Lamperti Transformation Research Articles

The truncated Euler–Maruyama method for CIR model driven by fractional Brownian motion

Recently, Hong et al. (2020) established the strong convergence rate of the backward Euler scheme for the Cox–Ingersoll–Ross (CIR) model driven by fractional Brownian motion with Hurst parameter H∈(1/2,1), which may effect the efficiency of computation. Taking advantage of being explicit and easily implementable, a positivity preserving explicit scheme is proposed in this paper. For overcoming the difficulties caused by the unbounded diffusion coefficient, an auxiliary equation with a constant diffusion coefficient obtained by proper Lamperti transformation is used. By means of Malliavin calculus, we show that the truncated Euler–Maruyama scheme applied to this auxiliary equation not only ensures the positivity of the numerical solution, but also has the H-order rate of the root mean square error over a finite time interval. Furthermore, by transforming back, an explicit scheme for the original CIR model is obtained and has the same convergence order. Finally, some numerical experiments are provided to illustrate the theoretical results.

Pathwise Convergent Approximation for the Fractional SDEs

Fractional stochastic differential equation (FSDE)-based random processes are used in a wide spectrum of scientific disciplines. However, in the majority of cases, explicit solutions for these FSDEs do not exist and approximation schemes have to be applied. In this paper, we study one-dimensional stochastic differential equations (SDEs) driven by stochastic process with Hölder continuous paths of order 1/2<γ<1. Using the Lamperti transformation, we construct a backward approximation scheme for the transformed SDE. The inverse transformation provides an approximation scheme for the original SDE which converges at the rate h2γ, where h is a time step size of a uniform partition of the time interval under consideration. This approximation scheme covers wider class of FSDEs and demonstrates higher convergence rate than previous schemes by other authors in the field.

Selfsimilar stochastic differential equations

Diffusion in a logarithmic potential (DLP) attracted significant interest in physics recently. The dynamics of DLP are governed by a Langevin stochastic differential equation (SDE) whose underpinning potential is logarithmic, and that is driven by Brownian motion. The SDE that governs DLP is a particular case of a selfsimilar SDE: one that is driven by a selfsimilar motion, and that produces a selfsimilar motion. This paper establishes the pivotal role of selfsimilar SDEs via two novel universality results. I) Selfsimilar SDEs emerge universally, on the macro level, when applying scaling limits to micro-level SDEs. II) Selfsimilar SDEs emerge universally when applying the Lamperti transformation to stationary SDEs. Using the universality results, this paper further establishes: a novel statistical-analysis approach to selfsimilar Ito diffusions; and the focal importance of DLP.

Advanced forecasting and disturbance modelling for model predictive control of smart energy systems

We describe a method for embedding advanced weather disturbance models in model predictive control (MPC) of energy consumption and climate management in buildings. The performance of certainty-equivalent controllers such as conventional MPC for smart energy systems depends critically on accurate disturbance forecasts. Commonly, meteorological forecasts are used to supply weather predictions. However, these are generally not well suited for short-term forecasts. We show that an advanced physical and statistical description of the disturbances can provide useful short-term disturbance forecasts. We investigate the case of controlling the indoor air temperature of a simulated building using stochastic differential equations (SDEs) and certainty-equivalent MPC using the novel short-term forecasting method. A Lamperti transformation of the data and the models is an important contribution in making this SDE-based approach work. Simulation-based studies suggest that significant improvements are available for the performance of certainty-equivalent MPC based on short-term forecasts generated by the advanced disturbance model: Electricity savings of 5%–10% while at the same time improving the indoor climate by reducing comfort violations by up to over 90%.

Exact simulation for multivariate Itô diffusions

AbstractWe provide the first generic exact simulation algorithm for multivariate diffusions. Current exact sampling algorithms for diffusions require the existence of a transformation which can be used to reduce the sampling problem to the case of a constant diffusion matrix and a drift which is the gradient of some function. Such a transformation, called the Lamperti transformation, can be applied in general only in one dimension. So, completely different ideas are required for the exact sampling of generic multivariate diffusions. The development of these ideas is the main contribution of this paper. Our strategy combines techniques borrowed from the theory of rough paths, on the one hand, and multilevel Monte Carlo on the other.

