A Class of Lévy SDE s: Kinematics and Doob Transformation

Transcription

1 UNIVERSITÀ DEGLI STUDI DI BARI Dottorato di Ricerca in Matematica XXII Ciclo A.A. 2009/20010 Settore Scientifico-Disciplinare: MAT/06 Probabilità Tesi di Dottorato A Class of Lévy SDE s: Kinematics and Doob Transformation Candidato: Andrea ANDRISANI Supervisore della tesi: Prof. N. CUFARO PETRONI Coordinatore del Dottorato di Ricerca: Prof. L. LOPEZ

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3 Contents Preface 5 1 Stochastic mechanics Introduction Kinematics in stochastic mechanics Velocities Diffusions Acceleration Dynamics in stochastic mechanics Lévy processes Introduction Definitions Lévy measures Lévy-Itô decomposition Infinite divisible distributions Stable and self-decomposable distributions Lévy processes as Markov processes Lévy processes as semimartingales SDE for Lévy processes and semimartingales Kinematics of solutions of Lévy SDE Introduction Forward and Backward Stochastic Velocities Introductory remarks Forward and Backward Stochastic Velocities Example: The Free Case Continuity Equation and Acceleration

4 4 Contents 3.4 The Ornstein-Uhlenbeck Equation What about dynamics? Schrödinger equation and Markov processes Introduction Analytic tools The forward and backward semigroups Dirichlet forms and semigroups Doob type transformations The Hamilton operator associated to X t Examples Conclusions 97 A Proof of formula (3.2.8) 99

5 Preface In the present thesis I study a possible extension of stochastic mechanics from the usual Wiener process to generic Lévy processes as source of fluctuations. Stochastic mechanics started with the pioneering work of Nelson in the sixties, [48, 49], with subsequent contributions from Carlen, Guerra, Morato, Yasue, Zambrini among others. For a large class of Markov process it defines typical kinematics quantities like forward and backward stochastic velocities, together with stochastic acceleration. All these definitions are reduced to those of classical kinematics if processes are actually deterministic. Then, defining a dynamics either by means of the second Newton s law, or by a suitable variational principle, it s possible to show that for processes obeying a SDE of the type dx t = b(x t, t)dt + dw t where W t is a Wiener process and b(x, t) a suitable function, the density function ρ(x, t) of X t can be given through a wave function ψ(x, t), solution of the Schrödinger equation of quantum mechanics i t ψ = Hψ, by means of the relation ρ(x, t) = ψ(x, t) 2, often called in physics the Born principle. The potential function appearing in the Hamilton operator H will then be the same that determines the stochastic acceleration in the second Newton s law. Since Nelson s publications there was a debate if to every solution of a Schrödinger equation it was possible to associate a diffusion process in the way described above. The positive answer was given by Carlen [7] in 1985, and we will try to trod along the same path. Lévy processes can be considered as a generalization of the well known Wiener processes. They are defined by few conditions: they are processes with independent and stationary increments, and they are also stochastically continuous. Among them we find many classes of well known processes as the Poisson, the Cauchy, the Delta, the t-student processes, and - of course - the

6 6 Preface Wiener process, which is the unique Lévy process with Gaussian distributed increments. The Lévy processes have several other nice properties: all of them are Markov processes, and in particular Feller processes; its always possible to find versions of them which have cad-lag (continuous on the right with left limits) paths; besides if they are integrable with null expectation then they are martingales. In general, however, all Lévy processes are semimartingales, i.e. processes that can be used as integrators in Ito integrals. It is important to remark that between Lévy processes and infinite divisible distributions there is a one-to-one correspondence. Finally we remember that the only Lévy process with a.s.-continuous paths is the Wiener process. Lévy processes were first studied by the French mathematician Paul Lévy ( ), and are now widely applied in a variety of mathematical fields, from beam halos [12, 13] to mathematical finance [53]. Some references about Lévy processes can be found in Sato [59], Applebaum [5] and Protter [54] among others. The Doob transformation [20, 60, 56] - that will be instrumental about an association of processes to wave functions solutions of a Lévy-Schrödinger equation - acts on a Markov process X t with infinitesimal generator L in the following way: given a positive function h in the domain of L with Lh = 0, then we define a new Markov process Xt h having as infinitesimal generator L h such that L h g = h 1 L(hg). We then introduce the Doob-type transformations, namely transformations of the same kind, but with h a generic not vanishing function. The aim of this research essentially is to check a Cufaro and Pusterla conjecture [14, 15]. They suggested that from a stochastic mechanics with fluctuations generated by a Lévy (in general not a Wiener) process, it would be possible to deduce an evolution equation similar to the Schrödinger one, with a free part of the Hamilton operator H where the infinitesimal generator of a Wiener process is replaced by the infinitesimal generator of the Lévy process. In formula, given ψ a square integrable complex function, [Hψ](x) := 1 2 ψ(x) [ψ(x + y) ψ(x)] l(dy) + V (x)ψ(x) where for a Borel set B R n, 0 / B, l(b) measures the mean number of jumps with size in B that the Lévy process performs in the unit time. l is called a Lévy measure.

7 7 Initially I focused my attention on the Lévy-type processes, i.e. the solutions of the SDE dx t = b(x t, t)dt + Z t, where Z t is a generic Lévy process. However after defining the kinematics for these processes, and after assuming that forces in the dynamic equations were always given by the gradient of a potential V, i have found some difficulty to link a process solution of the dynamic equation with the wave functions solutions of a Lévy-Schrödinger equation by means of the Born relation. In order to bypass this standstill Prof. Morato pointed out that in the Wiener case it was possible to find the Hamilton operator from the forward and backward generators of a diffusion process also by means of a Doob-type transformation, without apparently involving a dynamics [1, 2, 3, 47]. She then suggested to use these mathematical tools also in the Lévy-Schrödinger case, and i eventually found the way to link a suitable class of Markov processes with the solutions of the Lévy-Schrödinger equations under scrutiny. In this way it was possible to construct Markov processes connected with the solutions of the Lévy-Schrödinger equation hypothesized by Cufaro and Pusterla, and in particular I have found that - apart from the trivial free case - these processes are not Lévy-type processes, even if they are closely connected to them. These results show that there is a deep connection between Lévy processes and the newly proposed Lévy-Schrödinger equations, and we are interested now in implementing this association by means of a dynamics principle, as done in the past by Nelson, Guerra, Morato and Yasue in the Wiener case. Future research will then be devoted to reach this aim together with a more detailed study of these processes based on the solutions of the Cufaro and Pusterla version of the Schrödinger equation. This thesis is structured as follows: in the first chapter there is a brief review about stochastic mechanics. In the second chapter I introduce the Lévy processes along with their main properties. In the third chapter I have studied the kinematics of Lévy-type processes ruled by dx t = b(x t, t)dt + dz t, the expressions for the forward and backward stochastic velocities associated to it, together with their acceleration. I have analyzed, as particular example, the Ornstein-Uhlenbeck equation dx t = cx t + dz t, when (Z t ) t 0 is a stable Lévy process with 0 < α 2 and I have found explicit expressions for its kinematic quantities. In the fourth chapter I start with a short review of Markov processes focusing my attention on the associated semigroups and their infinitesimal generators. I also give a quick review on Dirichlet forms which are useful in the discussion. Then by means of a Doob-type transformation I associate a Markov process (X t ) t R - satisfying some mild conditions - to

