Oussama Amine
Regularization Effects for Certain Dynamical Systems through
Gaussian Noises
Thesis submitted for the degree of Philosophiae Doctor
Department of Mathematics
Faculty of Mathematics and Natural Sciences
2020
Series of dissertations submitted to the
Faculty of Mathematics and Natural Sciences, University of Oslo No. 2279
ISSN 1501-7710
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reproduced or transmitted, in any form or by any means, without permission.
Cover: Hanne Baadsgaard Utigard.
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Preface
This thesis is submitted in partial fulfillment of the requirements for the degree of Philosophiae Doctor at the University of Oslo. The research presented here was conducted at the University of Oslo and at Queen’s University, under the supervision of Professor Frank Proske and associate Professor Abdol-Reza Mansouri. This work was supported by the Faculty of Mathematics and Natural Sciences at the University of Oslo, partially, through the Stochastics of Renewable Energy Markets (STORE) project.
The thesis is a collection of four papers, that span 3 years of work with the aim of studying, using probabilistic as well as analytic tools, the phenomenon of regularization by noise. The common theme thus is that of the effect of noise on dynamical systems generated through ordinary differential equations and more specifically the role that certain perturbations play in improving the behavior of these systems even when the original non-perturbed version is known to be ill-posed. The papers are preceded by an introductory chapter on the background and motivation for our current work as a whole, as well as for each of the papers separately. It also provides a general idea on where the work is situated in the field of regularization by noise. The four papers have resulted from joint work with Professor Proske, while the last one was crystallized, shaped and improved through discussions with Professor Mansouri. The first paper is also joint work with Msc Emmanuel Coffie and Doctor Fabian Harang. The second and third papers are also joint work with Doctor David Banos.
Acknowledgements
From UiO I would like to thank: Professor Arne B. Huseby for his help during my first year as a PhD student, Professor Tom Lindstrøm who has been a source of inspiration throughout my years at UiO, Professor Snorre H. Christiansen for all the nice discussions over the many cups of coffee, Professor Sergey Neshveyev for his wonderful lectures, associate Professor Ulrik S. Fjordholm for the very informative seminar on the transport and continuity equations and his help with understanding some of the concepts related to them, associate Professor Salvador Ortiz-Latorre, Doctor David R. Banos, Professor Fred E. Benth, Professor Giulia Di Nunno, Martin Helsø for his help with LATEX , Professor Anders R. Swensen, Professor Sven O. Samuelsen, Professor Nils L. Hjort and the late Professor Hans P. Langtangen who is deeply missed.
From Queen’s university I would like to thank everybody I met there during my visit, especially associate Professor Bahman Gharesifard, Professor Andrew Lewis, Professor Ram Murty, Professor James A. Mingo, Professor Fady Alajaji, associate Professor Francesco Cellarosi, associate Professor Thomas Barthelmé, Professor Oleg I. Bogoyavlenskij, associate Professor Ivan Dimitrov, Professor Gregory G. Smith, Professor Mike Roth, associate Professor Serdar Yüksel, Professor Atabey Kaygun, Emine Yildrim and Jennifer Read.
From Ukraine I thank my friend Yuriy Prykhodko for his hospitality and generosity. Also Doctor Georgii V. Riabov, Professor Andrey A. Dorogovtsev, Doctor Olga Izyumtseva, Professor Andrey Pilipenko, Professor Oleg Klesov and Katerina for making my research stay in Kiev memorable.
I would also like to thank several other people I met during my PhD years.
Especially Professor Yaozhong Hu, Professor Youssef Ouknine and Professor Shiqi Song for helping me improve my understanding of some aspects of stochastic analysis. Thanks to Professor Nils Berglund for the nice discussions we had in Sarajevo. I thank also Professor Massimiliano Gubinelli for the long discussions on mathematics and other topics as well as for his generosity in answering my questions in L’Aquila. From Germany also, I thank Martin Bauer for all the interesting discussions as well as Doctor Lukas Wresch for his help with understanding the papers of Davie and Shaposhnikov as well as for making several suggestions for improving the introduction to this thesis. I also thank my co-author Emmanuel Coffie and wish him good luck in his PhD studies.
I thank as well Nacira Agram for all the nice discussions we had as well as for her support during my years of study.
My supervisors: I thank Professor Bernt Øksendal who was the reason for my interest in the field of stochastic analysis after I was fortunate to take a course taught by him. I also thank him for all the discussions we had during the past 3 years.
To Professor Abdol-Reza Mansouri, I can’t express how grateful I am for the opportunity that I had to work with you during my stay at Queen’s. Not only did I change as a mathematician after my stay there, but I have changed and grown as a human being. Thank you for accepting to supervise me as well as for giving me the opportunity to work with you.
Last, but not least, I thank my main supervisor Professor Frank Norbert Proske who made all of this possible the moment he accepted to take me as his student, and even before that. It is no exaggeration when I say that every word in this work is either directly or indirectly due to you. You have been an inspiration to me not only in relation to mathematics but, most importantly on the human level, your generosity and selflessness are truly rare. I want you to know that I will always be grateful to you dear Frank.
My friends: I thank Yassine for making life in Oslo more enjoyable and all the nice moments we spend together discussing about everything.
I thank Lara for the great time we spent learning new languages.
I also would like to thank Ijlal for being available when I needed someone to talk to as well as for sparking my interest in the field of security and cryptography.
I am truly happy to have met you!