Stochastic differential equations driven by fractional Brownian motion with locally Lipschitz drift and their implicit Euler approximation

In this paper, we study a class of one-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst parameter $ H \gt \frac{1}{2}$. The drift term of the equation is locally Lipschitz and unbounded in the neighbourhood of the origin. We show the existence, uniqueness and positivity of the solutions. The estimates of moments, including the negative power moments, are given. We also develop the implicit Euler scheme, proved that the scheme is positivity preserving and strong convergent, and obtain rate of convergence. Furthermore, by using Lamperti transformation, we show that our results can be applied to stochastic interest rate models such as mean-reverting stochastic volatility model and strongly nonlinear Aït-Sahalia type model.

Multifractal processes: Definition, properties and new examples

We investigate stochastic processes possessing scale invariance properties which we refer to as multifractal processes. The examples of such processes known so far do not go much beyond the original cascade construction of Mandelbrot. We provide a new definition of the multifractal process by generalizing the definition of the self-similar process. We establish general properties of these processes and show how existing examples fit into our setting. Finally, we define a new class of examples inspired by the idea of Lamperti transformation. Namely, for any pair of infinitely divisible distribution and a stationary process one can construct a multifractal process.

Lamperti transformation of scaled Brownian motion and related Langevin equations

Recently, scaled Brownian motion has attracted considerable attention in the context of single particle tracking experiments displaying anomalous fractional dynamics. Its probability density function coincides with the one for fractional Brownian motion. On the other hand, scaled Brownian motion displays weak ergodicity breaking. In this paper we show that by applying the so-called Lamperti transformation we are able to transform the scaled Brownian motion into ergodic process. This allows us to estimate the distribution and moments of the studied process having only one, appropriately long trajectory. We apply the same method to the scaled fractional Brownian motion and stable Lévy motion. It appears that the method works also in the case of long-range correlations and heavy-tailed distributions. We confirm our theoretical results using numerical simulations.

Optimal strong convergence rate of a backward Euler type scheme for the Cox–Ingersoll–Ross model driven by fractional Brownian motion

In this paper, we investigate the optimal strong convergence rate of numerical approximations for the Cox–Ingersoll–Ross model driven by fractional Brownian motion with Hurst parameter H∈(1∕2,1). To deal with the difficulties caused by the unbounded diffusion coefficient, we study an auxiliary equation based on Lamperti transformation. By means of Malliavin calculus, we prove that the backward Euler scheme applied to this auxiliary equation ensures the positivity of the numerical solution, and is of strong order one. Furthermore, a numerical approximation for the original model is obtained and converges with the same order.

On the law of homogeneous stable functionals

LetAbe theLq-functional of a stable Lévy process starting from one and killed when crossing zero. We observe thatAcan be represented as the independent quotient of two infinite products of renormalized Beta random variables. The proof relies on Markovian time change, the Lamperti transformation, and an explicit computation performed in [38] on perpetuities of hypergeometric Lévy processes. This representation allows us to retrieve several factorizations previously shown by various authors, and also to derive new ones. We emphasize the connections betweenAand more standard positive random variables. We also investigate the law of Riemannian integrals of stable subordinators. Finally, we derive several distributional properties ofArelated to infinite divisibility, self-decomposability, and the generalized Gamma convolution.

Lamperti transformation - Cure for ergodicity breaking

Recent results in single-particle experiments show that many complex systems display ergodicity breaking. In this paper we demonstrate how to transform a non-ergodic anomalous diffusion process in order to recover the ergodicity property. In the introduced method we use the so-called Lamperti transformation. Our approach enables us to perform statistical inference using only one recorded trajectory of the analyzed process even in the case when the original process displays ergodicity breaking. The method can be applied to examine many important examples of non-ergodic anomalous diffusion models, including space-time fractional Fokker-Planck equations, Lévy walks or subordinated fractional Brownian motions.