8 8 Preface a self-adjoint, bounded from below operator H in the Hilbert space of complex, square integrable functions. The function ψ that I used in the Doob-type transformation shows then a double nature: it is a solution of the Schrödingertype equation i t ψ = Hψ, where H plays the role of the Hamilton operator, and verifies also the Born relation ρ(x, t) = ψ(x, t) 2, where ρ is the density function of (X t ) t R. I consider finally some examples showing that H is the usual Hamilton operator of quantum mechanics if (X t ) t R is a Wiener process, and that there are processes - derived by general Lévy processes - that admits as H the operator hypothesized by Cufaro and Pusterla in [14, 15]. The conclusions of my inquiry are summarized in the chapter five, while an Appendix relates a few calculation details. I wish to express my gratitude to Prof. Nicola Cufaro Petroni, who introduced me in the world of stochastic mechanics and of Lévy processes, and who patiently supervised my thesis, and to Prof. Laura Morato of University of Verona, who guided my researches with suggestions, constructive criticisms, adjourned me on the state of the art about various arguments treated in this thesis, and helped me in making my permanence in Verona comfortable and enjoyable. I wish also to mention Prof. De Martino and Prof. De Siena of University of Salerno, and Prof. Pusterla of University of Padova, with whom i spent pleasant time with stimulating conversations both on mathematics and physics and about facts of life. Finally, I express my thanks to the Department of Informatica - Università degli Studi di Verona - that accomodated me during my out of office PhD period.

9 Chapter 1 Stochastic mechanics 1.1 Introduction Stochastic mechanics was born as a tentative to understand microscopic world and predictions of quantum mechanics in an objective prospective, i.e. by means of trajectories, so attributing elements of physical reality to particle. Actually, stochastic mechanics says much more, since it proves that dynamics in quantum phenomena is still governed by classical Newton s laws, if one takes the accuracy of change the definitions of kinematics quantity as velocity and acceleration, in order to take account of the fluctuations to which microscopic objects are supposed to be subjected. Stochastic mechanics was born from Nelson s works [48, 49, 50], even if Fènyes [27] was the first to interpret quantum mechanics in terms of stochastic processes. Is also to be mentioned an article of Bohm and Vigier [6], who hypothesize the presence of a noise affecting the motion of microscopical particles. Other important contribution to the theory came from Albeverio, Carlen, Föllmer, Guerra, Morato, Yasue, Zambrini, and so on [1, 2, 7, 28, 32, 62, 63, 65, 66]. The theory can be considered as a success, since it essentially gives the same physical predictions of the Schrödinger equation [7]; besides can take into account spin presence [19, 26], symmetric properties of Bose particles [50], and it finds applications in the fields theory contests [32, 68], if one choices opportune manifolds in which stochastic processes live (here for simplicity i ll suppose that all processes takes value in R n ). However, like any other realistic theory compatible with quantum mechanics, it presents some non-local effects, disliking a part of physics community. The same Nelson posed some

10 10 Chap. 1 - Stochastic mechanics objections to his theory as a physical theory [51], even if some of them were confuted [9]. From a mathematical point of view, noises in stochastic mechanics are essentially given by Wiener processes. However, there are works in which more general Lévy processes are involved: between them we cite Garbaczewski and Olkiewicz [30, 31], Cufaro and Pusterla [14, 15], and Ichinose et al. [36, 37]. The chapter is organized as follow; in Section 1.2 i first define, following Nelson definitions [49], forward and backward mean velocity for a wide class of stochastic processes, and i investigate about their mutual relation in particular for diffusions processes. Then i define the meam acceleration. In Section 1.3 i show how can Schrödinger equation can be deduced once dynamics, by means of the second Newton s law or by variational methods, are applied to a wide class of Markov diffusion processes. 1.2 Kinematics in stochastic mechanics Velocities In this section we define the probabilistic counterpart of the typical kinematic quantities of deterministic mechanics, i.e. position, velocities and acceleration. We will essentially follow the approach given by Nelson in [49, 50]. As the position, we will consider an adapted stochastic process (X t ) t I, defined in filtered probability space (Ω, F, P, (F t ) t I ), where I R is an interval, that takes value in R n and such that 1. E{ X t } <, i.e. X t L 1 (Ω), t I. 2. the function t X t L 1 (Ω) is continuous. Then we define the mean forward derivative (or mean forward velocity) as the process DX t := 1 lim t 0 + t E {X t+ t X t F t } (1.2.1) when this limit exists in L 1 (Ω) for every t I. Some explanations occur for this definition. Since some basilar processes in probability like Wiener processes have high irregular patterns, even if they are continuous almost everywhere, in defining the concept of time derivation for stochastic processes it would be too restrictive to limit to a process with