Teachers: I wouldn’t be here writing these lines without the help, sacrifice and generosity of all the teachers that I had throughout my years of study. I take the opportunity to mention two who influenced me deeply in relation to mathematics. Professor Chebbaki and Professor Farouj, thank you very much for all you have taught me.
Family: A big hug to all of my family, my father and mother as well as my two beautiful brothers. I am grateful to have you in my life.
Oussama Amine Oslo, June 2020
List of Papers
Paper I
Amine, Oussama; Coffie, Emmanuel; Harang, Fabian Andsem and Proske, Frank Norbert “A Bismut-Elworthy-Li Formula for Singular SDE’s Driven by a Fractional Brownian Motion and Applications to Rough Volatility Modeling”.
To appear in Communications in Mathematical Sciences.
Paper II
Amine, Oussama; Baños, David and Proske, Frank Norbert “Regularity Properties of the Stochastic Flow of a Skew Fractional Brownian Motion”.
To appear in Infinite Dimensional Analysis Quantum Probability and Related Topics.
Paper III
Amine, Oussama; Baños, David and Proske, Frank Norbert “C∞ regularization by Noise of Singular ODE’s”. Submitted for publication.
Paper IV
Amine, Oussama; Mansouri, Abdol-Reza and Proske, Frank Norbert “Well- posedness of the Deterministic Transport Equation with Singular Velocity Field Perturbed along Fractional Brownian Paths”. Submitted for publication.
Contents
Preface iii
List of Papers vii
Contents ix
1 Introduction 1
1.1 The classical approach: Zvonkin-Veretennikov and the PDE
approach way . . . 5
1.2 The path-by-path approach: Davie’s way . . . 8
1.3 An alternative approach: On Compactness and Flows . . . 13
1.4 Summary of Papers . . . 16
References . . . 18
Papers 22 I A Bismut-Elworthy-Li Formula for Singular SDE’s Driven by a Fractional Brownian Motion and Applications to Rough Volatility Modeling 23 I.1 Introduction . . . 23
I.2 Bismut-Elworthy-Li formula . . . 26
I.3 Application: Stock price model with stochastic volatility . 43 I.A Appendix . . . 45
References . . . 53
II Regularity Properties of the Stochastic Flow of a Skew Fractional Brownian Motion 57 II.1 Introduction . . . 57
II.2 Main Result . . . 59
II.A Appendix . . . 67
References . . . 75
III C∞-Regularization By Noise of Singular ODE’s 77 III.1 Introduction . . . 77
III.2 Framework and Setting . . . 87
III.3 A New Regularizing Process . . . 93
III.4 Construction of the Solution . . . 97
III.5 Infinitely Differentiable Flows for Irregular Vector Fields . 124 III.A A Compactness Criterion for Subsets of L2(Ω) . . . 131
III.B Technical Estimates . . . 133
References . . . 139 IV Well-posedness of the Deterministic Transport Equation
with Singular Velocity Field Perturbed along Fractional
Brownian Paths 145
IV.1 Introduction . . . 146 IV.2 From SDE’s to random ODE’s . . . 154 IV.3 Applications to the Transport equation (TE) and Continuity
equation (CE) . . . 180 IV.4 Further improvement of the spatial regularity of solutions of
the transport equation with singular velocity fields perturbed along fractional Brownian paths with Hurst parameter
"H ↓0" . . . 188 IV.A Appendix . . . 190 References . . . 203
Chapter 1
Introduction
In the present work, we show how certain types of universal random perturbations affect certain basic dynamical systems that are a priori ill-posed in their original form. More precisely1 our contribution can be described through the following example. Consider the following ordinary differential equation
dX(t,x)
dt =b(t, X(t, x)) X(0, x) =x
or in integral form
X(t, x) =x+ t
0 b(s, X(s, x))ds when x∈Rd and t∈[0, T].
Let us take the Lagrangian point of view and "follow" a hypothetical particle starting from a spatial position x at time 0. If we think of b(.) as a function outputting an instantaneous velocity as a function of its argument, then when asking a question about the existence of a solution to the above equation we are in fact asking questions about the existence of a path traced by a particle starting at time 0 from x and changing its velocity as function of its position through b. More precisely what we are asking for is a map X :Rd → C
[0, T];Rd that sends each initial position to a trajectory X(., x). The Cauchy-Lipschitz theory tells us that this map exists, locally in time, if bis locally Lipschitz in the spatial argument uniformly in time. Moreover the solution map, called the flow, X is locally Lipschitz in x. The appearance of Lipschitz regularity with respect to bothb and X is not arbitrary and one can show that in general the flow inherits its spatial regularity from b. This Lipschitz regularity, of b at least, is crucial for the uniqueness of the solutions to the equation above. The fact that uniqueness can also follow from the Lipschitz regularity of a candidate flow map2 is not obvious at first sight but can be shown as was done in the work of Van Kampen [20].
This is in fact optimal and examples can be constructed to show this. In fact, consider the following initial value problem
dX(t)
dt = X(t)23 X(0) = 0
does not admit a solution map sincet→ 0 and t→ t3/27 are solutions to the above equation starting from 0. In fact there exists a whole family of solutions
1For a more concrete enumeration of our contribution see Section 1.4.
2See 1.2.3 for a definition.
indexed by c namely
Xc(t) =
0 , if t≤c (t−c)3/27 , if t≥c for any c∈ [0,∞].
