Unbounded Drift Research Articles

Localization Uncertainty Estimation for Autonomous Underwater Vehicle Navigation

High-precision positioning capability is a crucial technology for achieving accurate navigation in autonomous underwater vehicles (AUVs). However, due to severe electromagnetic wave attenuation underwater and the unavailability of the global positioning system (GPS), inertial-navigation-based dead reckoning is considered the primary method for underwater positioning. Unfortunately, errors accumulated during the navigation process lead to unbounded drift, and filtering-based methods have been used to mitigate the errors, but with limited success. In this paper, we propose a precise underwater dead-reckoning mathematical model that recursively calculates the ground truth and corresponding errors based on an AUV’s motion model, and we derive empirical formulas. Compared to related methods, this approach not only models the cumulative errors of relative noise measurements, but also provides recursive expressions with corresponding statistical moments. The experimental results demonstrate that this formula significantly reduces positioning errors in underwater navigation tasks.

Sandwiched SDEs with unbounded drift driven by Hölder noises

AbstractWe study a stochastic differential equation with an unbounded drift and general Hölder continuous noise of order $\lambda \in (0,1)$ . The corresponding equation turns out to have a unique solution that, depending on a particular shape of the drift, either stays above some continuous function or has continuous upper and lower bounds. Under some mild assumptions on the noise, we prove that the solution has moments of all orders. In addition, we provide its connection to the solution of some Skorokhod reflection problem. As an illustration of our results and motivation for applications, we also suggest two stochastic volatility models which we regard as generalizations of the CIR and CEV processes. We complete the study by providing a numerical scheme for the solution.

On radial solutions for some elliptic equations involving operators with unbounded coefficients in exterior domains

We study existence and multiplicity of radial solutions for some quasilinear elliptic problems involving the operator $L_N=\Delta - x\cdot \nabla$ on $\mathbb{R}^N\setminus B_1$, where $\Delta$ is the Laplacian, $x\cdot \nabla$ is an unbounded drift term, $N\geq 3$ and $B_1$ is the unit ball centered at the origin. We consider: (i) Eigenvalue problems, and (ii) Problems involving a nonlinearity of concave and convex type. On the first class of problems we get a compact embedding result, whereas on the second, we address the well-known question of Ambrosetti, Brezis and Cerami from 1993 concerning the existence of two positive solutions for some problems involving the supercritical Sobolev exponent in symmetric domains for the Laplacian. Specifically, we provide\linebreak a new approach of answering the ABC-question for elliptic problems with unbounded coefficients in exterior domains and we find asymptotic properties of the radial solutions. Furthermore, we study the limit case, namely when nonlinearity involves a sublinear term and a linear term. As far as we know, this is the first work that deals with such a case, even for the Laplacian. In our approach, we use both topological and variational arguments.

Sup-norm adaptive drift estimation for multivariate nonreversible diffusions

We consider the question of estimating the drift for a large class of ergodic multivariate and possibly nonreversible diffusion processes, based on continuous observations, in sup-norm loss. Nonparametric classes of smooth functions of unknown order are considered, and we suggest an adaptive approach which allows to construct drift estimators attaining optimal sup-norm rates of convergence. Reversibility structures and related functional inequalities are known to be key tools for these estimation problems. We can discard such restrictions by making use of mixing properties which are satisfied for the very general class of processes under consideration. Analysing diffusions, the scalar case is very distinct from the general multivariate setting. Therefore, we treat scalar and multivariate processes separately which leads to in several aspects improved univariate results. While we consider drift estimation on bounded domains for exponentially β-mixing multivariate processes, for scalar diffusion processes we work under minimal assumptions that allow estimation of unbounded drift terms over the entire real line, and we provide classical minimax results (including lower bounds) which cannot be obtained under state-of-the-art conditions in the multivariate case. In addition, we prove a Donsker theorem for the classical kernel estimator of the invariant density in the scalar setting and establish its semiparametric efficiency.

Moderate deviations for the Langevin equations: Strong damping and fast Markovian switching

In this paper, we obtain a moderate deviations principle (MDP) for a class of Langevin dynamic systems with a strong damping and fast Markovian switching. To facilitate our study, first, analysis of systems with bounded drifts is dealt with. To obtain the desired moderate deviations, the exponential tightness of the solution of the Langevin equation is proved. Then, the solution of its first-order approximation using local MDPs is examined. Finally, the MDPs are established. To enable the treatment of unbounded drifts, a reduction technique is presented near the end of the paper, which shows that Lipschitz continuous drifts can be dealt with.

Sensitivity analysis with respect to a stochastic stock price model with rough volatility via a Bismut–Elworthy–Li formula for singular SDEs

In this paper, we show the existence of unique Malliavin differentiable solutions to SDE‘s driven by a fractional Brownian motion with Hurst parameter H<12 and singular, unbounded drift vector fields, for which we also prove a stability result. Further, using the latter results, we propose a stock price model with rough and correlated volatility, which also allows for capturing regime switching effects. Finally, we derive a Bismut–Elworthy–Li formula with respect to our stock price model for certain classes of vector fields.

