Lamperti Transformation in Stochastic Processes

• Lamperti Transformation is a method that converts self-similar processes into Markov processes by using a time-change mechanism, linking scaling properties with stationary increments.
• It extends to real-valued, multidimensional, and Banach-space processes, facilitating analysis in areas such as SDEs, Gaussian fields, and population dynamics.
• The transformation underpins numerical schemes and statistical inference methods, providing robust tools to study state-dependent diffusions and self-similarity exponents.

The Lamperti transformation is a structural operation connecting self-similar processes to Markov processes with stationary increments, or more generally, constant diffusion or multiplicative invariance properties. Originally formulated by J. Lamperti in 1967 for positive self-similar Markov processes, the transformation has since been extended to encompass real- and Banach-space-valued processes, Markov additive processes, Gaussian fields, multidimensional processes, and discrete as well as continuous time. It is central in the probabilistic analysis of scaling limits, population dynamics, stochastic differential equations (SDEs) with state-dependent coefficients, and in statistical inference for self-similarity exponents.

1. Classical One-Dimensional Lamperti Transformation

The classical Lamperti transformation relates positive self-similar Markov processes (pssMp) to Lévy processes via a specific random time-change. For a process $X$ on $(0,\infty)$ that is self-similar of index $\alpha>0$, i.e.,

$$\{c X_{c^{-\alpha} t}: t\ge0\} \stackrel{\mathrm{law}}{=} \{ X_t: t\ge0\} \text{ under } P_{c x},$$

there exists a (possibly killed) Lévy process $\xi$ such that

$$X_t = \exp\big( \xi_{\tau(t)} \big ), \quad \tau(t) = \inf \left\{ s > 0 : \int_0^s \exp(\alpha \xi_u) du > t \right\}.$$

The correspondence is bijective: every pssMp is the exponential of a Lévy process subordinated by the "Lamperti clock," and every real Lévy process yields a pssMp under this transformation (Caballero et al., 2010, Berestycki et al., 2011, Siri-Jégousse et al., 2024).

The Lamperti transform also underlies explicit solutions, extinction behavior, and uniqueness results for jump-type SDEs with self-similar structure, and is a building block for more general constructions via measure changes, h-transforms, and time-inversions.

2. Extensions to Real-Valued, Multitype and Banach-Valued Processes

The transformation generalizes to real-valued self-similar Markov processes (ssMp), possibly with sign changes, via the Lamperti–Kiu transform. A real-valued ssMp $X$ is represented as

$$X_t = \exp(\xi_{\phi^{-1}(t)}) \cdot \mathrm{sgn}(X_{\phi^{-1}(t)}),$$

where $(\xi,J)$ is a Markov additive process (MAP), with $J$ tracking the sign, and $(0,\infty)$0 the log-magnitude. The time-change is as before, but modulo the absolute value for sign changes (Chaumont et al., 2011, Kyprianou, 2015).

In Banach space settings, an analogous scheme is constructed using the norm and a suitable "polar decomposition" mapping $(0,\infty)$1 into (norm, direction) coordinates. The time-change is then governed by powers of the norm, and the additive component is a Markov process on the unit sphere (Siri-Jégousse et al., 2024, Kyprianou et al., 27 Jun 2025, Alili et al., 2016). This provides a canonical mapping between multidimensional self-similar Markov processes and MAPs, with explicit construction in terms of the norm-dependent geometry (e.g., $(0,\infty)$2-norm in the orthant yields a simplex-valued modulator).

For bivariate Markov chains and interacting systems (e.g., population models), the Lamperti transform is defined in terms of power time-changes of multitype random walks or Lévy processes, resulting in multidimensional time-changes and generalized MAP representations (Fittipaldi et al., 2022, Haas et al., 2016, Chaumont, 2014).

3. Lamperti Transformation for SDEs and Numerical Schemes

A central analytical application is the reduction of scalar or multidimensional SDEs with state-dependent diffusion coefficients to SDEs with constant, typically unit, diffusion coefficients. Given an SDE,

$(0,\infty)$3

the Lamperti transform is $(0,\infty)$4 with $(0,\infty)$5. By Itô's formula,

$(0,\infty)$6

a unit-diffusion equation (Halidias et al., 2020, Ulander, 2023). This is the foundation of the Lamperti semi-discrete (LSD) and Lamperti-splitting schemes for numerically approximating SDEs. Importantly, the invertibility and regularity of $(0,\infty)$7 preserve boundaries and positivity, making these methods robust for bounded state-space or positivity-constrained problems.