The intuition for this phenomenon is that the non-Lipschitzianity of the vector field allows particles that can stay at rest for an arbitrary length of time without moving . This can also be explained by saying that the push-forward of a certain initial mass of particles becomes infinite or zero at a latter point in time. The reason for this is, of course, the fact that
t→ t23
is not locally Lipschitz in any open set containing the point 0. On the other hand, on any open set not containing 0 the behavior described above cannot take place and therefore developing a new, and necessarily relaxed, notion of solution to the differential equation above that can give a canonical way to select a specific solution among the possibly many at problematic points like 0 above while leaving the unproblematic ones untouched might be possible.
Thus, if we are willing to relax the notion of a solution to the above equation then there might be hope, and this is in fact the seminal work of DiPerna-Lions [7] on Lagrangian flows. Roughly speaking, the flow map that we seek need not satisfy the equation at every point but only almost everywhere with respect to some measure. The result holds for b such that:
b∈L1
(0, T) ;Wloc1,p
and div(b) ∈L1((0, T) ;L∞)
for some p ≥1. The previous result was extended in, again a seminal work of, Ambrosio [2].
In contrast to the Lipschitz case, the transfer of regularity is not anymore obvious and the work [10] shows that no transfer of Sobolev differentiability is to be expected in general.
The intuitive picture that one can draw to summarize the previous discussion is that unless the vector field enjoys some sort of spatial differentiability, the dynamical system given by 1 can exhibit a whole spectrum of “pathologies” e.g.
coalescence, particles splitting or being created from nothing.
Let us try a different approach by adding an additional velocity field that is of a very generic type, take the simplest namely a constant velocity field, then3 for a fixed initial position we have
X(t) =x0 + t
0 b(X(s))ds+ t
0 λds X(t) =x0 +
t
0 b(X(s))ds+λt
3This example is taken from [11].
This in turns implies, supposing in what follows that b is continuous. that Y(t) =x0 +
t
0 b(Y(s) +λs)ds, where we have used the very useful transformation
Y(t) =X(t)−λt Define the function B by
B(s) := d
dtB(s) := b(s) b(s) +λ,
then
Y(t) =x0+ t
0 b(Y(s) +λs)ds
=x0+ t
0 (λ+b(Y(s) +λs))·(B(Y(s) +λs))ds
=x0+ t
0
d
dtB(Y(s) +λs)ds
=x0−B(x0) +B(Y(t) +λt).
(1.1)
Now, suppose that there exist two solutions, say X1 and X2, then
|Y1(t)−Y2(t)|= |B(Y1(t) +λt)−B(Y2(t) +λt)|
If b is in addition bounded, then for some very large instantaneous velocity λ we have that B(s)< L << 1. Hence B is Lipschitz with Lipschitz constant strictly less than 1 and
|Y1(t)−Y2(t)| < L|Y1(t)−Y2(t)|
This heuristic argument shows that for, a bounded continuous vector field, b to give rise to a well-posed ODE, an instantaneous perturbation of the form dλ(t) :=λdt is sufficient.
At least two problems can be observed with the previous approach:
1. If b is truly singular e.g. discontinuous, then a constant instantaneous speed does not help with the singularities that b might have and in turn transfer to the dynamical system i.e. the flow. Hence something that has, for a lack of a better description, infinite speed would be ideal for the task at hand. This should happen for both signs as well to avoid a totally nonsensical situation.
2. Although not mentioned explicitly, the argument uses in an essential way the one-dimensional character of the equation to define B. Hence
another operator has to play the role that B played above. Moreover, and with the previous point in mind, truly singular vector fields can exhibit complex types of singularities in the multidimensional setting and thus the perturbation that we should consider ought to have some property of instantaneous change of direction in addition to that of speed.
The story so far leads us to the “best” candidate4, namely Brownian motion λ(t) := B(t). A slight inconvenience, at first sight at least, is that we cannot write dB(t) = ˙B(t)dt but if we work with dB(t) directly, the equation in integral form can be written as
X(t) =x0 + t
0 b(X(s))ds+B(t).
We cannot of course write the operator B as before but if we take one step back from its definition above we can see that a similar conclusion can be attained with following operator instead
A: [0, T]×Rd → Rd (t, x)→ At(x) :=
t
0 b(x+B(s))ds
In order to conclude we have to show that, for any two solutions X1 and X2, we have
|X1(t)−X2(t)|=|Y1(t)−Y2(t)|= |At(Y1(s)+B(s))−At(Y2(s)+B(s))| ≤L|Y1(s)−Y2(s)| for 0 ≤s≤t≤ T.
The task is now related to the operator A and its properties e.g. regularity, and how it interacts with solutions to the equation at hand.5
In fact defining new functionals through the perturbation (t→ B(t)) that enjoy better regularity than that enjoyed by b in the original problem is a recurring theme. We make a small digression to see how different variations of the above idea, namely the use of the regularizing effect resulting from the addition of (t→B(t)), results in well-posedness through auxiliary operators.
4When restricting the discussion to continuous perturbations. If one is willing to sacrifice continuity then there exists results in the setting when the perturbation is given by the paths of a Levy process (see for example [14]).
