune system on the tumor cells.

By supposing that m0 < p0− δ0, equation (1.1) can be adimensionalized [4]. This process

yields the following equation: ˙ X = X − X 2 K − ˆ β X2 1 + X2 . (1.2)

The adimensional positive constant ˆβ models the rate at which the IS acts on the tumor. The latter is an example of a biological parameter whose approximation with a constant, even in the framework of a simplified model like the one here considered, often proves to be inadequate. The choice taken in [5] in order to account for the for stochastic variations of this parameter is to add a Gaussian white noise to the constant parameter ˆβ. Thus, the fluctuating

β(t) = ˆβ + σ ξ(t)

is considered, where ξ(t) is a white noise of unitary intensity. Equation (1.2) then reads as follows: ˙ X = X −X 2 K − ˆ β X2 1 + X2 − σ X2 1 + X2ξ(t) . (1.3)

The white noise is a particular stochastic process, which is a mathematical abstraction of a process with null autocorrelation time (see Appendix A.5.1 for the definition and the interpretation of the autocorrelation time). However, the formalization of the concept of white noise requires the introduction of the theory of generalized stochastic processes, which are the stochastic analogue of distributions for usual functions. Only in that precise sense, the white noise could be seen as the derivative of the standard Brownian motion, which at the same time is well-known to be an almost surely not differentiable process, in the usual sense.

If the white noise is approximated by stochastic processes with small yet non null auto-correlation time τ , then the so-called Wong-Zakai theorem assures that equation (1.3) can

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1.1. White noise effects on a tumor-growth model

be formally rewritten, in the limit τ → 0, as the following Stratonovich SDE (see Appendix):

dX = X − X 2 K − ˆ β X2 1 + X2 ! dt + σ X 2 1 + X2 ◦ dWt.

Although apparently reasonable, the introduction of white noise in the model here analyzed has far-reaching consequences on the biological meaning of the model itself. Indeed, due to the intrinsic unboundedness of the Gaussian white noise, the new fluctuating param-eter β(t) can often take negative values. Within the present context, it means that the immune system may generate tumor cells, instead of killing them. This is of course some-thing which is biologically unrealistic. In figure 1.1, we show an example on the time interval [0, 1] of the approximated values which the fluctuating parameter β(t) takes during a simu-lation of the model, where the time-step dt = 10−3 was used. The standard deviation σ of the white noise is chosen to be only a tenth of the mean parameter ˆβ: however, it suffices for generating a parameter which is negative almost 40% of the times.

Figure 1.1 graphically shows that the need of controlled variations on the main parameter ˆ

β becomes essential in order not to lose the physical meaning of the model. Thus, in the next section, we introduce three classes of bounded noises and their main statistical properties. Examples of their use on pharmacokinetic models will be developed in later chapters.

0

0.5

1 −0.015

0

0.015 β

(t) dt

time

Figure 1.1: Plot of an approximate realization of the infinitesimal stochastic parameter β(t)dt = ˆ

βdt + σdWt, where ˆβ = 1.8, σ = ˆβ/10 = 0.18, and dt = 10−3. With dWt is intended a Brownian

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Chapter 1. Introduction to Bounded Noises for Modelling Biological Processes

1.2 Different methods for generating Bounded Noises

In this section we formalize the idea of bounded noise, and provide some examples of bounded noises which will be of capital importance throughout all the following chapters. Formal proofs of their properties will be supplied in detail in Chapters 2, 3, and 4. However, in order to provide a good overall view, we bring forward here some of the main properties and results concerning the bounded noises of our interest.

Definition 1.2.1. Let _{Ω, F , P, (F}t)_t≥0

be a filtrated probability space. An adapted stochastic process X : Ω × [ 0 , ∞ ) → R is called bounded, or a bounded noise, if it takes values on a bounded real interval I with probability one:

∃ B > 0 : P |Xt| < B ∀ t ≥ 0 = 1 .

There are different ways one could generate bounded noises. In the following, we illus-trates three different approaches. Later, we shall give formal mathematical proofs of the boundedness of the noises here introduced.

1.2.1 Method of bounded function: Sine-Wiener noise

A first simple method consists in applying a bounded deterministic function f : R → I to a stochastic process Yt. Borrowing a common idea in the recent literature of bounded noise

[6, 7], we shall consider the case where the bounded function f is the trigonometric Sine function, and the process Y is a (rescaled) Wiener process. In all this thesis, the standard Wiener process will be always denoted by the capital letter W .

We then define the Sine-Wiener noise as Xt= B sin r 2 τ Wt ! . (1.4)

In this case, there is no need of proving the boundedness of X, since it is straightforward from the definition. The positive constant B represent the width of the noise.

As far as its autocorrelation function is concerned (Appendix A.5.1), an explicit form is available (see [6]): RXX(s, t) = B2 2 1 − exp −4s τ · exp −t − s τ . (1.5)

We can then safely conclude that the parameter τ appearing in the definition of the Sine-Wiener noise is the characteristic autocorrelation time of the process. The stationary density

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1.2. Different methods for generating Bounded Noises of the Sine-Wiener noise is instead the following:

pst

SW(x) =

1

π√B2_{− x}2 . (1.6)

It can be obtained in many ways, but a rather interesting and - as far as we know - new way will be showed in the next chapter. In figure 1.2 we plot its shape for B = 1. The plot reveals that the stationary density of the Sine-Wiener noise diverges near the boundaries ±B. It is a bimodal symmetric density, which has 0 as its less likely value.

−0.5 0 0.5 1 0 0.5 1 1.5 2

Figure 1.2: Stationary probability density (1.6) of the Sine-Wiener noise, for B = 1. It diverges at the boundaries ±1.

1.2.2 Cai-Lin family

Another less straightforward way of generating bounded noises is by means of Stochastic Differential Equations, which are usually written as

dXt= µ(Xt)dt + σ(Xt)dWt. (1.7)

For any needed clarification, see Appendix A.4. In this section, we concentrate on the case where the drift µ(x) is linear, null in 0, with negative sign if x > 0 and positive sign if x < 0. Following [8, 9], we assume that

µ(x) = −αx , α > 0 .

Then, if we want the solution of SDE to be bounded in the interval I = [−B, B], it is necessary that the diffusion σ vanishes at the boundaries of I. The choice adopted in [8, 9, 10] is the following:

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Chapter 1. Introduction to Bounded Noises for Modelling Biological Processes Such choices, together with the introduction of suitable parameters, lead to the following SDE: dXt = − 1 θ Xtdt + s B2_{− X} t2 θ(δ + 1) dWt. (1.9)

The family of noises defined through equation (1.9) will be called Cai-Lin noises family. The real parameters B, θ, and δ are subject to the conditions

B > 0 , θ > 0 , δ > −1 . (1.10)

Of course, B represents the width of the unique strong solution of (1.9), provided of course that the initial condition of the SDE lives in I. Proof of this fact will be supplied in Chapter 3. Parameters δ and θ are instead related to the stationary density and the autocorrelation time of the noise, respectively. The stationary probability density of Cai-Lin noise (which will be derived in Chapter 4) reads as follows:

pst

CL(x) = Z

−1

(B2− x2₎δ_, _(1.11)

where Z in the normalization constant, Z = B2δ+1 √ π Γ(1 + δ) Γ(1.5 + δ) . (1.12) −0.5 0 0.5 1 0 0.5 1 1.5 _δ=5 δ=1 δ=0 δ=−0.5

Figure 1.3: Stationary probability density (1.11) of the Cai-Lin noise, for B = 1 and different values of δ.

