Lamperti Representation Research Articles

General Self-Similarity Properties for Markov Processes and Exponential Functionals of Lévy Processes

Positive self-similar Markov processes are positive Markov processes that satisfy the scaling property and it is known that they can be represented as the exponential of a time-changed Lévy process via Lamperti representation. In this work, we are interested in what happens if we consider Markov processes in dimension 1 or 2 that satisfy self-similarity properties of a more general form than a scaling property. We characterize them by proving a generalized Lamperti representation. Our results show that, in dimension 1, the classical Lamperti representation only needs to be slightly generalized. However, in dimension 2, our generalized Lamperti representation is much more different and involves the exponential functional of a bivariate Lévy process. We briefly discuss the complications that occur in higher dimensions. We present examples in dimensions 1, 2 and 3 that are built from growth-fragmentation, self-similar fragmentation and Continuous-state Branching processes in Random Environment. Some of our arguments apply in the context of a general state space and show that we can exhibit a topological group structure on the state space of a Markov process that satisfies general self-similarity properties, which allows to write a Lamperti-type representation for this process in terms of a Lévy process on the group.

Discretization of the Lamperti representation of a positive self-similar Markov process

This paper considers discretization of the Lévy process appearing in the Lamperti representation of a strictly positive self-similar Markov process. Limit theorems for the resulting approximation are established under some regularity assumptions on the given Lévy process. Additionally, the scaling limit of a positive self-similar Markov process at small times is provided. Finally, we present an application to simulation of self-similar Lévy processes conditioned to stay positive.

Fluctuation theory for spectrally positive additive Lévy fields

A spectrally positive additive Levy field is a multidimensional field obtained as the sum $\mathbf {X}_{\mathrm {t}}={\mathrm {X}}^{(1)}_{t_{1}}+{\mathrm {X}}^{(2)}_{t_{2}}+\dots +{\mathrm {X}}^{(d)}_{t_{d}}$, ${\mathrm {t}}=(t_{1},\dots ,t_{d})\in \mathbb {R}_{+}^{d}$, where ${\mathrm {X}}^{(j)}={}^{t} (X^{1,j},\dots ,X^{d,j})$, $j=1,\dots ,d$, are $d$ independent $\mathbb {R}^{d}$-valued Levy processes issued from $\mathbf {0}={}^{t}(0,0,\dots ,0)$, such that $X^{i,j}$ is non decreasing for $i\neq j$ and $X^{j,j}$ is spectrally positive. It can also be expressed as $\mathbf {X}_{\mathrm {t}}=\mathbb {X}_{\mathrm {t}}\cdot {\mathbf {1}}$, where ${\mathbf {1}}={}^{t}(1,1,\dots ,1)$ and $\mathbb {X}_{\mathrm {t}}=(X^{i,j}_{t_{j}})_{1\leq i,j\leq d}$. The main interest of spaLf’s lies in the Lamperti representation of multitype continuous state branching processes. In this work, we study the law of the first passage times $\mathbf {T}_{\mathrm {r}}$ of such fields at levels $-{\mathrm {r}}$, where ${\mathrm {r}}\in \mathbb {R}_{+}^{d}$. We prove that the field $\{(\mathbf {T}_{\mathrm {r}},\mathbb {X}_{\mathbf {T}_{\mathrm {r}}}),{\mathrm {r}}\in \mathbb {R}_{+}^{d}\}$ has stationary and independent increments and we describe its law in terms of this of the spaLf $\mathbf {X}$. In particular, the Laplace exponent of $(\mathbf {T}_{\mathrm {r}},\mathbb {X}_{\mathbf {T}_{\mathrm {r}}})$ solves a functional equation leaded by the Laplace exponent of $\mathbf {X}$. This equation extends in higher dimension a classical fluctuation identity satisfied by the Laplace exponents of the ladder processes. Then we give an expression of the distribution of $\{(\mathbf {T}_{\mathrm {r}},\mathbb {X}_{\mathbf {T}_{\mathrm {r}}}),{\mathrm {r}}\in \mathbb {R}_{+}^{d}\}$ in terms of the distribution of $\{\mathbb {X}_{\mathrm {t}},{\mathrm {t}}\in \mathbb {R}_{+}^{d}\}$ by the means of a Kemperman-type formula, well-known for spectrally positive Levy processes.