5In fact we show in Paper III that the operator Aas well as higher order operators that generalize it "encode" the regularizing effect thatB, or in the case of Paper IV a generalization ofB, generate as a function of the strength of their erraticity as measured by the property oftwo sided strong local non-determinism. This brings to mind the result of [19] where the regularization effect of the operatorx→
f(x−γ(s))a(s)ds, where γ is a smooth curve and a is a smooth cutoff function in a neighbourhood of 0, were linked to geometric properties of the curve namely the degree of curvature, in a specific sense, that γ possess’. A geometric approach, as in [19], to the study of the regularization effect of rough perturbations, through the operatorA, on singular vector fieldsb, seems to be, though very interesting, beyond reach at the moment.
The classical approach: Zvonkin-Veretennikov and the PDE approach way
1.1 The classical approach: Zvonkin-Veretennikov and the PDE approach way
Note that there is a subtle issue that we have not addressed yet, namely the fact that B is a stochastic process, (t, ω) → B(t, ω) a function of time and randomness, and thus must be defined on some space that can accommodate the two terms “stochastic” and “process”, namely a filtered probability space.
Moreover, common wisdom tells us that in order to say anything useful about its properties, the law of B must enter the picture one way or another. This raises the question of whether or not we have the same notion of solution, as the one we had prior to the addition of B. As a consequence the meaning of
"well-posedness" risks being changed in the presence of all these extra ingredients that come with the introduction of B.
The traditional approach, and probably most natural one given the historical context, is to think of the solution of the B-perturbed equation as a stochastic differential equation i.e. the solution, if it exists of course, is again itself a stochastic process that is compatible with the filtered probability space that came with B. This is called in the literature a strong notion of solution. The other notion being that of weak solution and here it is allowed that the solution be defined on a new filtered probability space with a new Brownian motion, say B defined on this new space. Thus, if we think of the perturbation B as on of the inputs to the problem and the solution X as its output, the notion of causality B →X is lost in the weak formulation.
Similarly, a weak and strong notions of uniqueness exist and we can say that we have weak uniqueness if for any two (weak) solutions (X1, B1) and (X2, B2) the law of X1 and X2 are the same, while we have strong uniqueness if for any two solutions (X1, B) and (X2, B) we have X1 = X2 almost surely.
These notions are related through the Yamada-Watanabe theorem
Theorem 1.1.1. Weak existence + path-wise uniqueness =⇒ Strong uniqueness.
Now with the lingo in place we can state the first result in the area of regularization by additive perturbation. Note that this is just part of the result of Zvonkin-Veretennikov as the original results and proofs yield existence and uniqueness.
Theorem 1.1.2([22], [21]). If b is bounded measurable then the equation X(t) =x0+
t
0 b(X(s))ds+B(t)
interpreted as a stochastic differential equation, has a unique strong solution.
Note on the proof strategy: Since we have changed the character of the original equation, we might as well use the tools that come with this new notion of solution. In our context it is Itô’s lemma for stochastic processes which says
that for any function u(t, x) in C1,2 we have u(t, X(t)) =u(0, x0)+
t
0 ∂tu(s, X(s))ds+
t
0 ∂xu(s, X(s))dXs+1 2
t
0 ∂xxu(s, X(s))ds i.e.
u(t, X(t)) =u(0, x0) + t
0 ∂tu(s, X(s))ds+ t
0 ∂xu(s, X(s))b(X(s))ds+
t
0 ∂xu(s, X(s))dB(s) + 1 2
t
0 ∂xxu(s, X(s))ds where, for any functiong with some appropriate conditions, the term
t
0 g(s)dB(s)
is the Itô integral, defined as the L2-limit of finite linear combinations with respect to the increments of the Brownian motions. Note that this is just the integral form of the classical chain rule with the exception of the additional term
1 2
t
0 ∂xxu(s, X(s))ds,
which can be interpreted as a correction due to the “infinite” speed that B has.
Another useful tool that we get in this framework is the Itô isometry E |
T
0 g(s)dB(s)|2
= E T
0 |g(s)|2ds
The last ingredient is a result from the theory of parabolic PDE’s:
Theorem 1.1.3. [22, Theorem 2] Consider the following Cauchy problem
∂
∂tu(t, x) =b(x)ux(t, x) + 1
2uxx(t, x) u(T, x) =x
(1.2) Then there exists T > 0 such that the above problem admits a unique (weak) solution and u∈ Wp1,2([0, T]×U) for any bounded domain U and for all p >1.
Moreover there exists a constant ρ >0 such that
|u(t, x)−u(t, y)| ≥ρ|x−y|.
With the necessary ingredients in place, we can now describe the idea behind the Zvonkin-Veretennikov approach. The strategy, based on the Yamada- Watanabe theorem, yields the existence of a unique strong solution from the existence of a weak solution and the property of path-wise uniqueness. The
The classical approach: Zvonkin-Veretennikov and the PDE approach way solution u to (1.1.3) is going to play the role of the regularizing transformation here and its regularity, related to the Brownian perturbation, is the key ingredient for the proof of well-posedness. We sketch a proof for this point namely that of path-wise uniqueness and in doing so we suppose that b, in addition to being bounded, is Hölder continuous with a certain Hölder exponent θ. This simplifies some parts of the proof since it implies that the solution is twice continuously differentiable in x with a second derivative that is Hölder continuous (see [8]).