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1.2. Different methods for generating Bounded Noises

As formula (1.11) suggests, δ is the only parameter which actually affects the stationary probability density of the noise. Positive values of δ yield a symmetric unimodal density with its peak in zero, while negative values yield a bimodal density diverging at the boundaries ± B. In figure 1.3 the phenomenological bifurcation induced by parameter δ is shown.

To conclude with the Cai-Lin noise, it only remains to provide a formula of its char-acteristic autocorrelation time τ . The precise form of the autocorrelation function of the noise is probably not analytical. However, as we shall see in Chapter 4, the characteristic autocorrelation time of the noise can be found. It is a function of the only parameter θ. More precisely, the following simple relation holds:

τ_CL = θ . (1.13)

1.2.3 Tsallis-Borland family

Again by resorting to SDEs, we can generate another family of bounded noises. In the case here considered, the diffusion of the process never vanishes. On the contrary, it is equal to a positive constant. Thus, the boundedness of the solution of the SDE can only be consequence of a sort of infinite deterministic force at the boundaries of the requested state space, which tends to push the solution away from the boundaries, when they are approached.

The example we consider in this thesis is the following [7, 11]:

dXt= − 1 θ B2_X t B2_{− X} t2 dt + B r 1 − q θ dWt. (1.14)

We will refer to it as to the Tsallis-Borland noise (family).

The quantity B must be positive and represents the width of the noise. Parameters θ and q must instead satisfy at least the following conditions, in order for the coefficients of the SDE to be well defined:

θ > 0 , q < 1 . (1.15)

As already happened with the Cai-Lin noise, parameter θ is only related to the charac-teristic autocorrelation time τ_{T B} of the process. An explicit formula of τ as a function of θ is however not available. Nevertheless, the autocorrelation time τ can be well approximated over the whole range of q values as follows [11]:

τ_{T B} ' 2

5 − 3q θ . (1.16)

In our simulations, we will fix the value of τ according to what needed, and we will then use relationship (1.16) to automatically choose the matching value of θ.

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Chapter 1. Introduction to Bounded Noises for Modelling Biological Processes The quantity q is instead linked to the stationary density of the process. Although condition (1.15) is of course sufficient to obtain well-defined coefficients of the SDE, questions of existence, uniqueness and boundedness are strongly related to the particular value of q and not as trivial as they might appear at first sight. We will go into details on this issue in Chapter 2. At present, let us only write that, for the values of q for which solutions of the SDE (1.14) are well defined, the stationary density of the process is given by

pst

T B(x) = Z

−1

(B2 − x2₎1−q1 _. _(1.17)

The previous expression is similar to the stationary density (1.11) of the Cai-Lin noise. The main difference with respect to the latter is that here the exponent (1 − q)−1 can only take positive values, due to the necessary condition q < 1. Therefore, we only get unimodal densities with their mode in 0. In figure 1.4, we plot expression (1.17) for different values of q < 1. However, let us remark that in the following chapters we shall see that not all values of q are appropriate for the generation of suitable bounded noises.

−0.5 0 0.5 1 0 0.5 1 1.5 q=0.8 q=0.5 q=0 q=−5

Figure 1.4: Stationary probability density (1.17) of the Tsallis-Borland noise, for B = 1 and different values of q < 1.

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Chapter 2 Unboundedness and non-Uniqueness

of Tsallis Process for q < 0

The bounded noises introduced at the end of Chapter 1 will be used later in this work both for validating new algorithms and, above all, to investigate the consequences of their use in pharmacokinetic models. Since, however, the coefficients of the SDEs in study do not satisfy Lipschitz conditions, it becomes important to carry out an investigation of basic questions such as their existence and uniqueness, and the boundedness of their solutions. In this chapter, we concentrate on the Tsallis-Borland noise, finally discovering that the value of the parameter q can have rather unexpected consequences on the behavior of the trajectories of the noise.

The analysis of such questions for the Tsallis-Borland process requires the introduction of specific stochastic tools presented in the next section. These tools are usually exploited in order to investigate whether the trajectories of a stochastic process X can attain specific values or not. In our analysis, the points that we shall consider are the boundaries ±B. Understanding whether the boundaries of a process are actually reached or not by the process itself is not only a point of interest on its own, but also turns out to be useful in order to establish the global existence and uniqueness of some SDEs. This is why we begin with the introduction of the aforementioned tools, and only later move to the analysis of strong existence, uniqueness and boundedness of the Tsallis-Borland process.

2.1 Scale function and speed measure

We limit ourselves here to display the results of interest for us, by addressing the specific literature for proofs of the theorems. See, for example, Karatzas-Shreve [12, Chap. 5.5] and

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Chapter 2. Unboundedness and non-Uniqueness of Tsallis Process for q < 0 Revuz-Yor [13, Chap. 7.3] for references.

Definition 2.1.1. Let X be a stochastic process whose state space is an interval I ⊂ R which can be open, semi-open or closed. Suppose the process starts with a deterministic condition: X0 ≡ x0 ∈

◦

I. For x ∈ I, we call

Tx= inf{ t ≥ 0 | Xt= x }

the first hitting time of x.

In general, we shall write l and r for the left and the right endpoints of I, so thatI = (l, r).◦ Definition 2.1.2. A function s : I → R is called a scale function for the process X (with X0 ≡ x0) if it is strictly increasing and, for any a < x0 < b ∈ I, it holds

P Ta < Tb = s(b) − s(x0) s(b) − s(a) . Equivalently, if P Tb < Ta = s(x0) − s(a) s(b) − s(a) .

Remark. If s(x) is a scale function, also ˜s(x) = αs(x) + β is a scale function for any α > 0 and β ∈ R, as a quick check reveals. Thus, for example, we can arbitrarily set s(x0) = 0.

Monotony then implies s(a) < 0, s(b) > 0.

The size of the scale function in a point x represents a sort of quantification of the “accessibility” of that point, when the process starts from x0. The bigger |s(x)| is with

respect to |s(y)|, the less probable x is reached before y.

Definition 2.1.3. A process X is said to be on natural scale if its scale function is the identity. In this case, roughly speaking, we could say that the tendency of the process to move left is the same than to move right.

An immediate consequence of the definition of scale function is the following property. Proposition 2.1.4. Let X be a stochastic process and s(x) its scale function. Then the process eX = s(X) is on natural scale.

Proof. Let ˜x0 be the starting point of eX0, and ˜a, ˜b two points in the state space s(I) of

e

X. Thanks to the strictly increasing property of s, there exist unique a, b, x0 ∈ I such that

˜

a = s(a), ˜b = s(b), and ˜x0 = s(x0). Then, again for monotony

P TXe ˜ a < T e X ˜ b = PT X a < T X b = s(b) − s(x0) s(b) − s(a) = ˜_{b − ˜}_x 0 ˜_{b − ˜}_a .