Ergodic aspects of some Ornstein–Uhlenbeck type processes related to Lévy processes

This work concerns the Ornstein–Uhlenbeck type process associated to a positive self-similar Markov process (X(t))t≥0 which drifts to ∞, namely U(t)≔e−tX(et−1). We point out that U is always a (topologically) recurrent ergodic Markov process. We identify its invariant measure in terms of the law of the exponential functional Iˆ≔∫0∞exp(ξˆs)ds, where ξˆ is the dual of the real-valued Lévy process ξ related to X by the Lamperti transformation. This invariant measure is infinite (i.e. U is null-recurrent) if and only if ξ1∉L1(P). In that case, we determine the family of Lévy processes ξ for which U fulfills the conclusions of the Darling–Kac theorem. Our approach relies crucially on a remarkable connection due to Patie (Patie, 2008) with another generalized Ornstein–Uhlenbeck process that can be associated to the Lévy process ξ, and properties of time-substitutions based on additive functionals.

Strong convergence of the symmetrized Milstein scheme for some CEV-like SDEs

In this paper we study the rate of convergence of a symmetrized version of the Milstein scheme applied to the solution of the one dimensional SDE $$X_t = x_0 + \int_{0}^t{b(X_s)ds}+\int_{0}^t{\sigma |X_s|^\alpha dW_s}, \;x_0>0,\;\sigma>0,\; \alpha\in[\tfrac{1}{2},1).$$ Assuming $b(0)/\sigma^2$ big enough, and $b$ smooth, we prove a strong rate of convergence of order one, recovering the classical result of Milstein for SDEs with smooth diffusion coefficient. In contrast with other recent results, our proof does not relies on Lamperti transformation, and it can be applied to a wide class of drift functions. On the downside, our hypothesis on the critical parameter value $b(0)/\sigma^2$ is more restrictive than others available in the literature. Some numerical experiments and comparison with various other schemes complement our theoretical analysis that also applies for the simple projected Milstein scheme with same convergence rate.

Optimal Scale Invariant Wigner Spectrum Estimation of Gaussian Locally Self-Similar Processes Using Hermite Functions

This paper investigates the mean square error optimal estimation of scale invariant Wigner spectrum for the class of Gaussian locally self-similar processes, by the multitaper method. In this method, the spectrum is estimated as a weighted sum of scale invariant windowed spectrograms. Moreover, it is shown that the optimal multitapers are approximated by the quasi Lamperti transformation of Hermite functions, which is computationally more efficient. Finally, the performance and accuracy of the estimation is studied via simulation.

Ergodicity of scalar stochastic differential equations with Hölder continuous coefficients

It is well-known that for a one dimensional stochastic differential equation driven by Brownian noise, with coefficient functions satisfying the assumptions of the Yamada–Watanabe theorem (Yamada and Watanabe, 1971, [31,32]) and the Feller test for explosions (Feller, 1951, 1954), there exists a unique stationary distribution with respect to the Markov semigroup of transition probabilities. We consider systems on a restricted domain D of the phase space R and study the rate of convergence to the stationary distribution. Using a geometrical approach that uses the so called free energy function on the density function space, we prove that the density functions, which are solutions of the Fokker–Planck equation, converge to the stationary density function exponentially under the Kullback–Leibler divergence, thus also in the total variation norm. The results show that there is a relation between the Bakry–Émery curvature dimension condition and the dissipativity condition of the transformed system under the Fisher–Lamperti transformation. Several applications are discussed, including the Cox–Ingersoll–Ross model and the Ait-Sahalia model in finance and the Wright–Fisher model in population genetics.

Ergodic properties of Lévy flights coexisting with subdiffusion and related models

We analyze ergodic properties of two different stationary processes resulting from a combination of Lévy flights and subdiffusion. The first one comes from the Lamperti transformation of subordinated γ-stable process. The second one is a sequence of increments of subordinated Lévy process. We prove that both processes are mixing and ergodic. We also study increments and the Lamperti transformation of time-changed fractional Brownian motion. It appears that these models of anomalous diffusion are also ergodic and mixing. We believe that the derived results will help to verify ergodicity in single particle tracking experiments.