(Sketch) proof of path-wise uniqueness: Indeed, let ube the solution to the previous Cauchy problem, then by applying Itô’s formula we get
u(t, X(t)) =u(0, x0) + t
0 ux(s, X(s))dB(t)
for any solutionX to the SDE. Taking two arbitrary solutions and applying the Itô isometry we get that
E
(u(t, X1(t))−u(t, X2(t)))2
=E t
0 (ux(s, X1(s))−ux(s, X2(s)))2ds
On the other hand, by Hadamard formula we have that ux(s, X1(s))−ux(s, X2(s))=
1
0 uxx(s, ξX1(s) + (1−ξ)X2(s))dξ
(X1(s)−X2(s))
≤C(X1(s)−X2(s))
by Theorem 1.1.3 and the Hölder continuity assumption on b.
Thus we get E
(u(t, X1(t))−u(t, X2(t)))2
≤CE t
0 (X1(s)−X2(s))2ds
On the other hand, the second part of Theorem 1.1.3 gives us that
|u(t, x)−u(t, y)| ≥ρ|x−y| for some ρ >0.
Hence E
(X1(t)−X2(t))2
≤ 1 ρ2E
(u(t, X1(t))−u(t, X2(t)))2
≤ C ρ2
t
0 E
(X1(s)−X2(s))2 ds
Gronwall’s lemma implies that E
(X1(t)−X2(t))2
= 0
and hence X1(t) =X2(t) a.s. for allt∈[0, T], and by continuity we get that the
two processes X1 and X2 are equal a.s.
A common characteristic of the different approaches that are based on the Zvonkin-Veretennikov approach is the change of focus from the term involving the singular term b to something related to the perturbed equation but that is of better regularity e.g. the solution u of the above Cauchy problem. Thus the problem is reduced to one of studying the properties of solutions to certain types of PDE’s as in Theorem 1.1.3. This strategy is very general and can be used to not only show the well-posedness of the SDE but also to show the regularity of the solution with respect to the initial condition. The PDE approach, however, comes at a cost and that is the dependence of the whole method on the Markovianity of the perturbation.
1.2 The path-by-path approach: Davie’s way
The blessing as well as the curse of the previous approach is in the modification that happened to the notion of solution of our original ODE. Suppose we really insist on solving our perturbed equation in the space of continuous functions C
[0, T];Rd
without any mention of “stochastic process”. Then we can formulate the following question:
Question: Let b be a bounded measurable drift and choose a realization of B i.e. fix an ω and consider the continuous function (t→ B(t) :=B(t, ω)) for that ω, does the following ODE
X(t) =x0+ t
0 b(X(s))ds+B(t)
have a unique solution? This question was posed by N. V. Krylov and was communicated6, through I. Gyongy, to A. M. Davie who answered it positively in his seminal work [5]. In this work he showed that
Theorem 1.2.1. For any bounded Borel measurable b : [0, T]×Rd → Rd and x0 ∈Rd for almost all Brownian paths, the equation
X(t) =x0 + t
0 b(s, X(s))ds+B(t) has exactly one solution in the space of continuous paths.
The previous result has been further improved in the crucial work of [17].
Namely, the author shows
Theorem 1.2.2. For any bounded Borel measurable b: [0, T]×Rd →Rd and for almost all Brownian paths, the following problem
X(t, x) =x+t
0 b(s, X(s, x))ds+B(t) X(0, x) =x
has a unique solution ((t, x)→ X(t, x)).
6The first written statement we found of this problem is in [15] page 29 by N. V. Krylov himself.
The path-by-path approach: Davie’s way The proof of this beautiful result relies on an, almost forgotten, result of Egbert van Kampen [20]. We reproduce the proof of the original result of Van Kampen using the exposition in [9] then we sketch a slight extension immediate from [17] in order to show the principle at play.
Theorem 1.2.3(Van Kampen 1936). Consider the following ODE dX(t)
dt =b(t, X(t))
X(t0) =x0, (1.3)
where b is a continuous function on [0, T]×Rd on R:= [t0, t0+a]×[x0− b, x0 +b]. Suppose further that there exists a function η : [t0, t0+a]×[t0, t0+ a]×[x0−β, x0+β] with β < b such that
1.2.3.1. For every fixed (t1, x1), X(t) :=η(t, t1, x1) is a solution of dX(t)
dt = b(t, X(t)) X(t1) =x1
1.2.3.2. η is uniformly Lipschitz in x1.
1.2.3.3. η(t3, t1, xt1) = η(t3, t2, η(t2, t1, xt1)) whenever the expressions are defined.
Then η(t, t0, x0) defines the unique solution of the 1.3 on R.
Proof. (Sketch)[9] Pick a solution X(t) of 1.3, we shall show that on a small time intervalX(t) =η(t, t0, x0).
Using the continuity of b on R, we can find a small γ such that for all t0 ≤s, t1, t≤t0 +γ we have, by Lipschitz regularity of η,
|η(t, t1, X(t1))−η(t, s, X(s))| ≤L|X(t1)−η(t1, s, X(s))| for some positive constant L.
Now fix t∈ [t0, t0+γ] and define, for s∈[t0, t0 +γ]
σ(s) :=η(t, t0, x0)−η(t, s, X(s)) Then to conclude, we have to show that
σ(t) :=η(t, t0, x0)−X(t) = 0.
Indeed, we have
|σ(t1)−σ(s)| =|η(t, t1, X(t1))−η(t, s, X(s))| ≤L|X(t1)−η(t1, s, X(s))|. On the other hand, since both η(t1, s, X(s)) and X(t1) are solutions, we have
η(t1, s, X(s)) =X(s) + t1
s
b(u, η(u, s, X(s)))du
and
X(t1) =X(s) + t1
s
b(u, X(u))du.