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2.1. Scale function and speed measure

We now give another definition useful in the following, and provide a first result concerning the computation of the scale function.

Definition 2.1.5. The speed measure associated with the process X is the measure on I

m(dx) = 2dx

s0_{(x) σ}2_(x), x ∈ I .

Suppose now a stochastic differential equation is given. We require the following two hypotheses on its coefficients:

σ2(x) > 0 for x ∈I◦ (2.1) ∀ x ∈I , ∃ ε > 0 :◦ Z x+ε x−ε 1 + |µ(y)| σ2_(y) dy < ∞ . (2.2)

Of course, if the coefficients are continuous functions in I, then the local integrability condition (2.2) is trivially fulfilled, and the only important condition becomes (2.1). Let us observe that both conditions are anyway fulfilled in the case of the Tsallis-Borland and the Cai-Lin noises.

Theorem 2.1.6. Let dXt = µ(Xt)dt + σ(Xt)dWt be a SDE where µ and σ satisfy (2.1,

2.2). Suppose a weak solution of the SDE exists. Then, it holds s(x) = Z x c exp − Z y c 2 µ(z) σ2_(z) dz dy , c ∈ I .◦ (2.3)

Before stating the main theorem of this section, which will illustrate the great usefulness of the scale function in order to establish whether the bounds of the state space of the process are reached or not, let us make some remarks which indirectly explain the origin of formula (2.3).

If X satisfies the SDE in theorem 2.1.6, then the process Y = s(X) satisfies the following SDE (see Itˆo’s formula, Thm A.3.2):

d Yt= µ(Xt)s0(Xt) + 1 2σ 2_(X t)s00(Xt) dt + σ2(Xt)s0(Xt) dWt.

If it is imposed that the drift of the last SDE is null, then the function s(x) provided in (2.3) is obtained. Such a strategy is carried out so that the new process Y is a (local) martingale. In this case, in fact, the theory provides a lot of useful instruments which concern stopping times, stopping theorems and local times. We will not introduce such

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Chapter 2. Unboundedness and non-Uniqueness of Tsallis Process for q < 0

tools here, but only mention the fact that these, indeed, are the instruments required for the investigation of the finiteness of the random times Tl and Tr (which, more precisely, are

stopping times).

Furthermore, let us observe another thing. Formula (2.3) is somewhat similar to formula (A.40), which supplies the expression of the stationary density of a SDE. It can be easily checked that the relation s0 = (σ2_pst₎−1 _{holds. The stationary probability density can be}

then deduced from the expression of the scale function:

pst(x) = 1

σ2_{(x) s}0_(x).

Notice that, except for a multiplicative factor, the last expression coincides with the density of the speed measure, w.r.t. Lebesgue measure. Indeed, the connection between the stationary behavior of a SDE and the speed measure has been deepened in the specific stochastic literature. It is not purpose of this thesis to carry out an in-depth analysis of all these topics. However, presenting them and emphasizing the connections with well-know quanti-ties appeared helpful for us, in order to give at least a small intuition of their meaning and their usefulness.

Let us now come back to our purposes, and state the main result of this section, which we take from [12, Chap. 5, Prop. 5.22]. Even if l or r are not in I, we shall anyway use the following notation:

s(l) = lim

x→l+s(x) , s(r) = lim_x→r−s(x) .

Theorem 2.1.7. Let conditions (2.1, 2.2) hold, and let X be a weak solution of the corre-sponding SDE, with deterministic initial condition x0∈

◦

I. Furthermore, call T the random time

T := infnt ≥ 0 X_t∈/ ◦

I = (l, r)o= Tl∧ Tr.

We can then distinguish four cases: (i) if both |s(l)| , |s(r)| = ∞, P T = ∞ = 1 = P inf t≥0Xt = l = P sup_t≥0 Xt = r ; (ii) if both |s(l)| , |s(r)| < ∞, P lim t→TXt= l = s(r) − s(x0) s(r) − s(l) = 1 − P limt→TXt= r ;

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2.1. Scale function and speed measure (iii) if |s(l)| < ∞, |s(r)| = ∞, P lim t→TXt= l = 1 = P sup_t≥0 Xt< r ; (iv) if |s(l)| = ∞, |s(r)| < ∞, P lim t→TXt= r = 1 = P inft≥0Xt> l .

Although it might appear complex, the meaning of the previous theorem is rather simple. In case (i), the process never hits the boundaries l and r, since T = ∞. In case (ii), the process hits these boundaries with a certain probability if T is finite, otherwise approaches them in infinite time with the same probability. In case (iii) or (iv), the process can approach only one between l and r: again, if T if finite it means that the process hits the boundary, otherwise it only tends to the latter in infinite time.

Except that in case (i), no claim about the finiteness of T can be made through the previous result. However, as we have just highlighted, the finiteness of T is of capital importance in order to understand, in the last three cases, if the process actually hits one of the endpoints of I, or only tends to them in infinite time. In this regard, another tool is required, which exploits the speed measure (Definition 2.1.5).

Definition 2.1.8. By denoting with s(x) the scale function and m(dx) the speed measure of a process X, we set

v(x) := Z x

c

s(x) − s(y) m(dy) , x ∈ I . (2.4)

Notice that v(x) is always positive, and its finiteness does not depend on the choice of c. If e is one of the endpoints of I, it can be shown that |s(e)| = ∞ ⇒ v(e) = ∞, while the converse implication does not hold. Thus, for example, in case (i) of theorem 2.1.7, where T is a.s. infinite, we must have v(l) = v(r) = ∞. Similarly, the almost surely finiteness of T is strongly related to the finiteness of v. The following characterization can be found in [12, Prop. 5.32, Chap. 5].

Proposition 2.1.9. Under the usual notation and assumptions of foregoing results, the almost surely finiteness of T is equivalent to one of the following mutually exclusive condi-tions:

(a) both v(l) < ∞ and v(r) < ∞,

(b) v(l) < ∞ and v(r) = ∞, but also s(r) = ∞, (c) v(r) < ∞ and v(l) = ∞, but also s(l) = −∞.

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2.2 Application to the Tsallis-Borland case

We can now exploit the just introduced tools to deal with the problem of strong existence and uniqueness of the Tsallis-Borland noise, which will naturally pose the problem of its boundedness. Since a relevant difference is present between the two cases q < 0 and q ≥ 0, each case is treated separately. Let us preliminarily observe that we can assume for our purposes that the constant B appearing in the Tsallis-Borland equation

dXt= − 1 θ B2_X t B2_{− X} t2 dt + B r 1 − q θ dWt

equals one. In fact, if the change of variable Y = X/B is considered, the new process Y is discovered to satisfy the original equation with B = 1. Thus, B only represents a scale factor for the noise, and can then be supposed to be equal to one. This way, the expressions of the drift and the diffusion of Tsallis-Borland equation are

µ(x) = −1 θ x 1 − x2 , (2.5) σ(x) ≡ σ = r 1 − q θ , (2.6)

for x ∈ I = (−1, 1). Of course, each time we speak about positive values of q, it is implicitly understood that the condition

q < 1

holds. Otherwise, the the diffusion itself of the SDE would not be well-defined.