11 1.2 - Kinematics in stochastic mechanics 11 smooth paths, in order to perform a classical derivation for each trajectory t X t (ω), at varying of ω Ω. On the other hand, a definition of the type DX t := lim E{X t+ t X t }/ t would be too coarse and we will eventually loose a lot of useful information. Indeed, in this case, the derivative would be a simple number, instead of a process, and we would loss the fluctuation component of it. So we need something between these two options and equation (1.2.1) is what we need. In fact, if for simplicity we suppose that (X t ) t I is a Markov process, then for t I { } DX t = lim E (Xt+ t X t ) t 0 + t X t and given x R n, with lim t 0 + E{(X t+ t X t )/ t X t = x} we perform a limit in mean of (X t+ t X t )/ t for t which goes to 0 +, but just for trajectories that pass at x at time t. Remarks i) Definition (1.2.1) is not uncommon in probability theory and stochastic processes: if (X t ) t 0 is a time-homogeneous Markov process, and if (A, D(A)) is the infinitesimal generator 1 associated to (X t ) t 0, then for every function f D(A), the process [f(x t )] t 0 admits a mean forward derivation and Df(X t ) = [Af](X t ) However it must not to be confused with Malliavin derivation [46], which is another type of derivation in probability. ii) Every martingale (M t ) t 0 w.r.t. (F t ) t 0 admits a mean forward derivation and DM t = 0 iii) If (X t ) t I admits a L 1 derivation, i.e. exists in L 1 (Ω) the limit 1 lim t 0 + t E{X t+ t X t } (1.2.2) then (X t ) t I admits a forward mean derivation which is equal to (1.2.2). 1 see Chap. 4 for a brief review about infinitesimal generators associated to Markov processes

12 12 Chap. 1 - Stochastic mechanics Notice that from the mean forward derivation (DX t ) t I we can reconstruct the original process (X t ) t I, at least of a martingale. Indeed, given a, b I, we can prove [49] that { b } E {X b X a F a } = E DX s ds F a a So if t 0 is the lower extreme of I, defined (1.2.3) we have t M t = X t DX r dr t 0 t s E {M t F s } = E{X t DX r dr F s } = X s t 0 DX r dr = M s t 0 We have said that conditional expectation smooths the irregularities of the patterns, and so that we can perform a derivation. However this isn t enough to neglect a distinction between right and left derivatives; so, together with the mean forward derivative, that essentially is a right derivative, we have to consider the mean backward derivative, i.e. the left side counterpart. In order to do this, we need to define another family of σ-algebras F t F, at varying of t I, such that F s F t, if s t So the family ( F t ) t I is a backward filtration, and, as the element of F t can be interpreted as the events that have occurred in the past, until time t, now the elements of Ft may be considered the events that will happen in the future beyond time t. So, given a process (X t ) t I which satisfies 1 and 2, but adapted to ( F t ) t I, then the mean backward derivative or velocity DX t is defined as DX t := 1 lim t 0 + t E{X t X t t F t } (1.2.4) with the limit taken in L 1 (Ω) as before. Obviously an analogue of (1.2.3) is true even for the mean backward derivation. Other velocities can be defined. If (X t ) t I is a process that admits forward and backward mean velocities, we can take the arithmetic mean of DX t and DX t, and define a new velocity, called the current velocity v t

13 1.2 - Kinematics in stochastic mechanics 13 v := DX t + DX t 2 while if we take the difference, divided by 2, we get the so called osmotic velocity u t u := DX t DX t 2 In a confrontation with deterministic mechanics, current velocity takes the role of ordinary velocity; for example it changes sign for time reversion. On the other hand the osmotic velocity is essentially specific for stochastic mechanics; it is meaningless in ordinary mechanics where it will usually be zero. Contrary to current velocity, the osmotic velocity doesn t change the sign in time reversion. Obviously, there is a strict correlation between forward and backward stochastic mean velocity; first, taken (X t ) t I a Markov process, if we consider the time-reversed process ( X t ) t Ĩ, where t Ĩ if t I and X t = X t, which is still Markov, then D X 1 t = lim t 0 + t E{ X t+ t X t X t } 1 = lim t 0 + t E{X t t X t X t } = [ DX] t So, a part some minus sign, the forward mean velocity of a Markov process is the backward mean velocity of its time-reversed process, and vice-versa. Besides this obvious relation, there is the following theorem which is very important: Theorem Let (X t ) t I and (Y t ) t I be two processes defined on the same filtered probability space (Ω, F, P, (F t ) t I ) and continuous in L 2 (Ω). Suppose moreover that DX t and DY t both exist and are continuous in L 2 (Ω). Then for every a, b I, a b, we have E{X b Y b X a Y a } = b a E{Y t DX t + X t DYt }dt. (1.2.5) Proof. To prove (1.2.5), divide [a, b] in m subinterval of length δ = (b a)/m. Then defined t 0, t 1,..., t m with t 0 = a, t i = δ + t i 1 and t m = b we have that

14 14 Chap. 1 - Stochastic mechanics m 1 lim m i=1 m 1 lim m i=1 m 1 E{X b Y b } E{X a Y a } = lim E{X ti+1 Y ti X ti Y ti 1 } = m { E (X ti+1 X ti ) Y t i + Y ti 1 2 { Xti+1 X ti Y ti + Y E ti 1 t i+1 t i 2 i=1 + X t i+1 + X ti (Y ti Y ti 1 ) 2 + X t i+1 + X ti 2 b a } = } Y ti Y ti 1 δ = t i+1 t i E{Y t DX t + X t DYt }dt There is another integration by part formula; however, in order to show it, we have to introduce the quadratic variation processes and the quadratic covariation processes. Let (X t ) t I a stochastic process in a filtered probability space (Ω, F, P, (F t ) t I ). Then, if t 0 is the lower extreme of I, we define the quadratic variation process ([X, X] t ) t I as a matrix process with components k 1 [X i, X j ] t = Xt lim (X i δ 0 + t l+1 Xt i l )(X j t l+1 X j t l ) (1.2.6) l=0 when the limit exists in L 1 (Ω) Here t 0 t 1... t k = t realize a partition of [t 0, t], while δ = max i {t i+1 t i }. From (1.2.6) one can verify that in one dimension [X, X] is an increasing process P-a.s., so it can be used as integrator for a Lebesgue-Stieltjes integral. We note that in literature the differential term d[x, X] t is often indicated with the symbol (dx t ) 2, which makes clear the choice of the name. As a simple example, in almost every books on stochastic processes is proved that for a one dimensional Wiener process (W t ) t 0 [W, W ] t = t while if (N t ) t 0 is a Poisson process, then [N, N] t = N t If (Y t ) t I is another stochastic process in (Ω, F, P, (F t ) t I ), then one can define the quadratic covariation process between (X t ) t I and (Y t ) t I as