4. Gaussian Fields, Random Fields, and Discrete Analogues

The Lamperti transformation acts as a logarithmic change of variables, mapping self-similar fields to stationary (homogeneous) random fields. For a one-parameter $(0,\infty)$8-self-similar process $(0,\infty)$9,

$\alpha>0$0

is stationary, and the inverse map is $\alpha>0$1. This extends to $\alpha>0$2-parameter random fields and multi-self-similar Gaussian fields, including fractional Brownian sheets, subfractional and bifractional motions (Davydov et al., 2017, Molchan, 2021, Shokrollahi et al., 5 Jan 2026). Lamperti's transformation, often called the "logarithmic transform," is the main tool for translating between self-similarity and translation-invariant (stationary) structure.

In discrete time, a Lamperti map acts between stationary random fields indexed by $\alpha>0$3 and component-wise self-similar fields indexed on the multiplicative lattice, yielding explicit AR(1)-type characterizations of stationarity, and making concrete the bridge between stationary increments and self-similarity (Voutilainen et al., 2023).

5. Applications: Branching Processes, Multitype Interactions, and Statistical Inference

Continuous-state branching processes (CB, CBI), multitype branching, and interacting branching particle models admit representations as time-changed Lévy (subordinator) processes via the Lamperti transform. The time-change is determined by cumulative population size or other additive functionals, leading to explicit functional equations for population sizes and their scaling limits (Caballero et al., 2010, Chaumont, 2014, Fittipaldi et al., 2022). The genealogical structure of measure-valued branching processes is similarly tied to the Fleming–Viot process and the $\alpha>0$4-coalescent via the infinite-dimensional Lamperti transform (Siri-Jégousse et al., 2024).

The transformation is further exploited for consistent estimation of self-similarity exponents in processes with or without stationary increments. By mapping $\alpha>0$5-self-similar (possibly non-stationary) processes to stationary processes, empirical methods exploit ergodicity in the Lamperti domain to construct robust estimators for $\alpha>0$6 and related parameters, even in the presence of complicated dependence (Wu et al., 25 Feb 2026).

6. Markov Additive Processes, Duality, and Deeper Factorizations

The Lamperti–Kiu transformation formalizes the equivalence between self-similar Markov processes and Markov additive processes (MAPs) with an explicit space-time change. This structure has led to explicit Wiener–Hopf factorizations in the stable process setting, yielding new fluctuation identities, space-time invariance laws, pathwise dualities, and Cramér-type asymptotics (Kyprianou, 2015, Chaumont et al., 2011). There are now fully explicit characterizations of the matrix exponents and ladder height processes for these MAPs, supporting further analysis in the context of stable Lévy processes and their boundary behavior, with notable applications in potential theory.

Duality principles and Doob $\alpha>0$7-transforms arise naturally in the extended Lamperti framework: the dual MAP $\alpha>0$8 under suitable reversibility conditions corresponds to a spatial inversion and time change in the self-similar process, revealing deep symmetries between growth and decay processes, conditioning, and entrance laws (Alili et al., 2016).

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References:

• See (Halidias et al., 2020) for concrete SDE reduction and LSD numerical schemes.
• See (Shokrollahi et al., 5 Jan 2026, Molchan, 2021, Davydov et al., 2017), and (Voutilainen et al., 2023) for Gaussian processes, random fields, and discrete time analogues.
• For deep factorizations, duality, and MAP constructions, see (Kyprianou, 2015, Chaumont et al., 2011, Alili et al., 2016), and (Kyprianou et al., 27 Jun 2025).
• For measure-valued and infinite-dimensional settings, see (Siri-Jégousse et al., 2024).
• For interacting and multitype processes, see (Fittipaldi et al., 2022) and (Chaumont, 2014).
• For statistical estimation via the transformation, see (Wu et al., 25 Feb 2026).

The Lamperti transformation is thus a universal tool, unifying self-similarity and stationary increments, enabling explicit analysis of scaling limits, pathwise representations, numerical schemes, and inferential methods across diverse classes of stochastic processes.