2.2.1 The case where q ≥ 0

We begin with the case q ≥ 0, although some general remarks are also valid for the case q < 0. While the diffusion (2.6) is extremely regular, the same cannot be asserted for the drift, which presents two asymptotes in ±1. Such a characteristic does not enable us to exploit standard theorems which require Lipschitz conditions on the coefficients. However, the fact that the drift is locally Lipschitz in each point of the interior of I allows us to make the following considerations.

For ε > 0, define µε as the continuous extension of µ outside the interval (−1 + ε, 1 − ε):

µε(x) =      µ(x) if |x| ≤ 1 − ε µ(1 − ε) if x ≥ 1 − ε µ(−1 + ε) if x ≤ −1 + ε (2.7)

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2.2. Application to the Tsallis-Borland case

Such a µε is of course Lipschitz on R. Thus, the stochastic differential equation

dXt= µε(Xt)dt + σdWt (2.8)

has a unique global strong solution, for any starting point x0 ∈ R.

Now take x0 ∈

◦

I and ε sufficiently small such that |x0| < 1 − ε. We denote by X(ε) the

solution of (2.8) corresponding to this ε and the initial condition x0. Further, we set

τε= inf t ≥ 0

|Xt| ≥ 1 − ε .

Of course, till the random time τε, the process X(ε) is also solution of the original

Tsallis-Borland equation (TBE) with drift µ. This not only shows local existence for this equation till the boundary of I is not reached, but also shows uniqueness of the solution, which is intrinsically a local problem, till the same time. However, if we show that with probability one a solution of the TBE does never reach the endpoints of I, we also obtain a global solution of the same equation. At this point, we can take advantage of all the instruments introduced in Section 2.1.

Lemma 2.2.1. A version of the scale function associated with the Tsallis-Borland equation is given by s(x) = Z x 0 1 − z2− 1 1−q _dz _(2.9)

Proof. We must compute expression (2.3), when µ(z) = −1 θ z 1 − z2 and σ 2 (z) ≡ 1 − q θ . Let

us fix c = 0 and first calculate the internal integral: Z y 0 −2µ(z) σ2_(z) dz = − 1 1 − q Z y 0 −2z 1 − z2 dz = − 1 1 − q ln 1 − y 2_. Then s(x) (2.3)= Z x 0 exp Z y 0 −2µ(z) σ2_(z) dz dy = Z x 0 exp − 1 1 − q ln 1 − y 2 dy = Z x 0 1 − y2− 1 1−q _{dy .}

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Theorem 2.2.2. Suppose 0 ≤ q < 1 holds. Then, for any random variable X0 with state

space ◦

I = (−1, 1), there exists a unique, strong solution of the Tsallis-Borland equation dXt= − 1 θ Xt 1 − Xt2 dt + r 1 − q θ dWt (2.10)

with initial condition X0. Moreover, such an equation never leaves

◦ I. Proof. For the sake of convenience, let us set α = 1

1−q. So is α ≥ 1 under our assumption

that 0 ≤ q < 1. In the notation of theorem 2.1.7, we denote by T the first exit time from ◦ I. We have s(1) = Z 1 0 1 (1 − z2₎α dz ≥ C Z 1 0 1 (1 − z)α = ∞ , where C = min z∈[0,1](1 + z) −α = 2−α. Analogously, we have s(−1) = −∞.

Thanks to theorem 2.1.7 (i), we conclude that T is almost surely infinite, namely that any stochastic process X solution of (2.10) with initial condition X0 ∈

◦

I never reaches the boundaries ±1 of I. At this point, we can exploit that the concerned equation has a local unique strong solution starting from any point in I. Since, however, any solution never◦ leaves I, we also obtain the existence of a unique global strong solution of the SDE.◦

2.2.2 The more problematic case q < 0

It now remains to study the case where q < 0. In this case, it holds

α = 1 1 − q < 1 . Thus, it is obtained s(1) = Z 1 0 1 (1 − z2₎α dz < ∞

and s(−1) = −s(1) > −∞. Under such conditions, we know that for a suitable p ∈ (0, 1) depending on the starting condition x0, it holds

P lim

t→TXt= −1 = p , P limt→TXt = 1 = 1 − p .

On the set Ω0 = {T < ∞} we can conclude that X hits the boundaries ±1 with positive

probability, since the process is continuous and the condition lim

t→TXt= ±1 then simply reads

XT = ±1. The point is to quantify P[T < ∞].

A rather intuitive reasoning can probably be made about the complement of Ω0, namely

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2.2. Application to the Tsallis-Borland case

of the equation would be, for sufficiently large t, in an arbitrarily small neighbourhood of one of the two boundaries, say +1. The stationary behavior of the process would then show positive masses on +1, which is not our case. This reasoning thus seems to suggest that, if q < 0, the event {T = ∞} can only happen with null probability.

For a mathematical treatment of this last fact, we should resort to the results displayed in Section 2.1 about the function v defined through speed measure and scale function (Propo-sition 2.1.9).

Theorem 2.2.3. Suppose q < 0 holds. Then, any solution of the Tsallis-Borland equation (2.10) starting in I will almost surely reach one of the endpoints ±1 in finite time.◦

Proof. We want of course to exploit the aforementioned proposition. Let us then calculate expression (2.4) for our SDE. A quick check reveals that the speed measure associated to our SDE is m(dy) = 2dy σ2_s0_(y) = C 1 − y 2α dy , 0 < α < 1 , α = 1 1 − q . Then, by neglecting the constants, we have

v(1) = Z 1 0 s(1) − s(y) m(dy) = Z 1 0 Z 1 y 1 − z2−α dz 1 − y2α dy (1+z)−α≤1 ≤ Z 1 0 Z 1 y (1 − z)−αdz 1 − y2αdy = Z 1 0 (1 − z)1−α 1 − α y 1 1 − y2α dy (∗) = 1 1 − α Z 1 0 (1 − y)1−α 1 − y2α dy = 1 1 − α Z 1 0 (1 − y) 1 + yα dy ≤ 2 α 1 − α < ∞

It is clear the the hypothesis 0 < α < 1 was crucial in the passage labelled with (∗). Because of symmetry, we also have v(−1) = v(1) < ∞. Thanks to point (a) of Proposition 2.1.9, the thesis follows. Moreover, we also have in this special case that the mean of the first hitting time of one of the two boundaries is itself finite.

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The previous result involves not trivial consequences. Till a solution of the SDE remains in (−1, 1), we know that strong uniqueness and strong local existence hold for the equation. If, instead, the boundary is reached, the drift itself (2.5) of the equation loses its meaning. This would be in general not an insurmountable problem: other processes present singular points which can be reached under particular conditions, being then the process instanta-neously reflected. In our case, however, there appears to be no reason for which the process, once reached the boundary, should be reflected in the interior of I instead of outside I. If the reflection did not take place, the boundedness and the uniqueness itself of the solution would be completely lost. In other words, even though we have not provided a mathematical proof of this fact, it is clear that if q < 0 the Tsallis-Borland SDE loses in general its boundedness and, as a consequence, its uniqueness.