This yields , when t1 → s, that
|X(t1)−η(t1, s, X(s))|= o(1)|t1 −s|. Hence
|σ(t1)−σ(s)| =Lo(1)|t1−s|.
This means that (s→ σ(s)) is differentiable and, since σ(t0) = 0, it is identically 0.
If the the "flow"7 η is not Lipschitz regular then by inspecting the above proof we can conjecture that a vector field with better regularity than just continuity might compensate for it. This is in fact the case and the following argument in [17] shows that if the vector field is β-Hölder regular and the flow* is α-Hölder regular such that α(β + 1) > 1 then we can recover a similar result to Van Kampen’s.
Letb now be uniformly β-Hölder and η satisfies the conditions in 1.2.3 except for the the Lipschitz regularity which we substitute now with a uniformα-Hölder regularity. Suppose further that α(β+ 1)> 1 then, using the same notation of the previous proof, we have
|σ(t1)−σ(s)| =|η(t, t1, y(t1))−η(t, s, y(s))| ≤M|y(t1)−η(t1, s, y(s))|α for some constant M. On the other hand, by the boundedness of
|y(t1)−η(t1, s, y(s))|= t1
s
b(u, y(u))du− t1
s
b(u, η(u, s, y(s)))du
≤ t1
s
|b(u, y(u))−b(u, η(u, s, y(s)))|du
≤ C|t1 −s|.
(1.4)
Using the same reasoning but with the Hölder continuity of b instead, we have
7We will abuse the use of the word flow to denote a family of solutions enjoying the properties of 1.2.3. We will denote such occurrences by a flow*.
The path-by-path approach: Davie’s way
|y(t1)−η(t1, s, y(s))|= t1
s
b(u, y(u))du− t1
s
b(u, η(u, s, y(s)))du
≤ t1
s
|b(u, y(u))−b(u, η(u, s, y(s)))|du
≤ t1
s
|y(u)−η(u, s, y(s))|βdu
≤ C|t1−s|1+β.
(1.5)
Hence
|σ(t1)−σ(s)|=Co(1)|t1 −s|1+δ, where δ := α(β + 1)−1> 0.
This means that (s→σ(s)) is differentiable and, since σ(t0) = 0, it is identically 0.
The previous manipulations show that even if the vector field does not fall into the framework of the Cauchy-Lipschitz regularity, a regular enough flow* might compensate for it. This suggests an intuitive principle which says that uniqueness can happen anywhere on the interpolating spectrum between a Lipschitz vector field and, automatically, continuous flow on the one hand and a continuous vector field plus a Lipschitz flow* on the other, with anything in the middle as well, provided the two regularities add up in an appropriate manner. However, the previous discussion should be put together with the following two points:
1. A continuous vector field is very far from the result of Davie and the arguments above, if they are to work, should be modified in order to accommodate this. A manipulation similar to the example at the beginning of the introduction might be necessary in order to move the discussion from b to some operator based on it.
2. If b is discontinuous, then it is difficult to imagine that there might exist a flow* with any kind of regularity if there is no "external" mechanism for regularity transfer.
The above two points are in fact the key to the method used in [17] in the context of
X(t) =x+ t
0 b(s, X(s))ds+B(t) with b bounded and Borel measurable.
Now looking back at (1.4) and (1.5) we see, supposing η exists, that
|X(t1)−η(t1, s, X(s))|= t1
s
b
u, X(s) +B(u)−B(s) + u
s
b(r, X(r))dr
du
− t1
s
b
u, X(s) +B(u)−B(s) + u
s
b(r, η(r, s, X(s)))dr
du
= t1
s
b(u, B(u) +ζ1(u))du− t1
s
b(u, B(u) +ζ2(u))du, (1.6) where
ζ1(u) :=X(s)−B(s) +u
s b(r, X(r))dr ζ2(u) :=X(s)−B(s) +u
s η(r, s, X(s))dr are two Lipschitz functions.
The following two remarks allow us to conclude uniqueness:
1. The averaging operator is almost Lipschitz (Lemma 3.6 [17]).
2. The equation above, when interpreted as a SDE, admits a stochastic flow that is α-regular for any α∈ (0,1) (Proposition 2.3 [18]).8
These are essentially the two ingredients which make up the proof in [17]
modulo some complications related to "almost Lipschitz" in point 1 above.
The original proof in [5] used in an essential way the regularizing effect of the averaging operator A defined above as
A: [0, T]×Rd →Rd (t, x) →At(x) :=
t
0 b(s, x+B(s))ds
but not the existence of a stochastic flow solving the SDE interpretation of the perturbed equation. This made the proof much more difficult since the regularizing effect of the averaging operator by itself is not sufficient to yield the result, and one needs better estimates of this regularization effect along solutions of the equation. This is essentially the passage from Lemma 3.1 and Lemma 3.2 to Lemma 3.5 and its almost sure version Lemma 3.6 in [5].
It is worth mentioning that a purely deterministic path-by-path approach is possible and this is the work of [4] where only the the regularizing effect of the averaging operator is used in combination with a non-linear Young integration theory with respect to increments of the averaging operator. This comes, however, at the cost of very stringent requirements at the level of the vector fieldb, namely b is in the Besov space B∞,∞α which for α ∈ (0,1) is just the space of Hölder
8This holds only for b∈ Lq
[0, T], Lp
Rd
with dp + 2q <1 but the boundedness of b permits the use of such a result.