15 1.2 - Kinematics in stochastic mechanics 15 k 1 [X i, Y j ] t = X a0 Y a0 + lim (X i δ 0 + t l+1 Xt i l )(Y j t l+1 Y j t l ) (1.2.7) l=0 From (1.2.6) and (1.2.7) one can verify easily that [X, Y ] t = [Y, X] t and [X, Y ] t = 1 2 ([X + Y, X + Y ] t [X, X] t [Y, Y ] t ) So, being the difference of two increasing process, the quadratic covariation is a finite variation process, and it can be also used as an integrator function for Lebesgue-Stieltjes integrals. d[x, Y ] t is often denoted as dx t dy t. For a more detailed discussion about quadratic variations and its properties, see Protter [54] for example. At this point, given the hypothesis of Theorem 1.2.2, and supposing that X and Y admits an integrable quadratic covariation in [a, b], we have that E{X b Y b X a Y a } = b a E{Y t DX t + X t DY t }dt + E{[X, Y ] b [X, Y ] a } (1.2.8) The proof of (1.2.8) follows easily by the proof of the Theorem by noting that (X ti+1 Y ti+1 X ti Y ti ) = X ti (Y ti+1 Y ti ) + Y ti (X ti+1 X ti ) + (Y ti+1 Y ti )(X ti+1 X ti ) Then by (1.2.5) and (1.2.8) we obtain that b a E{X t DYt }dt = b a E{X t DY t }dt + E{[X, Y ] b [X, Y ] a } (1.2.9) Before going into the definition of the acceleration, we want to explicit these relations between forward and backward velocities for an important class of stochastic processes, the smooth Markovian diffusion processes, that we will analyze in the following paragraph. In the following we will see an explicit expression for u t, for some particular diffusion processes, that will make clear the name of it.

16 16 Chap. 1 - Stochastic mechanics Diffusions We define a smooth Markov diffusion as follow: given a probability space (Ω, F, P), an a.s. continuous Markov process (X t ) t I, with value in R n, is a smooth diffusion with coefficient σ ij (x, t), b i (x, t) and b i (x, t) for 1 i, j n, if the mean forward and backward velocities of (X t ) t I exist and DX t = b(x t, t), DXt = b(x t, t) (1.2.10) d[x, X] t = σ(x t, t)dt P a.s. (1.2.11) Here σ ij, b i and b i are all smooth functions, σ ij is bounded and is the entries of a strictly positive definite square matrix. Essentially by Lévy theorem 2 it can be shown that they are solutions of SDE of the type dx t = b(x t, t)dt + σ 1 2 (Xt, t) dw t (1.2.12) where (W t ) t I is a Wiener process in R n, w.r.t. the filtration (F t ) t I, with F t := σ{x s : s I, s t}, and σ 1 2 is a square matrix function such that for every x R n and t I Besides (X t ) t I satisfies σ 1 2 (x, t) σ 1 2 (x, t) = σ(x, t) dx t = b(x t, t)dt + σ 1 2 (Xt, t)d W t where now ( W t ) t I is a reversed Wiener process, i.e. a Wiener process w.r.t. the backward filtration ( F t ) t I, where F t := σ{x u : u I, u t}. Notice the following important result due to Föllmer [28], which states when a solution of the stochastic differential equation dx t = b(x t, t)dt + dw t (1.2.13) actually is a diffusion in the sense given above (namely, when a process X t, solution of (1.2.13) admits an integrable mean backward derivative, for a given 2 Lévy theorem states that a stochastic process (X t) t 0 actually is a standard Wiener process if and only if it is a continuous local martingale with [X, X] t = t.

17 1.2 - Kinematics in stochastic mechanics 17 function b). It results that if the process (b(x t, t)) t I satisfies the following relation { } E b(x t, t) 2 dt < I then (X t ) t I admits a mean backward derivation, and it is a diffusion in the sense given above 3. Now given f C 0 (Rn I), i.e. a infinite derivable function with compact support, by Itô s formula for continuous processes f(x t ) f(x t0 ) = t 0 f(x s ) dx s t 0 2 i,jf(x s )d[x i, X j ] s (1.2.14) repeated indexes stay for a sum and by the well known properties of quadratic covariation [54], we have Df(X t, t) = [ t + b(x t, t) + 1 ] 2 σij (X t, t) i,j 2 f(x t, t) (1.2.15) and Df((X t, t)) = [ t + b(x t, t) 1 ] 2 σij (X t, t) i,j 2 f(x t, t) (1.2.16) The operators b i i σij 2 i,j, bi i 1 2 σij 2 i,j are called respectively the forward and backward diffusion operators. They give insight about the time evolution of ρ(dx, t), the distribution function of X t. Let us for simplicity suppose that σ ij = νδ ij, with ν > 0. For every function f C0 (Rn I) i.e. a function with compact support in I for the time variable t by applying (1.2.5) for g = 1, we have d 0 = I dt E {f(x t, t)} dt = E {Df(X t, t)} dt I and by (1.2.15) 3 A more detailed discussion about Föllmer s results will be given in Remark

18 18 Chap. 1 - Stochastic mechanics 0 = I E {[ t + b + ν ] } 2 f(x t, t) dt = R n I [ t + b + ν 2 ] f(x, t) ρ(dx, t)dt By integrating by part we obtain that the measure ρ is a weak solution of the following equation t ρ = 1 ρ (bρ) (1.2.17) 2 called the forward Fokker-Planck equation. Of course ρ will also satisfy a similar equation for the backward process. This is the backward Fokker-Planck equation: t ρ = ν 2 ρ ( bρ) (1.2.18) (1.2.17) is a parabolic equation, with coefficient b which is a smooth function. From the theory of parabolic equations, (see [25] for example), we know that (1.2.17) admits as solutions probability measures which are absolutely continuous, w.r.t. the Lebesgue measure, for every t I, and their density functions, that we call ρ(x, t), are smooth both in x and t. It can be shown that equations analogous to (1.2.17) and (1.2.18) are satisfied also by the forward and the backward transition probability functions respectively, p(x, s; dy, t) and p(y, t; dx, s) associated to (X t ) t I. We remember that given s, t I, s t and x, y R n, for a Borel set B p(x, s; B, t) := P(X t B X s = x), p(y, t; B, s) = P(X s B X t = y) Now, using the results of the previous section, we will explicit the relation between the coefficients b i and b i, for the diffusion (X t ) t I. Let us take two functions f, g C 0 (Rn ). Then it s easy to prove that the processes (f(x t )) t 0 and (g(x t )) t 0 verify the hypothesis of Theorem We can apply (1.2.9) and we obtain I E{f(X t ) Dg(X t )}dt = I E{f(X t )Dg(X t )}dt + E { I d[f(x), g(x)] t } (1.2.19)