2.3 Physical interpretation of the apparent paradox

All these issues could appear even stranger but, at the same time, quite captivating if the dynamics of the process is given a physical meaning.

Let us consider the linear motion, along the x axis, of a material point P of mass m, on which the following forces act:

1. a viscous force Fv(t) = −γ x0(t),

2. a stochastic force Fs(t) = γ

√

2β ξ(t), where ξ is a white noise,

3. a conservative force, sum of two repulsive forces centered in l = −1 and r = 1, Fc(x) = Fl(x) + Fr(x) ,

such that its potential U = Ul + Ur is infinite at the centers of repulsion. In other

words, we require that lim

x→−1±Fl(x) = ±∞ , _x→+1lim±Fr(x) = ±∞ .

For the sake of convenience, we shall write Fi(x) = γ ϕi(x), i = l, r. The Newton’s equation

for the motion of the point P thus reads as follows:

mx00 = −γ x0 + Fc(x) + Fs(t) . (2.11)

The connection between (2.11) and the TBE needs an intermediate step, which consists in assuming that the damping coefficient γ is so large and the mass m is so small that

m

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2.3. Physical interpretation of the apparent paradox

In this case, one can adopt the first type Kramer’s approximation by neglecting the contribute of the acceleration, which yields the following approximate equation:

x0 = ϕl(x) + ϕr(x) +

p

2β ξ(t). (2.13)

The previous notation can simply be considered a physical notation for SDEs. Indeed, since the coefficient of the white noise ξ(t) does not depend on x, both by means of Itˆo’s calculus or of Stratonovich one the following stochastic differential equation is obtained:

dxt= ϕl(xt) + ϕr(xt) dt +

p

2β dWt. (2.14)

If the conservative forces ϕl(x) and ϕr(x) are chosen to be

ϕl(x) =

1

x + 1, ϕr(x) = 1

x − 1, (2.15)

then the obtained SDE is the Tsallis-Borland equation, where θ = 1/2 and β = 1 − q. The potentials Uland Ur of the conservative forces are infinite at their respective centers

of repulsion. Indeed, it holds

Ul(x) = − ln |x + 1| , Ur(x) = − ln |x − 1| ,

which yields

U (x) = Ul(x) + Ur(x) = − ln

1 − x2 . (2.16)

Thus, if started in (−1, 1), the point P is inside a potential well of infinite height. If the point could reach one of the endpoints ±1, it would mean that the only stochastic force Fs could enable the point P to overcome an infinite potential well (of course, we are only

referring at the case where β > 1, i.e. q < 0). Such a consequence is nonphysical. Probably, the only explanation to such an apparent paradox is that equations (2.13) and (2.14) are only an approximation of the true Newton equation. Assumption (2.12) has indeed been transmuted into m/γ = 0.

This approximation is in general very good, but appears to fail in the particular case where the conservative forces are chosen as in (2.15). In the sense that we will shortly illustrate, this case seems to represent a sort of bifurcation between two different physical behaviors. Indeed, let us consider the following conservative forces for α > 0:

ϕα_l(x) = sgn(x + 1) |x + 1|α , ϕ α r(x) = sgn(x − 1) |x − 1|α (2.17)

which, for α = 1, reduce to the conservative forces previously employed. The potential U of ϕl+ ϕr reads as follows: Uα(x) = − Z ϕα_l + ϕα_r =      −|x + 1| 1−α + |x − 1|1−α 1 − α if α 6= 1 − ln |1 − x2_| _{if α = 1} (2.18)

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It can be noticed that U (±1) is infinite if α > 1, while it is finite if α < 1. In the discriminating case α = 1, moreover, U (x) is a potential well of infinite height. In this sense, this case appears to be more similar to α > 1 than to α < 1. However, notice that this infinite is of logarithmic order, thus inferior than the polynomial order of the case α > 1.

Analogous studies to the one carried out in Subsections 2.2.1, 2.2.2 would show that the SDE corresponding to the conservative forces ϕα

i is bounded for α > 1, unbounded for

α < 1. This is rather intuitive, since it means that the motion of the point P is bounded if the potential well is infinite, and unbounded otherwise. If, instead, α is exactly equal to one, the physical behavior of P is not uniquely determined: if β < 1, then the motion is actually bounded; if instead β > 1, then the stochastic force√2β ξ(t) can exceed the repulsive force, thus allowing the point P to overcome the infinite potential well.

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Chapter 3 Cai-Lin Strong Existence,

Uniqueness, and Boundedness

In Chapter 2 we have dealt with the main mathematical questions concerning the Tsallis-Borland SDE. In this brief chapter, we investigate the same problems for the Cai-Lin SDE

dXt = − 1 θ Xtdt + s 1 − Xt2 θ(δ + 1) dWt. (3.1)

Of course, the choice B = 1 has been made, since the constant B only represents a scale factor for the Cai-Lin SDE (as it was already observed for the Tsallis-Borland SDE).

Some technical results must be exploited, but we shall discover in the end that the Cai-Lin equation does not present irregular and unexpected behavior as the Tsallis-Borland equation does. Although the sign of the parameter δ affects the behavior of the Cai-Lin trajectories near the boundaries ± 1, it has no consequences on the uniqueness of the SDE, and the boundedness of its solution.

3.1 Pathwise uniqueness

Throughout the course of this section, we need particular results on existence and uniqueness of SDEs with H¨older coefficients. Similar results are for example provided in [13, Chap.2 Par.3].

The diffusion of (3.1) is of course meaningful only in the interval I = [−1, 1]. For a formal treatment of the equation, it is nonetheless convenient to extend this coefficient to the real domain, in a suitable way. Once we have seen that the new defined equation has solutions which never leave the interval I, it will become clear that the Cai-Lin SDE can be written

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Chapter 3. Cai-Lin Strong Existence, Uniqueness, and Boundedness

as in (3.1), and that the particular extension of σ(x) has actually played no role. For similar reasons, it can be convenient to modify in a continuous way the drift of the equation outside the interval I, in order to obtain a bounded function. So, the equation mainly studied in this section is the following:

dXt= 1 θµ(Xt) dt + β σ(Xt) dWt, (3.2) where µ(x) =      1 for x ≤ −1 −x for |x| ≤ 1 −1 for x ≥ 1 (3.3) and σ(x) = ( √ 1 − x2 _{for |x| ≤ 1} 0 for |x| ≥ 1 . (3.4)

The constant β is instead β = [θ (δ + 1)]−12_.

We begin with a lemma concerning the regularity of the function p|x| , which our diffusion is a modification of.

Lemma 3.1.1. The following inequality holds: p|x| − p|y| 2 ≤ |x − y| _{∀ x, y ∈ R .} (3.5)

Proof. First of all, we can assume both x and y to be non-negative. In fact, if both variables were negative, the left- and the right-hand side of (3.5) would remain the same by replacing x with −x and y with −y. If, instead, only one of the two variables were negative, the same change would increase the value of the RHS and would left unchanged the one of the LHS of (3.5). Thus, we can safely assume x, y ≥ 0.