How fast planar maps get swallowed by a peeling process

The peeling process is an algorithmic procedure that discovers a random planar map step by step. In generic cases such as the UIPT or the UIPQ, it is known [15] that any peeling process will eventually discover the whole map. In this paper we study the probability that the origin is not swallowed by the peeling process until time $n$ and show it decays at least as $n^{-2c/3}$ where \[ c \approx 0.1283123514178324542367448657387285493314266204833984375... \] is defined via an integral equation derived using the Lamperti representation of the spectrally negative $3/2$-stable Lévy process conditioned to remain positive [12] which appears as a scaling limit for the perimeter process. As an application we sharpen the upper bound of the sub-diffusivity exponent for random walk of [4].

On the remarkable Lamperti representation of the inverse local time of a radial Ornstein-Uhlenbeck process

We give a description, in terms of ``pseudo-stable increasing process'', of the Lamperti process associated with the inverse local time of a radial Ornstein-Uhlenbeck process. Following Bertoin-Yor, we also express, in two particular cases, the law of the perpetuity associated with this inverse local time.

A Lamperti-type representation of continuous-state branching processes with immigration

Guided by the relationship between the breadth-first walk of a rooted tree and its sequence of generation sizes, we are able to include immigration in the Lamperti representation of continuous-state branching processes. We provide a representation of continuous-state branching processes with immigration by solving a random ordinary differential equation driven by a pair of independent Levy processes. Stability of the solutions is studied and gives, in particular, limit theorems (of a type previously studied by Grimvall, Kawazu and Watanabe and by Li) and a simulation scheme for continuous-state branching processes with immigration. We further apply our stability analysis to extend Pitman's limit theorem concerning Galton-Watson processes conditioned on total population size to more general offspring laws.

Pricing Asian Options Under a Hyper-Exponential Jump Diffusion Model

We obtain a closed-form solution for the double-Laplace transform of Asian options under the hyper-exponential jump diffusion model. Similar results were available previously only in the special case of the Black-Scholes model (BSM). Even in the case of the BSM, our approach is simpler as we essentially use only Itô's formula and do not need more advanced results such as those of Bessel processes and Lamperti's representation. As a by-product we also show that a well-known recursion relating to Asian options has a unique solution in a probabilistic sense. The double-Laplace transform can be inverted numerically via a two-sided Euler inversion algorithm. Numerical results indicate that our pricing method is fast, stable, and accurate; and it performs well even in the case of low volatilities.

The Lamperti representation of real-valued self-similar Markov processes

In this paper, we obtain a Lamperti type representation for real-valued self-similar Markov processes, killed at their hitting time of zero. Namely, we represent real-valued self-similar Markov processes as time changed multiplicative invariant processes. Doing so, we complete Kiu's work [Stochastic Process. Appl. 10 (1980) 183-191], following some ideas in Chybiryakov [Stochastic Process. Appl. 116 (2006) 857-872] in order to characterize the underlying processes in this representation. We provide some examples where the characteristics of the underlying processes can be computed explicitly.

Proof(s) of the Lamperti representation of continuous-state branching processes

This paper uses two new ingredients, namely stochastic differential equations satisfied by continuous-state branching processes (CSBPs), and a topology under which the Lamperti transformation is continuous, in order to provide self-contained proofs of Lamperti’s 1967 representation of CSBPs in terms of spectrally positive Lévy processes. The first proof is a direct probabilistic proof, and the second one uses approximations by discrete processes, for which the Lamperti representation is evident.