19 1.2 - Kinematics in stochastic mechanics 19 Then from Itô formula (1.2.14), from (1.2.11) and the properties of quadratic covariation, by (1.2.19) we get I E{f(X t ) Dg(X t )}dt = that we can rewrite as I E{f(X t )Dg(X t )}dt + ν I E{ f(x t ) g(x t )}dt E{f(X t ) Dg(X t )} = E{f(X t )Dg(X t )} + νe{ f(x t ) g(x t )} We make now use of (1.2.15) and (1.2.16) for Dg and Dg. We have ν ] f [ b g R n 2 g ρdx = f [b g + ν ] R n 2 g ρdx + ν f g ρdx R n (1.2.20) But the last integral on the right of (1.2.20), on integration by parts, is equal to ν R n f [ g + ρ ] ρ g ρdx where we set ρ/ρ = 0 if ρ = 0. Then we can rewrite (1.2.20) as R n f ν ] [( [ b g 2 g ρdx = f b ν ρ ) g ν ] R n ρ 2 g ρdx and finally, taking into account that f and g are arbitrary, we obtain b = b ν ρ ρ (1.2.21) This is the relations between forward and backward velocities that we are searching for. By (1.2.21), for the osmotic velocity we have u = b b 2 = ν 2 ρ ρ = ν log ρ 1 2 (1.2.22) So the presence of the typical osmotic pressure term ρ/ρ explains the name for u. About the current velocity v we have

20 20 Chap. 1 - Stochastic mechanics v = b + b 2 = b ν 2 ρ ρ (1.2.23) Note that, by summing forward and backward Fokker-Plank equations (1.2.17) and (1.2.18), and taking into account (1.2.23), we get t ρ = (vρ) (1.2.24) So ρ satisfies also a continuity equation and the role played by the current velocity density is given by v. This explain the terminology for v Acceleration The last quantity to define for completing the kinematics for stochastic processes is the mean acceleration. It is defined by Nelson [50] as a := D DX t + DDX t 2 (1.2.25) Actually other consistent possible choices for a are possible 4. However this definition is justified from variational methods (see [32, 50, 62, 63, 65, 66, 67]). For a diffusion process, supposing b and ρ smooth enough, by (1.2.15) and (1.2.16), a reads a = D b + Db 2 = t ( b + b 2 ) (b b + b b) ( b b 2 ) and making use of current and osmotic velocity v and u we get a = t v + v v u u u (1.2.26) 1.3 Dynamics in stochastic mechanics Once obtained the kinematics, now we can turn to the dynamics. As in deterministic mechanics, dynamics can be defined in stochastic mechanics in two ways: by the second Newton s law and by a variational method. I will show only the first one. About the variational methos in stochastic mechanics, see [32, 48, 62, 63, 66, 67]. 4 See Zambrini [67] about the choice (DDX t + D DX t)/2 for the definition of mean acceleration.

21 1.3 - Dynamics in stochastic mechanics 21 Let us suppose that a mechanical system is represented by a process (X t ) t 0 defined on a probability space (Ω, F, P), that takes value in R n, and adapted with respect to a forward and backward filtration. So X t gives the coordinates of the system at time t. If the system is subject to a field of forces given by a function F, dynamical principle in stochastic mechanics imposes that F (X t, t) = ma = m D DX t + DDX t (1.3.1) 2 where m is the mass of the system. Let us see what are the conseguences of relation (1.3.1). For simplicity suppose that F is a conservative force, i.e. that there exists a smooth function V such that F = V. We apply then (1.2.26) in (1.3.1) so that t v = V m + u u v v + u (1.3.2) So the dynamics equation (1.3.1) reduces to an equation for the current and osmotic velocities, that can be solved only if coupled with this other equation t u = ( v) (v u) (1.3.3) that can be derived by (1.2.22) and by the continuity equation (1.2.24). So at the question what is the diffusion process, for a given initial distribution and initial forward drift, with its mean acceleration fixed by the dynamics equation (1.3.1)?, one can answer by solving equations (1.3.2) and (1.3.3) for given initial conditions of u and v, which by (1.2.22) and (1.2.23) are determined by the initial values of ρ and b. Then by the relation b(x, t) = v(x, t) + u(x, t) which easily follows from the definitions of u and v, one can find the expression for the forward drift b for every time t I. At this point the stochastic differential equation dx t = b(x t, t)dt + dw t is defined, and its solution 5 is the diffusion searched. The couple of equations (1.3.2) and (1.3.3) is non linear, so it could be a difficult task to resolve them. However we will now show how to replace them 5 Actually a weak solution, (see Section for details), since no Wiener process is initially given.

22 22 Chap. 1 - Stochastic mechanics with a singular linear complex one. We have seen in (1.2.22) that the osmotic velocity u can be written as the gradient of a function R u = ρ 2ρ = log ρ1/2 = R (1.3.4) with R = log ρ 1/2. If we suppose 6 that even v can be written as a gradient of a smooth function S, i.e. then we have that v = S (1.3.5) and (1.3.2) and (1.3.3) become u u = 1 2 ( R 2 ) v v = 1 2 ( S 2 ) t S = V ( R 2 S 2 ) + R (1.3.6) t R = S S R (1.3.7) At this point, if we define the complex function ψ = e R+iS (1.3.8) it is straightforward to see that ψ verifies a complex partial differential equation, namely the Schrödinger equation i t ψ = ψ + V ψ (1.3.9) To see this, just substitute (1.3.8) into (1.3.9), divide by ψ itself and then take the real and imaginary part of it; they will correspond respectively to (1.3.6) and (1.3.7). The potential term V that appear in (1.3.9) is the same potential that gives rise to the field of forces F in (1.3.1). We note that ψ 2 = ρ from (1.3.8) and (1.3.4), while from (1.3.4) and (1.3.5) we have 6 Relation (1.3.5 actually can be deduce by variational principles.