Without loss of generality, let us further assume that x ≥ y holds. Then: √ x −√y 2 = x + y − 2√x√y x≥y ≤ x + y − 2y = x − y = |x − y | .

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3.1. Pathwise uniqueness

Lemma 3.1.2. Let σ be the function defined in (3.4). Then, there exists a constant C > 0 such that the following inequality holds:

|σ(x) − σ(y)|2 ≤ C |x − y| _{∀ x, y ∈ R .} (3.6)

Proof. Suppose first x, y ∈ I = [−1, 1]. Then, thanks to Lemma 3.1.1 |σ(x) − σ(y)|2 (3.4)= √ 1 − x2 ₋p_{1 − y}2 2 3.1.1 ≤ 1 − x2 − (1 − y2) = | x − y | | x + y | ≤ C | x − y |, (3.7) where C = max|x + y|x, y ∈ I = 2 .

If, now, both x and y are not in I, the claim is trivial, since the LHS of (3.6) in zero. Finally, if for example x ∈ I and y /∈ I, by assuming y > 1 we get

|σ(x) − σ(y)|2 = |σ(x) − σ(1)|2

(3.7)

≤ C | x − 1 | ≤ C | x − y | . (3.8)

If, on the contrary, y < −1 held, it would be enough to consider σ(−1) instead of σ(1) in the first row of (3.8).

The following result is due to Revuz and Yor.

Lemma 3.1.3 (Thm 3.5, [13]). Let µ and σ be the drift and diffusion coefficients of a SDE. Suppose that µ is Lipschitz continuous and that σ satisfies

|σ(x) − σ(y)|2 ≤ ρ ( |x − y| ) , (3.9)

where ρ : (0, ∞) → (0, ∞) is such that Z ε

0

1

ρ(z)dz = ∞ ∀ ε > 0 . Then, pathwise uniqueness holds for the SDE.

The previous lemma immediately supplies a result of uniqueness for our equation. Theorem 3.1.4. Pathwise uniqueness holds for the auxiliary equation (3.2).

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Proof. Of course, the drift µ(x) in (3.3) is Lipschitz (even the “original” one of equation (3.1) is Lipschitz). As far as σ is concerned, we can take the function ρ of Lemma 3.1.3 to be a multiple of the identity:

ρ(z) = C z .

Then, 1/ρ in not integrable in a neighbourhood of zero, and condition (3.9) of Lemma 3.1.3 is of course fulfilled thanks to Lemma 3.1.2. The thesis follows from the aforementioned Lemma 3.1.3.

3.2 Weak and strong existence

Once strong uniqueness has been ascertained, it is sufficient to prove the existence of a weak solution of the SDE in order to obtain the strong existence of the same SDE, thanks to the Yamada-Watanabe theorem (see Theorem A.4.5 in Appendix). For the proof of weak existence of SDE (3.2), the choice of making the drift coefficient bounded, as in (3.3), becomes extremely useful. In fact, the following result proved in [12, p.323] holds.

Theorem 3.2.1. Let _{µ, σ : R → R be bounded and continuous functions, and P}0 a

probability measure on B(R) such that Z

R

|x|2m

P0(dx) < ∞ (3.10)

for some m > 1. Then, there exists a weak solution of the SDE dXt= µ(Xt) dt + σ(Xt)dWt

corresponding to every initial X0 with distribution P0.

Theorem 3.2.2. There exists a unique strong solution of the SDE dXt=

1

θµ(Xt) dt + β σ(Xt) dWt,

with µ and σ as in (3.3, 3.4), for any given random variable X0 with state space I = [−1, 1].

Proof. If X0 takes only values in I, by denoting with P0 its probability law, we have that

P0(Ic) = 0 .

Condition (3.10) is then trivially fulfilled: Z R |x|2m_P0(dx) ≤ Z 1 −1 12m_P0(dx) = 1 .

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3.3. Boundedness of the original Cai-Lin equation

Since both µ and σ are bounded, we know from theorem 3.2.1 that a weak solution exists. Furthermore, pathwise uniqueness holds thanks to theorem 3.1.4. It only remains to apply Yamada-Watanabe theorem to prove the thesis.

In order to prove strong existence and uniqueness of the true Cai-Lin equation (3.1), it remains to show that the unique solution of the slightly different equation (3.2) never leaves the interval I.

As we shall see in Section 3.4, the boundedness of the Cai-Lin noise is, for a large range of δ (i.e. δ ≥ 0), a direct consequence of the fact that its trajectories do not attain the boundaries ± 1. However, for a small range of δ values, the boundaries can be reached: in this case proving directly that the solution is reflected in the interior of I is not straightforward. Therefore, for the sake of clarity, we first prove here the boundedness of the Cai-Lin equation for all δ > −1, through the so-called Comparison Theorem proposed in [13]. Then, in Section 3.4, we analyze the behavior of Cai-Lin equation near the boundaries of its state space.

3.3 Boundedness of the original Cai-Lin equation

Theorem 3.3.1 (Comparison theorem, [13]). Let

dXt= b(i)(Xt) dt + σ(Xt) dWt for i = 1, 2 (3.11)

be two stochastic differential equations, whose coefficients satisfy: i) |σ(x) − σ(y)|2 ≤ ρ ( |x − y| ), where ρ is as in Lemma 3.1.3 ,

ii) at least one between b(1) _{and b}(2) _{satisfies a Lipschitz condition, and the inequality}

b(1)_{(x) ≤ b}(2)_{(x) holds everywhere.}

Further, let X₀(i) be two random variables, with X₀(1) ≤ X₀(2) _{P-a.s., and X}(i) _{the solutions}

of (3.11) with starting conditions X₀(i), for i = 1, 2. Then,

P Xt(1) ≤ X (2)

t for all t ≥ 0 = 1 .

The foregoing theorem represents the last tool we need in order to prove the existence of a unique, strong, bounded solution of the original Cai-Lin SDE. In the following summarizing theorem, I represents the usual interval [−1, 1].

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Theorem 3.3.2. Let X0 be a random variable with P[X0 ∈ I ] = 1. Then, there exist a

unique strong solution of the Cai-Lin SDE

dXt = − 1 θ Xtdt + s 1 − Xt2 θ(δ + 1) dWt,

with initial condition X0, and such a solution does a.s. never leave the interval I:

P[Xt∈ I for all t ≥ 0] = 1 .

Proof. Let us consider the usual auxiliary equation (3.2), which we rewrite for convenience: dXt=

1

θµ(Xt) dt + β σ(Xt) dWt.

Such an equation has a unique strong solution starting from X0 (Thm. 3.2.2). We shall

denote this solution by X(1), the starting r.v. X0 in the statement of the theorem by X (1) 0 ,

and the coefficient µ(x) by b(1)(x). Now consider b(2)_{(x) as follows}

b(2)(x) =      1 for x ≤ −1 −x for −1 ≤ x ≤ 0 0 for x ≥ 0

and X₀(2) the degenerate random variable X₀(2) ≡ 1. Thanks to the previous theorems, there exists a unique strong solution of the following system:

   dYt = 1 θ b (2) (Yt) dt + β σ(Yt) dWt Y0 = X (2) 0 .