An alternative approach: On Compactness and Flows continuous functions. The reasons for such a requirement on b should be clear from the discussion above.
It is also worth mentioning here as well the remarkable work [12] which abstracts the idea of using the non-linear Young integral while at the same time takes into account the probabilistic side which did not enter in the approach of [4]. The many examples shown in [12] are witness to the strength of this point of view, however, it is not clear how this can be applied to the study of the finer properties of the stochastic flow and their relation to higher regularization effects one obtains when using more regularizing perturbations than justB.
1.3 An alternative approach: On Compactness and Flows
There seems to be a discrepancy between the purely deterministic approach and the one using probabilistic tools. We will try to give an explanation for this and in doing so we will introduce the method that the current thesis is based on.
If we look back at the proof of [5]’s result using [17]’s insight, we see that the regularizing effect of the averaging operator is only sufficient to conclude if it is combined with the regularity of the stochastic flow. Indeed if we look at the following expression
t1
s
b
u, Xx(s) +B(u)−B(s) + u
s
b(r, Xx(r))dr
du with Xx a solution to our differential equation. This is trivially equal to
t1
s
b
t2, Xx(s) +B(t2)−B(s) + t2
s
b(r, Xx(r))dr
dt2. Iterating once more
t1
s
b
t2, Xx(s) +B(t2)−B(s)+
t2
s
b
t3, Xx(s) +B(t3)−B(s) + t3
s
b(r, Xx(r))dr
dtt3
dt2. (1.7) Thus we see that there is a compounding regularization effect of Brownian perturbation through the averaging operator. Moreover there is a temporal structure reminiscent of the chronological calculus of A. Agrachev and R.
Gamkrelidze [1], and an approach that takes this temporal structure into account might result in a quantification of "regularization effects of higher order". The issue is that any attempt to study the effect of the perturbation B on the above expression as time progresses is hindered by the highly non-linear dependence on B. If only there was a way to linearize the above expression to untangle the effect ofB through all of the above nonlinearities.
Indeed there is one, and it is achieved by differentiating along the perturbations B using the tools of Malliavin calculus. Taking this point of view, it can be shown using, the equivalent of the chain rule for this new calculus, that when b is smooth and compactly supported, the following holds
DθXx(t) = t
θ
b(u, Xx(u))DθXx(u)du+I,
where b is the spatial derivative of b and I is the identitiy matrix. Here X is (unique strong) solution, by the smoothness of b, of
Xx(t) =x+ t
0 b(u, Xx(u))du+B(t),
where we have used the property that the Malliavin derivative of B(t) is the identity matrix I when θ < t. Iterating the above expression, we get the representation
DθXx(t) = ∞ k=0
θ<u1<···<uk<t
b(u1, Xx(u1))· · ·b(uk, Xx(uk))du1· · ·duk. A similar manipulation yields, for the derivative of the flow which exists by standard arguments in the smooth and compactly supported b case, that
∂
∂xXx(t) = ∞ k=0
0<u1<···<uk<t
b(u1, Xx(u1))· · ·b(uk, Xx(uk))du1· · ·duk. As both, DθX(t) and ∂x∂ Xx(t) are random variables, controlling their moments is a logical next step to seek, and indeed we get
E
| ∂
∂xXx(t)|p
≤C(r,b∞)×
∞ k=0
0<u1<···<uk<t
b(u1, x+B(u1))· · ·b(uk, x+B(uk))du1· · ·dukp
Lps(Ω,Rd×d)
(1.8) for r, s∈ [0,∞) such that sp= 2q for some integer q and 1r + 1s = 1, where we used the Cameron-Martin-Girsanov theorem to change Xtx to x+B(t) under the expectation. Decomposing the norm of the sum in terms of the norm of its one dimensional components, we see that the task of controlling the moments is reduced to establishing an estimate of the type
E
t0<u1<···<un<t
n
i=1
Dαibi(ui, x+B(ui))
du1· · ·dun
≤ C(n, d,bi)|t−t0|ξ
An alternative approach: On Compactness and Flows for some ξ ∈ (0,1), where bi are smooth compactly supported real valued functions on [0, T]×Rd, αi ∈ {0,1}d is a multi-index such that |αi|= 1, Dαi is the derivative in the spatial argument with respect to the index αi.
Indeed the following key estimate was established in [5] (see also [13] and [16] when b is in a larger class)
E
t0<u1<···<un<t
n
i=1
Dαibi(ui, x+B(ui))
du1· · ·dun
≤ Cnn
i=1bi∞(t−t0)n/2 Γ(n2 + 1) , where Γ is the Gamma function.
Thus we recover the higher order equivalent of the averaging operator, only this time the higher order temporal regularization effect is taken into account.
This is a key estimate that contains the true regularization power of theB paths and it is a key tool in Davie’s original work in order to establish the almost Lipschitz regularity of the averaging operator even when b is merely bounded.
Of course, the above reasoning is only valid when b is smooth but a strength of the above estimate is its dependence on b through only its b∞ norm. Thus, if we can show that the sequence Xn, of solutions to the SDE with vector fields bn, converges toX, and the same for the derivatives in x, then we might be able to close the loop.
Indeed, and using a compactness result developed in [6], one can show that the sequence Xn is relatively compact in L2(Ω), and even converges in L2(Ω) strongly to X. This is achieved using the estimate above but applied in the control of the Malliavin derivative of the approximating solution i.e. DtXn9. Moreover, using the control on the moments of ∂x∂ Xtx,n we get that the limit above is Sobolev differentiable.