23 1.3 - Dynamics in stochastic mechanics 23 b = (R + S) = R ψ ψ, b = (S R) = I ψ ψ

24 24 Chap. 1 - Stochastic mechanics

25 Chapter 2 Lévy processes 2.1 Introduction In this chapter we give a brief review about Lévy processes, focusing on their properties that will be of some utilities in the following. We will start with their definition, and with the definition of the associated Lévy measures. Then we will describe the infinite divisible distributions, in particular the stable and self-decomposable ones, and we will show the one-to-one correspondence between Lévy processes and these distributions. After some example of Lévy processes, we give the Lévy-Khintchine formula and the strictly related Lévy- Itô decomposition, which decompose a Lévy process in a sum of two independent processes, one with continuous path and the other with pure-jump path. Then, after remarking that Lévy processes are Markov processes, we focus our attention on the semigroups and the infinitesimal generators associated to them, in particular for the symmetric ones. Finally we give a brief review of semimartingales (every Lévy process is a semimartingale) defining their Itô integrals and the relative SDEs. 2.2 Definitions Definition Let (Ω, F, (F t ) t 0, P) a filtered complete probability space. A process (Z t ) t 0 on R n adapted to (F t ) t 0 is said a Lévy process if 1. Z 0 = 0 a.s. 2. Z t Z s is independent from F s, for 0 s t.

26 26 Chap. 2 - Lévy processes 3. Z t+h Z t = Z s+h Z s in distribution, for every 0 s, t, h. 4. Z t is stochastically continuous, i.e. Z t+ t Z t 0 in probability as t 0. Remark A process Z t satisfying just conditions 1-3 is said an additive process. It also possible to define a Lévy process without involving a filtration. Definition Let (Ω, F, P) a complete probability space. Then a process (Z t ) t 0 is said an intrinsic Lévy process if 1. Z 0 = 0 a.s. 2. Z t Z s is independent from Z v Z u, for 0 s t, 0 u v, and if [s, t] [u, v] =. 3. Z t+h Z t = Z s+h Z s in distribution, for every 0 s, t, h. 4. Z t is stochastically continuous, i.e. Z t+ t Z t 0 in probability as t 0. It is straightforward to see that an intrinsic Lévy process is also a Lévy process; in fact if (Z t ) t 0 is an intrinsic Lévy process, then given the filtrations F t := {Z s : 0 s t} and F t := F t N, where N is the σ-algebre of the sets of measure zero w.r.t. P, it results that (Z t ) t 0 is also a Lévy process w.r.t. both F t and F t. Remarks The filtration (F t ) t 0 is often called the natural filtration associated to (Z t ) t 0. Ft is the σ-algebra generated by F t when it is completed with the sets of measure zero. It can be proves that the filtration ( F t ) t 0 is right continuous, i.e. Ft = u>t F u. A right continuous filtration is very common, for example the natural filtrations associated to a large class of Markov process (once completed with the sets of measure zero) are of this type; besides, when a filtration is right continuous a lot of useful properties are satisfied. For this reason in the following we will tacitly assume that the filtrations contains all the sets of measure zero and are right continuous. From this definitions it s possible to prove [54, 59] that for every Lévy process Z t there is a version 1 of it which has cad lag paths, i.e. trajectories right continuous with left limits, P-a.s., and which is still a Lévy process. In the following we always assume that all Lévy processes are cad-lag. 1 A process (Z t) t 0 is a version of another process (Z t) t 0 if Z t = Z t P-a.s. for every t 0

27 2.3. LÉVY MEASURES Lévy measures Since Lévy processes have in general non-continuous paths, it would be interesting to study the distribution of their jumps, i.e. the discontinuities of their trajectories. These information can be obtained by particular measures derived from Lévy processes, called Lévy measures. Actually a Lévy measure can be defined independently from a Lévy process; let l be a positive measure on the measurable space 2 (R n 0, B(Rn 0 )), for n N 0. Then l will be called a Lévy measure if ( y 2 1 ) l(dy) < (2.3.1) R n 0 (a b := min{a, b}). Lévy measures are directly related to the Lévy processes first of all because they are connected to the average number of jumps performed by the process in a unit time interval. Let (Ω, F, P) be a probability space, (X t ) t 0 a generic cad-lag process with value in R n and ( t X) t 0 the jump-process associated to X t, i.e. t X := X t X t where X t := lim s t X s For a given Λ B(R n ) with 0 / Λ, we can define a new process Nt Λ = 0 and for t > 0 N Λ 0 so defined: N Λ t (ω) := # of jumps in (0, t] with size s X(ω) Λ (2.3.2) Another way to write the definition of Nt Λ is given by the following formula = 1 Λ ( s X) (2.3.3) N Λ t 0<s t where the sum is intended for the instants of time in which there is a jump of X t. Since Nt Λ is cad-lag, (2.3.2) and (2.3.3) are correctly defined because then, in a finite time interval, the number of jumps greater than an arbitrary ε > 0 in absolute value is always finite [38]. Nt Λ has the following properties: N0 Λ = 0 as said, it is positive, increasing and takes integer values. For these reasons it is called a counting process. Besides, in the case of a Lévy processes we have the following proposition: 2 With the symbol R n 0 or N 0 we indicate respectively the sets R n and N without the element zero.

28 28 Chap. 2 - Lévy processes Proposition If X t = Z t is a Lévy process, then the process Nt Λ Poisson process with Poisson parameter E{N1 Λ}. is a Proof. See [54] The Poisson parameter E{N1 Λ } shows up to be an additive set function of Λ which can be extended to a measure on the measurable space (R n 0, B(Rn 0 )). It can be proved that this measure satisfies (2.3.1) and then it is a Lévy measure l that we consider as the Lévy measure associated to Z t. Remarks Toghether with l we can define other measures on the space (R n 0, B(Rn 0 )); for example, for a fixed ω Ω, t 0, at varying of Λ B(R n ), 0 / Λ, the function N t (ω; ) : Λ N Λ t (ω) R (2.3.4) is another additive set function that can be extended to a measure on the space (R n 0, B(Rn 0 )). Other measures, but on (Rn 0 R +, B(R n 0 R +)), are derived for a fixed ω Ω from the functions N(ω;, ) : Λ (0, t] N Λ t (ω) R (2.3.5) These measures are called random measures because of their dependence from ω. In the following we will indicate them as N t and N, omitting their dependence from ω in order to simplify the notation, when no ambiguity can occur. Note that for positive and measurable functions f in R n, and Λ B(R n ) with 0 / Λ we can write f(y)n t (dy) = f(z s )1 Λ ( s Z) (2.3.6) Λ 0<s t while if f is a B(R n R + ) measurable function, then for the same Λ as above t 0 Λ f(y, s)n(dy, ds) = 0<s t f(z s, s)1 Λ ( s Z) (2.3.7) Since N Λ t is a Poisson process with parameter E{N1 Λ } = l(λ), we have E{N Λ t } = tl(λ) E{[N Λ t tl(λ)] 2 } = tl(λ)