The process X_t(2) ≡ 1 trivially fulfills the previous system, since b(2)(X_t(2)) ≡ σ(X_t(2)) ≡ b(2)(1) = σ(1) = 0 , and dX_t(2) = d 1 = 0. Thus, X(2) _{is the unique solution of that system.}

Being X(1) the solution of (3.2) defined at the beginning, all the hypothesis of Theorem 3.3.1 hold. Thus, we get

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3.4. Behavior of the solution near the boundaries

By means of a completely symmetrical reasoning, it is shown that X_t(1) ≥ −1 holds with probability one as well. Thus, the unique strong solution of (3.2) fulfills

P Xt(1)∈ I ∀ t ≥ 0 = 1 .

Finally, since the dynamics in I of the original Cai-Lin equation (3.1) coincides with that of equation (3.2) in I, and since X_t(1) starts in I and never leaves this interval, it follows that Xt≡ X

(1)

t is the unique strong solution of the Cai-Lin equation as well, and that it is

confined in I.

3.4 Behavior of the solution near the boundaries

We end the current chapter by briefly applying the results exploiting scale function and speed measure (see Section 2.1) to the Cai-Lin SDE

dXt= − 1 θXtdt + s 1 − Xt2 θ(δ + 1) dWt.

The goal is to establish whether or when the endpoints of the state space I = [−1, 1] are reached by a solution of the SDE. First, however, a comparison with the Tsallis-Borland case is crucial. In that case, the achievability of the boundaries had capital consequences on the uniqueness and boundedness of the solution. In the case of the Cai-Lin SDE, instead, the situation is different. Both the drift and the diffusion of the SDE are well defined in ±1. Indeed, the strong existence and uniqueness of this SDE has been proved at the beginning of this chapter, for any value of δ, θ and x0 ∈ (−1, 1). However, knowing whether ±1 are

attainable or not certainly helps the understanding of the noise dynamics, and can turn out to be useful in other contexts, see Section 4.4.

Lemma 3.4.1. The expression of the scale function of the Cai-Lin noise reads as follows: s(x) =

Z x

0

1

(1 − z2₎δ+1 dz . (3.12)

Proof. Same proof as for the Tsallis-Borland case (Lemma 2.2.1).

Through a comparison between the two scale functions (the one of Cai-Lin (3.12) and the one of Tsallis-Borland (2.9)) it is clear that they have the same expression, once the correspondence

1

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Chapter 3. Cai-Lin Strong Existence, Uniqueness, and Boundedness has been considered. In particular, we have

0 ≤ q < 1 ←→ δ ≥ 0 .

Thus, some results are straightforward from what has been proved for the Tsallis-Borland noise in the case 0 ≤ q < 1. For example:

Theorem 3.4.2. If δ ≥ 0 holds, any solution of the Cai-Lin SDE starting in I = (−1, 1)◦ does never reach the endpoints of I.

Proof. s(1) = ∞, s(−1) = −∞, plus the result of theorem 2.1.7 (i).

As far as the case −1 < δ < 0 is concerned, since in this case we have |s(1)|, |s(−1)| < ∞, we must again resort to the speed measure and the function v(x) introduced in Definition 2.1.8, as already done for the Tsallis-Borland SDE. A simple calculation reveals that the speed measure of Cai-Lin is

m(dy) = 2dy

σ2_{(y) s}0_(y) = C 1 − y 2δ

dy . (3.13)

Then, the following result is obtained.

Theorem 3.4.3. Suppose −1 < δ < 0 holds. Then, any solution of the Cai-Lin equation starting in I will almost surely reach one of the endpoints ±1 in finite time.◦

Proof. We simply check that v(1) and v(−1) are finite. The thesis then follows from Propo-sition 2.1.9. The calculations are very similar to the ones of Theorem 2.2.3.

v(1) = Z 1 0 s(1) − s(y) m(dy) = Z 1 0 Z 1 y 1 − z2−(δ+1) dz 1 − y2δ dy ≤ Z 1 0 (1 − z)−δ −δ y 1 1 − y2δdy (∗) = −1 δ Z 1 0 (1 − y)−δ 1 − y2δdy < ∞

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Chapter 4 A More Detailed Investigation on the

Noises

In this chapter we present some new results concerning the bounded noises of our interest. The chapter has two main objectives. The first one is to carry out a more detailed analysis of the properties of each noise. The second one is instead to compare these properties with one another, so as to detect common relationships or fundamental differences between them. In the final section, we even come to the definition of a bounded noise which has not been mentioned in previous chapters.

4.1 Time-dependent density of the Sine-Wiener noise

We first begin with the Sine-Wiener noise, which enjoys the particular property of having an explicit analytical expression. In Chapter 1, the form of its stationary density was supplied (1.6). Here, we show how this expression can be obtained as limit of the time-dependent density of the noise. Let us first recall the form of the noise:

Xt= B sin r 2 τ Wt ! . (4.1)

At each time t, the random variable Xtis then obtained by applying the bounded function

F (z) = B sin z to a Gaussian distributed random variable Z ∼ N (0, 2t/τ ). In order to find its density, we need to use the following lemma.

Lemma 4.1.1 (Transformation of densities). Let I, J ⊆ R be two real intervals, and let Z : Ω → I be a random variable on the probability space (Ω, F , P). Furthermore, let F : I → J

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Chapter 4. A More Detailed Investigation on the Noises

be a diffeomorphism of real intervals. If ρ : I → R≥0 is the density of Z, then the random variable

X = F (Z) : Ω → J has a density p : J → R≥0, whose expression is given by

p(x) = ρ (F

−1_(x))

|F0_(F−1_(x))| . (4.2)

Proof. We exploit the fact that a random variable X has density p(x) if and only if ∀ h ∈ C(J, R) , it holds E [h(X)] =

Z

R

h(x) p(x) dx . In our case, we get

E [ h(X) ] = E [(h ◦ F )(Z)] = Z R (h ◦ F )(z) ρ(z) dz (1) = Z R h(x) ρ(F−1(x)) 1 |F0_(F−1_(x))| dx = Z R h(x) p(x) dx

where p is given by (4.2) and the change of variable x = F (z) was applied in (1). Again through the characterization at the beginning, we conclude that p is the density of X.

We now want to exploit the previous lemma to find out the expression of the Sine-Wiener noise density. The first idea would be of course to apply the lemma to the random variable

Z ∼ N (0, 2t/τ )

and the function F (z) = B sin z. However, Z takes values on the whole real interval, and the sine function is not bijective if considered on it. Thus, to a certain extent, we have to limit the range of values of Z to the interval [−π/2, π/2], where the sine is bijective. The proof of the following theorem formalizes this idea.