Some explicit identities associated with positive self-similar Markov processes

We consider some special classes of Lévy processes with no gaussian component whose Lévy measure is of the type π ( d x ) = e γ x ν ( e x − 1 ) d x , where ν is the density of the stable Lévy measure and γ is a positive parameter which depends on its characteristics. These processes were introduced in [M. E. Caballero, L. Chaumont, Conditioned stable Lévy processes and the Lamperti representation, J. Appl. Probab. 43 (2006) 967–983] as the underlying Lévy processes in the Lamperti representation of conditioned stable Lévy processes. In this paper, we compute explicitly the law of these Lévy processes at their first exit time from a finite or semi-finite interval, the law of their exponential functional and the first hitting time probability of a pair of points.

The Upper Envelope of Positive Self-Similar Markov Processes

We establish integral tests in connection with laws of the iterated logarithm at 0 and at +∞, for the upper envelope of positive self-similar Markov processes. Our arguments are based on the Lamperti representation and on the study of the upper envelope of the future infimum due to the author (see Pardo in Stoch. Stoch. Rep. 78:123–155, [2006]). These results extend laws of the iterated logarithm for Bessel processes due to Dvoretsky and Erdős (Proceedings of the Second Berkeley Symposium, [1951]) and stable Levy processes with no positive jumps conditioned to stay positive due to Bertoin (Stoch. Process. Appl. 55:91–100, [1995]).

Conditioned stable Lévy processes and the Lamperti representation

By variously killing a stable Lévy process when it leaves the positive half-line, conditioning it to stay positive, and conditioning it to hit 0 continuously, we obtain three different, positive, self-similar Markov processes which illustrate the three classes described by Lamperti (1972). For each of these processes, we explicitly compute the infinitesimal generator and from this deduce the characteristics of the underlying Lévy process in the Lamperti representation. The proof of this result bears on the behaviour at time 0 of stable Lévy processes before their first passage time across level 0, which we describe here. As an application, for a certain class of Lévy processes we give the law of the minimum before an independent exponential time. This provides the explicit form of the spatial Wiener-Hopf factor at a particular point and the value of the ruin probability for this class of Lévy processes.

Conditioned stable Lévy processes and the Lamperti representation

By variously killing a stable Lévy process when it leaves the positive half-line, conditioning it to stay positive, and conditioning it to hit 0 continuously, we obtain three different, positive, self-similar Markov processes which illustrate the three classes described by Lamperti (1972). For each of these processes, we explicitly compute the infinitesimal generator and from this deduce the characteristics of the underlying Lévy process in the Lamperti representation. The proof of this result bears on the behaviour at time 0 of stable Lévy processes before their first passage time across level 0, which we describe here. As an application, for a certain class of Lévy processes we give the law of the minimum before an independent exponential time. This provides the explicit form of the spatial Wiener-Hopf factor at a particular point and the value of the ruin probability for this class of Lévy processes.

On the future infimum of positive self-similar Markov processes†

We establish integral tests and laws of the iterated logarithm for the upper envelope of the future infimum of positive self-similar Markov processes and for increasing self-similar Markov processes at 0 and +∞. Our proofs are based on the Lamperti representation and time reversal arguments due to Chaumont, L. and Pardo, J.C. (Prépublication (L'université de Paris 6), 2005). These results extend laws of the iterated logarithm for the future infimum of Bessel processes due to Khoshnevisan, D., Lewis, T.M. and Li, W.V. (On the future infima of some transient processes, Probability Theory and Related Fields, 99, 337–360, 1994).

The Lower Envelope of Positive Self-Similar Markov Processes

We establish integral tests and laws of the iterated logarithm for the lower envelope of positive self-similar Markov processes at 0 and $+\infty$. Our proofs are based on the Lamperti representation and time reversal arguments. These results extend laws of the iterated logarithm for Bessel processes due to Dvoretzky and Erdos (1951), Motoo (1958), and Rivero (2003).

The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes

We consider the asymptotic behavior of semi-stable Markov processes valued in ]0,∞[ when the starting point tends to 0. The entrance distribution is expressed in terms of the exponential functional of the underlying Levy process which appears in Lamperti's representation of a semi-stable Markov process.

Branching Processes with Immigration and Related Limit Theorems

Branching Processes with Immigration and Related Limit Theorems