29 2.3 - Lévy measures 29 that, in view of (2.3.3) and of (2.3.6), can be rewritten as { E 0<s t { } } 1 Λ ( s Z) = E 1 Λ (y)n t (dy) = t 1 Λ (y)l(dy) R n 0 R n 0 and [ E R n 0 ] 2 1 Λ (y)n t (dy) t 1 Λ (y)l(dy) R n = t 1 Λ (y)l(dy) 0 R n 0 These relations extend naturally from 1 Λ to every measurable function f in R n satisfying some mild conditions: Proposition For a Borel subset Λ with 0 / Λ and a function f on R n such that f1 Λ L 1 (R n 0, l(dy)) we have { E 0<s t f(z s )1 { Zs Λ} } { } = E f(y)n t (dy) = t f(y)l(dy) (2.3.8) Λ R n 0 and if f1 Λ L 2 (R n 0, l(dy)), then [ ] 2 E f(y)n t (dy) t f(y)l(dy) R n = t f 2 (y)l(dy) (2.3.9) 0 R n 0 R n 0 Proof. For a complete proof see [54], Chap. I. Another important property about Lévy measures is, given α > 0, that E{ Z t α } <, t 0 y α l(dy) < (2.3.10) y 1 when l is the Lévy measure associated to Z t [59]. As a corollary of this last property we have that if Z t has bounded jumps, then it has finite moments of any order. In this contest it is useful to note that by subtracting jumps higher than a certain quantity from a Lévy process one still obtain a Lévy process, as the poiint 2 of the following theorem shows: Theorem Let (Z t ) t 0 a Lévy process, and for Λ B(R n ), 0 / Λ, let (N Λ t ) t 0 the associted Poisson counting process. Then we have

30 30 Chap. 2 - Lévy processes 1. For every function f with finite value in Λ, the process t Λ f(y) N Λ t (dy) is a Lévy process. 2. The process t Z t Λ y N Λ t (dy), i.e. the process (Z t ) t 0 without the jumps with size in Λ, is still a Lévy process. Proof. See [54], Chap. I. 2.4 Lévy-Itô decomposition Every Lévy process can be decompose into the sum of a Wiener and a pure jump process, the last one being a superposition of compensated Poisson processes 3. Besides these two processes are independent. Theorem (Lévy-Itô decomposition). Let (Z t ) t 0 a Lévy process. Then the following decomposition holds Z t = W t + y[n t (dy) tl(dy)] + y 1 { te Z 1 y >1 yn 1 (dy) = W t + y[n t (dy) tl(dy)] + y 1 αt + 0<s t } + yn t (dy) y >1 Z s 1 { Zs >1} (2.4.1) where (W t ) t 0 is a Brownian motion, α = E{Z 1 yn 1 (dy)} and, for any set Λ, 0 / Λ the Poisson process (N Λ t ) t 0 is independent from (W t ) t 0. Besides N Λ t is independent from N Γ t if Λ and Γ are disjoint. Proof. For a complete proof see [59], Chap. IV. We just note that for point 2 of Theorem and for (2.3.10), the random variable Z 1 Y 1 y N t Λ (dy) is integrable. As a corollary of Theorem we have that the only Lévy process with continuous path is the Wiener process plus eventually a deterministic drift. Another theorem strictly connected with the Lévy-Itô decomposition is this: 3 If X t is a Poisson process with parameter λ, then the process X t tλ is a martingale and it is called compensated Poisson process.

31 2.5 - Infinite divisible distributions 31 Theorem Every Lévy process (Z t ) t 0 can be decompose in a sum Z t = M t + A t, where (M t ) t 0 and (A t ) t 0 are both Lévy processes; (M t ) t 0 is a martingale with bounded jumps, M t L p (Ω), p 1, while (A t ) t 0 is an adapted process with path of finite variation on compacts. Proof. See [54] for a detailed proof. Obviously from (2.4.1) we will have M t := W t + y[n t (dy) tl(dy)], A t := αt + yn t (dy) y 0 y >1 Lévy-Itô decomposition is essentially equivalent to the Lévy-Khintchine formula, that give the explicit expression of the characteristic function ˆν t (u) := E{e iuzt } (2.4.2) of a Lévy process (Z t ) t 0 with distribution ν t of Z t. We will see this in the next section in connection with the infinite divisible distributions. 2.5 Infinite divisible distributions A way to construct Lévy processes is starting from the infinite divisible distributions (idd), i.e. those distribution functions ν on R n such that for every m N there exists a new distribution ν m satisfying the following relation ν = ν m m := ν m... ν m }{{} m times (2.5.1) where the symbol stays for the convolution product between measures. There is in fact a one-to one correspondence between infinite divisible distributions and Lèvy processes laws. This relation can be better seen if we consider the characteristic functions associated to distributions and defined as [61] ˆν : z R n e iz x ν(dx) R n When ν is an idd, then it s straightforward to see from (2.5.1) that for every n N the function ˆν 1/n is still a characteristic function; actually this is true for every t R + in place of 1/n[59]. Then the link between Lévy processes and idd is given by the following proposition.