Theorem 4.1.2. Let W be a Gaussian variable, W ∼ N (0, t), and X = B sin (W ). Then, the density of X is given by:

p(x) =X

n∈Z

ρ_W(nπ + (−1)narcsin (x/B)) √

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4.1. Time-dependent density of the Sine-Wiener noise where ρ_W(x) = √1 2πt exp − x 2 2t is the density of W . Thus, if we set ρ_Z(z) =X n∈Z

ρ_W (nπ + (−1)nz), the density p(x) can be rewritten as

p(x) = ρZ( arcsin(x/B) )_√

B2_{− x}2 . (4.4)

Proof. First of all, we consider a random variable Z : Ω →−π₂,π₂ such that F (Z(ω)) = F (W (ω)) ∀ ω ∈ Ω ,

where F is the function F (z) = B sin(z). In other words, for each possible value of w ∈ R, we must find the only z ∈ [−π

2, π

2] which satisfies sin(z) = sin(w). This unambiguously defines

the r.v. Z.

Since our aim is to find out the density of Z, we have to list, for each z ∈−π₂,π₂ , all the w ∈ R which z is a representative of. They are:

• w = z + 2kπ for all k ∈ Z, which are the ones in the first or fourth quarter;

• w = (π − z) + 2kπ = −z + (2k + 1)π, which are the ones in the second or third quarter. In a single expression, we can write all of them as w = (−1)n_{z + nπ, n ∈ Z. Thus, the} density of Z takes the following form:

ρ_Z(z) =X

n∈Z

ρ_W ((−1)nz + nπ) . (4.5)

By applying Lemma 4.1.1 to the function F (z) = B sin(z), we get p(x) = ρZ( arcsin(x/B) )

|B cos(arcsin(x/B))| =

ρ_Z( arcsin(x/B) ) √

B2_{− x}2 .

Although expression (4.3) supplies an explicit form of the density of X = B sin (W ), the properties of this density do not appear very clear, at least at first sight. However, that series converges to an important mathematical function whose properties are known. This function is one of the versions of the Theta function of Jacobi.

Definition 4.1.3. The function ϑ3(z, q) = 1 + 2 ∞ X n=1 qn2cos(2nz) =X n∈Z qn2cos(2nz)

is the third version of the Jacobi-Riemann Theta function. The second variable q must be real and such that |q | < 1, whereas z can take any complex value. For any fixed q, the series converges on the whole complex plane.

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Chapter 4. A More Detailed Investigation on the Noises

As it can be seen in [14, Chap.16], the density (7.10) of the r.v. Z of the foregoing theorem can be expressed in terms of the function ϑ3.

ρ_Z(z) = X k∈Z ρ_W (z + 2kπ) +X k∈Z ρ_W (−z + (2k + 1)π) = 1 2π ϑ3 z 2, e −t/2 + ϑ3 π − z 2 , e −t/2 . (4.6)

This result allows us to write the time-dependent density of the Sine-Wiener noise in an elegant form, which also immediately provides its stationary density. Although the following result is now a simple consequence of all the previous results, we present it as a theorem due to its importance.

Theorem 4.1.4. The time-dependent density of the Sine-Wiener noise (4.1) has the following form: p_SW(t, x) = ϑ3 z 2, e −t/τ_{+ ϑ} 3 π−z₂ , e−t/τ 2π√B2_{− x}2 , (4.7) where z = arcsin(x/B).

Proof. At time t, the Sine-Wiener noise is the random variable X = B sin (W ), where W ∼ N 0,2t_τ. We can then apply Theorem 4.1.2 to this specific r.v. W, and exploit the form of ρ_Z found in 4.6: this step simply requires the substitution t ↔ 2t/τ .

The previous expression of p_SW(t, x) is not only important on its own. In fact, it can be exploited to write down the stationary density of the noise and also appraise its characteristic autocorrelation time. Let us notice that if time t tends to plus infinity, both addends of the numerator of (4.7) tend to the constant one, since

ϑ3(z, 0) = 1 ∀ z ∈ C .

Thus, the stationary density of the Sine-Wiener noise is pst_SW(x) = lim t→∞pSW(t, x) = 1 + 1 2π√B2_{− x}2 = 1 π√B2 _{− x}2 . (4.8)

Moreover, since in every application it is important to know how fast the stationary density is attained, we can notice that expression (4.7) also qualitatively answers this ques-tion. In fact, the convergence takes place at the same speed at which the exponential function e−t/τ attains zero (the ϑ3 function tends to 1 linearly in q, when q tends to 0). This

is a proof of the fact that the Sine-Wiener noise has a characteristic autocorrelation time equal to τ : the full expression of its autocorrelation function has been however presented in Chapter 1, (1.5).

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4.2. Relation with the Cai-Lin noise

4.2 Relation with the Cai-Lin noise

Let us now recall the SDE defining the Cai-Lin process, with B = 1:

dXt = − 1 θ Xtdt + s 1 − Xt2 θ(δ + 1) dWt. (4.9)

The form of its stationary density was already previously supplied without proof: pst

CL(x) = Z

−1

(1 − x2)δ. (4.10)

However, this expression can be easily derived from formula (A.40) in Appendix, which supplies the stationary density of a process defined through a SDE. This formula only exploits the drift and the diffusion coefficients of the SDE. In our case, they are:

         µ(x) = −1 θ x , σ(x) = s 1 − x2 θ(δ + 1). (4.11) Then, we get pst_CL(x) A.40= 1 Z σ2_(x) exp Z x 2 µ(y) σ2_(y) dy = 1 Z (1 − x2₎ exp (δ + 1) Z x _−2y 1 − y2 dy = 1 Z (1 − x2₎ exp (δ + 1) ln 1 − x 2 = 1 Z (1 − x 2₎δ_. _(4.12)

Remark. Formula (A.40) surely holds for µ, σ ∈ C2(−1, 1). Even though our σ is not differentiable at the endpoints ±1, the concerned formula has been nevertheless applied in order to find an expression for a candidate stationary density. Now, a check would immediately reveal that this expression satisfies the stationary Fokker-Planck equation of the Cai-Lin SDE.

Bounded noises: new theoretical developments and applications in Pharmacokinetics

University of Pisa

Department of Mathematics

Master’s Thesis

Bounded Noises: new theoretical

developments, and applications

in Pharmacokinetics

September 19, 2014

Author

Dario Domingo

Supervisors

Dr. Alberto d’Onofrio

Prof. Franco Flandoli

Contents

Introduction

Chapter 1

Introduction to Bounded Noises for

Modelling Biological Processes

1.1

White noise effects on a tumor-growth model

0

0.5

1

−0.015

0

0.015

β

(t) dt

time

1.2

Different methods for generating Bounded Noises

1.2.1

Method of bounded function: Sine-Wiener noise

1.2.2

Cai-Lin family

1.2.3

Tsallis-Borland family

Chapter 2

Unboundedness and non-Uniqueness

of Tsallis Process for q < 0

2.1

Scale function and speed measure

2.2

Application to the Tsallis-Borland case

2.2.1

The case where q ≥ 0

2.2.2

The more problematic case q < 0

2.3

Physical interpretation of the apparent paradox

Chapter 3

Cai-Lin Strong Existence,

Uniqueness, and Boundedness

3.1

Pathwise uniqueness

3.2

Weak and strong existence

3.3

Boundedness of the original Cai-Lin equation

3.4

Behavior of the solution near the boundaries

Chapter 4

A More Detailed Investigation on the

Noises

4.1

Time-dependent density of the Sine-Wiener noise

4.2

Relation with the Cai-Lin noise