Since we have an almost Lipschitz flow, we can use the methodology in [17]
to recover Davie’s original result i.e. uniqueness in the path-by-path sense.
Note that there is no mention of any auxiliary results from the theory of PDEs and all of the analysis is at the level of the flow. This is in fact one of the strengths of this method as the above method can be applied in situations where the approach based on PDE theory is inapplicable.
A weakness, at first sight at least, however, of the above approach is its reliance on the probabilistic properties of the perturbationB, namely through the above key estimate as well as the compactness criterion in L2(Ω). This should be contrasted with the purely deterministic approach mentioned in the previous section. This objection is a valid one, but only in theory, in practice, however, the following two points tell us something different
1. Although the results in [4] are formulated using the notion of an irregular path which is a deterministic notion indeed, the final result is stated with respect to fractional Brownian motion (fBm) which is a Gaussian process that generalizes Brownian motion to the case when the increments are not independent. This is due to the fact that, although the notion of an
9To be precise, we need to controlDt2Xtn−Dt1Xtn.
irregular path is simple to state, giving examples of such paths is difficult and is only possible in fBm case through recourse to its law.
2. For vector fields b which are not distributions, the approach in [4]
necessitates the continuity of b. The discussion above shows, clearly we hope, that the truly problematic regime is the one of discontinuous b e.g. b∈Lq
[0, T], Lp Rd
with dp + 2q ≤ 1.
This begs the following question: Can we establish a similar result to [5] for truly singular vector fields, in the case when the perturbation is given by fBm BH? The main contribution of this thesis is a positive answer to this question through the Van Kampen principle as was done in the Brownian motion case in [17]. We summarize our contributions in the next section.
1.4 Summary of Papers
Paper I The first paper is the genesis for an improved estimate in relation to the derivative(s) of the flow of the following SDE
X(t) =x+ t
0 b(u, X(u))ds+BH(t),0≤t≤ T, (1.9) where BH(t),0 ≤ t≤ T is a d−dimensional fractional Brownian motion with Hurst parameter H ∈(0, 12) and where the vector field b issingular in the sense that
b∈L1,∞∞,∞ :=L1(Rd;L∞([0, T];Rd))∩L∞(Rd;L∞([0, T];Rd)).
Equation (1.9) was the main object of study in [3], the main result being Theorem 1.4.1. Consider the following SDE
Xs,x(t) =x+
t
s
b(u, Xs,x(u))du+BH(t)−BH(s), Xs,x(s) =x,0≤ s≤ t≤T (1.10) Let s∈[0, T], b∈ L1,∞∞,∞ and k≥1. Then if H < C(d, k)< 12 there exists a unique (global) strong solution X·s,x of the SDE (1.10). Moreover, for every x∈ Rd, t∈ [s, T] Xts,x is Malliavin differentiable in the direction of the Brownian motion B in the representation
BH(t) = t
0 KH(t, s)Id×ddB(s). (1.11) Further, Xs,·(t) is locally Sobolev differentiable μ − a.e. That is, more precisely,
Xts,· ∈
p≥2
L2(Ω;Wk,p(U)) for bounded and open sets U ⊂ Rd.
Summary of Papers Our main contribution is an improvement of the constantC(d, k), appearing in the previous theorem from
2(3d−1)1 ∧ d(2k+1)1
to 2(d−1+2k)1 . This is a significant improvement, from multiplicative to linear dependence on d, of previous results.
The above estimate has also influenced as well as affected the key result in [3] namely Proposition 3.2, which is the equivalent estimate, in the case of fBm perturbations, of the estimate above in [5].
We then use our result to prove a Bismut-Elworthy-Li formula with respect to the unique strong solution of (1.10) (Theorem I.2.6). We then use this formula to solve the problem of of computing Greeks, the Δ to be precise, of financial claims in a Black-Scholes model with stochastic volatility who’s dynamics are given by (1.10) (Theorem I.3.1).
Paper II In this paper we prove the higher order differentiability of a stochastic flow associated with a stochastic differential equation driven by an additive multi-dimensional fractional Brownian noise, where the bounded variation part is given by the local time of the unknown solution process, that is
Xx(t) =x+αLt(Xx)·1d+BH(t), 0≤t≤ T (1.12) where α∈R, 1d is the vector with entries 1 and Lt(Xx), 0≤ t≤T is the local time of the unknown solution process, which one can define as
Lt(Xx) = lim
ε0
t
0 ϕε(Xsx)ds,
where the limit is in probability and ϕε approximates, in distribution, the Dirac delta function δ0 in zero. Here a commonly used approximationϕε is given by
ϕε(x) =ε−d2ϕ(ε−12x), ε > 0, (1.13) where ϕ is a d−dimensional Gaussian probability density.
More precisely, our key contribution is showing that solutions Xx(t), to the stochastic differential equation (1.12), for all 0≤ t≤ T,
(x−→Xx(t)) ∈
p≥2
L2(Ω;Wk,p(U)).
whenever H < 2(d−1+2k)1 .
Paper III In this paper we ask a question in the opposite direction, namely: Can we construct a noise that has infinite, in the sense of restoring classical differentiability, regularization power on
Xx(t) =x+ t
0 b(u, Xx(u))du+B(t), Xx(0) = x,0 ≤t≤T, (1.14)