32 32 Chap. 2 - Lévy processes Proposition There is a one-to one correspondence between idd s and Lévy processes; in particular if (Z t ) t 0 is a Lévy process on R n, then the distribution function of Z t is an idd, for every t 0, and called ˆν t the characteristic functions associated to Z t we have ˆν t = ˆν t 1 (2.5.2) On the other side, if ν is an idd on R n, there is a unique, up to identity in law 4 Lévy process (Z t ) t 0 such that Z 1 has ν as a distribution function. Proof. A complete proof of this proposition can be found in [59], Chap. II. Here we limited to a sketch of it. Let (Z t ) t 0 be a Lévy process and t 0; then for every m N, fixed h = t/m, we can write m 1 Z t = (Z h(k+1) Z hk ) k=0 So we can express Z t as a sum of m independent and identical distributed random variables, respectively for properties 3 and 4 of definition (2.2.1). So by (2.4.2) the characteristic function ˆν t of Z t will be given by ˆν t = ˆν m t/m (2.5.3) with ˆν t/m the characteristic function of Z t/m = Z h. This is enough to prove that the distribution function of Z t is infinitely divisible. Besides, from(2.5.3), for t Q +, i.e. t = n/m with n and m positive integers, we have for every m N ˆν t = ˆν n/m = ˆν 1/m n = (ˆν n 1 ) 1/m = ˆν t 1 Then, by means of stochastic continuity, it s possible to extend the above relation for every t 0. As for the reversed implication, given an idd ν, we first define the following family of probability measures: P 0 : B B(R n ) δ 0 (B) P t1,...,t m : B 1... B m B(R mn ) dν (tm t m 1) dν t 1 B 1... B m 4 Two processes (X t) t 0 and (Y t) t 0 are said identical in law if for every m N 0 and for every 0 t 1... t m the random vectors (X t1,..., X tm ) and (Y t1,..., Y tm ) have the same distribution. Denoting the two processes with X and Y, the identity in law is often indicated as X d = Y.

33 2.5 - Infinite divisible distributions 33 for every m N and 0 t 1... t m, where δ 0 is the Dirac measure while with ν t we intend the distribution function associated to the characteristic function ˆν t. Then by the Kolmogorov extension theorem [38] from these measures can be defined a probability measure P on the measurable space ((R n ) [0, ), B((R n ) [0, ) )), and in this probability space the coordinate process 5 is a Lévy process with P Z1 = ν. Idd s, and consequently Lévy processes, are very well characterized. For their specification few parameters are needed, as the following Lévy-Khintchine formula makes clear: Theorem Let ν be an idd on R n. Then ˆν(z) = exp { 12 z az + iγ z + [ e iz y 1 iz y1 { y 1} (y) ] } l(dy) R n 0 (2.5.4) where a is a symmetric nonnegative-definite n n matrix, γ R n and l is a Lévy measure. This representation of ˆν by a, l and γ is unique. Besides, if a is a symmetric nonnegative-definite n n matrix, γ R n and l is a Lévy measure, then (2.5.4) is the characteristic function of an idd. Proof. See [59], pag. 37. Definition We call (a, l, γ) the generating triplet of ν, or of (Z t ) t 0, the Lévy process associated to ν. Remarks The function 1 y 1 in (2.5.4) can be modified in different ways. For example by any measurable and bounded function c : R n R such that c(y) = 1 + o( y ) as y 0 c(y) = O(1/ y ) as y Then (2.5.4) is rewritten as { ˆν(z) = exp 1 2 z az + iγ c z + R n 0 [ e iz y 1 iz yc(y) ] l(dy) 5 The coordinate process (Z t) t 0 on ((R n ) [0, ), B((R n ) [0, ) )) is a process that to every function x (R n ) [0, ), i.e. to every associates Z t(x) := x(t). x : t [0, ) x(t) R n }

34 34 Chap. 2 - Lévy processes where γ c := γ + y[c(y) 1 { y 1} (y)] l(dy) R n 0 In this case the new generating triplet will be indicated with (a, l, γ c ) c. Actually, if l is symmetric, i.e. l(b) = l( B) for every set B B(R n 0 ), then (2.5.4) can be rewritten as ˆν(z) = exp { 12 } z az + iγ z + [cos(z y) 1] l(dy) Stable and self-decomposable distributions There are two particular classes of idd s, the stable and the self-decomposable ones. We define them now. Definition Let ν a distribution function on R n. ν is said stable if for every a > 0 there exists b > 0 such that. R n 0 ˆν(az) = ˆν(z) b, z R n (2.5.5) Remark Stable distributions are idd s. The corresponding Lévy processes are called stable Lévy processes. Examples of stable processes are given, as one can trivially verify, by the Wiener and the Cauchy processes. The Cauchy process with parameters γ R n and c > 0 is the process generated by the following idd in R n : ( ) c n + 1 ν(b) := π (n+1)/2 Γ 2 B 1 ( x γ 2 + c 2 ) (n+1)/2 dx B B(Rn ) where Γ is the Euler function. For the characteristic function we have ˆν(z) = e c z +iγ z If (Z t ) t 0 is a stable process, then it s easy to see that for every a > 0 there exists b > 0 such that Z at d = bzt (2.5.6) for every t 0. A process that verifies relation (2.5.6) is called self-similar; however stability and self-similarity are not exactly the same concept, since

35 2.5 - Infinite divisible distributions 35 there are self-similar processes that are not even Lévy processes (see [59]). Remark also that there is a fixed relation between the coefficient a and b of (2.5.5). In particular it results that exists a coefficient 0 < α 2 such that b = a 1/α α is said Hurst index of the distribution or Hurst index of the process. For a Wiener process α = 2, while for the Cauchy one α = 1. However the value α = 1 is also related to the degenerate Lévy process Z t = vt, with v R n. The Lévy-Khintchin formula for a symmetric and rotationally invariant 6 stable distribution function in R n is very simple: for 0 < α < 2 the generating triplet is (0, l, 0) with the Lévy measure l(dy) = dy y α+n (2.5.7) Remark that here l is absolutely continuous. After some calculation we have for the characteristic function ˆν(z) = e c z α (2.5.8) with c > 0. When α = 2 the distribution function is a Gaussian distribution, so that the generating triplet is (a, 0, 0) and the Lévy-Khintchin formula is ˆν(z) = e z az/2 We consider next the family of the self-decomposable distribution functions. Definition Let ν be a distribution function on R n : ν is said selfdecomposable if for every b 1 there exists a probability measure ρ b on R n such that ˆν(z) = ˆν(b 1 z)ˆρ b (z) where ˆρ b is the characteristic function of ρ b. Remark Every stable distribution function is also self-decomposable: indeed, given b > 1, we have ˆν (z) = ˆν ( bb 1 z ) = [ˆν ( b 1 z )] b 1/α = ˆν ( b 1 z ) ˆρ b (z) 6 A distribution ν in R n is rotationally invariant if ν(b) = ν(ub) for every Borel set B and every orthogonal transformation U in R n.