Testing Self-Similarity Through Lamperti Transformations

Self-similar processes have been widely used in modeling real-world phenomena occurring in environmetrics, network traffic, image processing, and stock pricing, to name but a few. The estimation of the degree of self-similarity has been studied extensively, while statistical tests for self-similarity are scarce and limited to processes indexed in one dimension. This paper proposes a statistical hypothesis test procedure for self-similarity of a stochastic process indexed in one dimension and multi-self-similarity for a random field indexed in higher dimensions. If self-similarity is not rejected, our test provides a set of estimated self-similarity indexes. The key is to test stationarity of the inverse Lamperti transformations of the process. The inverse Lamperti transformation of a self-similar process is a strongly stationary process, revealing a theoretical connection between the two processes. To demonstrate the capability of our test, we test self-similarity of fractional Brownian motions and sheets, their time deformations and mixtures with Gaussian white noise, and the generalized Cauchy family. We also apply the self-similarity test to real data: annual minimum water levels of the Nile River, network traffic records, and surface heights of food wrappings.

Inversion, duality and Doob $h$-transforms for self-similar Markov processes

We show that any Rd∖{0}Rd∖{0}-valued self-similar Markov process XX, with index α>0α>0 can be represented as a path transformation of some Markov additive process (MAP) (θ,ξ)(θ,ξ) in Sd−1×RSd−1×R. This result extends the well known Lamperti transformation. Let us denote by XˆX^ the self-similar Markov process which is obtained from the MAP (θ,−ξ)(θ,−ξ) through this extended Lamperti transformation. Then we prove that XˆX^ is in weak duality with XX, with respect to the measure π(x/∥x∥)∥x∥α−ddxπ(x/‖x‖)‖x‖α−ddx, if and only if (θ,ξ)(θ,ξ) is reversible with respect to the measure π(ds)dxπ(ds)dx, where π(ds)π(ds) is some σσ-finite measure on Sd−1Sd−1 and dxdx is the Lebesgue measure on RR. Moreover, the dual process XˆX^ has the same law as the inversion (Xγt/∥Xγt∥2,t≥0)(Xγt/‖Xγt‖2,t≥0) of XX, where γtγt is the inverse of t↦∫t0∥X∥−2αsdst↦∫0t‖X‖s−2αds. These results allow us to obtain excessive functions for some classes of self-similar Markov processes such as stable Levy processes.

Optimal prediction for positive self-similar Markov processes

This paper addresses the question of predicting when a positive self-similar Markov process $X$ attains its pathwise global supremum or infimum before hitting zero for the first time (if it does at all). This problem has been studied in [9] under the assumption that $X$ is a positive transient diffusion. We extend their result to the class of positive self-similar Markov processes by establishing a link to [3], where the same question is studied for a Levy process drifting to $-\infty $. The connection to [3] relies on the so-called Lamperti transformation [15] which links the class of positive self-similar Markov processes with that of Levy processes. Our approach shows that the results in [9] for Bessel processes can also be seen as a consequence of self-similarity.

An Age- and State-Dependent Nonlinear Prognostic Model for Degrading Systems

Nonlinearity and stochasticity are two important factors contributing to the degradation processes of complicated systems, and thus have to be taken into account in stochastic degradation modeling based prognostics. However, current studies almost always focus on age-dependent stochastic degradation models, most of which are linear, or can be transformed into linear models. In this paper, we propose a general age- and state-dependent nonlinear degradation model for prognostics. In the presented model, a diffusion process with age- and state-dependent nonlinear drift and volatility coefficients is utilized to characterize the dynamics and nonlinearity of the degradation progression. To derive the estimated remaining useful life distribution, the considered diffusion process is first converted into a diffusion process with age- or state-dependent nonlinear drift but constant volatility through Lamperti transformation. Then, based on a well-known time-space transformation, we obtain an analytical approximated remaining useful life distribution in the concept of the first passage time. Furthermore, a maximum likelihood estimation method for unknown parameters in the concerned model is presented on the basis of closed-form approximated degradation state transition density functions by the Hermite-expansion method. An illustrative example is provided to show how the obtained results can be applied to a specific age- and state-dependent nonlinear degradation model. Finally, the presented model is fitted to bearing degradation data. Comparative results suggest the necessity of age- and state-dependent nonlinear degradation modeling in prognostics.