Drift-implicit Euler scheme for sandwiched processes driven by Hölder noises

In this paper, we analyze the drift-implicit (or backward) Euler numerical scheme for a class of stochastic differential equations with unbounded drift driven by an arbitrary λ-Hölder continuous process, λ ∈ (0,1). We prove that, under some mild moment assumptions on the Hölder constant of the noise, the L^{r}({Omega };L^{infty }([0,T]))-approximation error converges to 0 as O(Δλ), Δ → 0. To exemplify, we consider numerical schemes for the generalized Cox–Ingersoll–Ross and Tsallis–Stariolo–Borland models. The results are illustrated by simulations.

On admissible singular drifts of symmetric α‐stable process

AbstractWe consider the problem of existence of a (unique) weak solution to the SDE describing symmetric α‐stable process with a locally unbounded drift , , . In this paper, b belongs to the class of weakly form‐bounded vector fields, the class providing the L2 theory of the non‐local operator behind the SDE, i.e. . It contains as proper sub‐classes other classes of singular vector fields studied in the literature in connection with this operator, such as the Kato class, the weak class and the Campanato–Morrey class (in general, such b makes invalid the standard heat kernel estimates in terms of the heat kernel of the fractional Laplacian). We show that the operator with weakly form‐bounded b admits a realization as (minus) Feller generator, and that the probability measures determined by the Feller semigroup (uniquely in appropriate sense) admit description as weak solutions to the corresponding SDE. The proof is based on detailed regularity theory of in , .

Strong convergence of the Euler-Maruyama approximation for SDEs with unbounded drift

In this work, we prove strong convergence on small time interval of order for arbitrarily small of the Euler-Maruyama approximation for additive Brownian motion with Hölder continuous drift satisfying a linear growth condition. The proof is based on direct estimations of functional of the Euler-Maruyama approximation. The order of convergence does not depend on the Hölder index of the drift, thus generalizing the results obtained in [10] to both Linear growth and to an optimal convergence order.

Heat kernel of supercritical nonlocal operators with unbounded drifts

Heat kernel of supercritical nonlocal operators with unbounded drifts

Probabilistic representation of integration by parts formulae for some stochastic volatility models with unbounded drift

In this paper, we establish a probabilistic representation as well as some integration by parts formulae for the marginal law at a given time maturity of some stochastic volatility model with unbounded drift. Relying on a perturbation technique for Markov semigroups, our formulae are based on a simple Markov chain evolving on a random time grid for which we develop a tailor-made Malliavin calculus. Among other applications, an unbiased Monte Carlo path simulation method stems from our formulas so that it can be used in order to numerically compute with optimal complexity option prices as well as their sensitivities with respect to the initial values or Greeks in finance, namely the Delta and Vega, for a large class of non-smooth European payoff. Numerical results are proposed to illustrate the efficiency of the method.

Total variation distance between two diffusions in small time with unbounded drift: application to the Euler-Maruyama scheme

We give bounds for the total variation distance between the solutions to two stochastic differential equations starting at the same point and with close coefficients, which applies in particular to the distance between an exact solution and its Euler-Maruyama scheme in small time. We show that for small $t$, the total variation distance is of order $t^{r/(2r+1)}$ if the noise coefficient $\sigma$ of the SDE is elliptic and $\mathcal{C}^{2r}_b$, $r\in \mathbb{N}$ and if the drift is $C^1$ with bounded derivatives, using multi-step Richardson-Romberg extrapolation. We do not require the drift to be bounded. Then we prove with a counterexample that we cannot achieve a bound better than $t^{1/2}$ in general.

L 1 and L ∞ stability of transition densities of perturbed diffusions

Abstract In this paper, we derive a stability result for L 1 {L_{1}} and L ∞ {L_{\infty}} perturbations of diffusions under weak regularity conditions on the coefficients. In particular, the drift terms we consider can be unbounded with at most linear growth, and the estimates reflect the transport of the initial condition by the unbounded drift through the corresponding flow. Our approach is based on the study of the distance in L 1 {L_{1}} - L 1 {L_{1}} metric between the transition densities of a given diffusion and the perturbed one using the McKean–Singer parametrix expansion. In the second part, we generalize the well-known result on the stability of diffusions with bounded coefficients to the case of at most linearly growing drift.

On uniqueness of probability solutions of the Fokker-Planck-Kolmogorov equation

The paper gives a solution to the long-standing problem of uniqueness for probability solutions to the Cauchy problem for the Fokker- Planck-Kolmogorov equation with an unbounded drift coefficient and unit diffusion coefficient. It is proved that in the one-dimensional case uniqueness holds and in all other dimensions it fails. The case of nonconstant diffusion coefficients is also investigated. Bibliography: 70 titles.

An improved Aleksandrov–Bakel’man–Pucci estimate for a second-order elliptic operator with unbounded drift

The classical Aleksandrov–Bakel’man–Pucci estimate (ABP estimate) for a second-order elliptic operator in nondivergence form is one of the fundamental tools for the bounds of subsolutions. Cabre improved the ABP estimate by replacing a constant factor, the diameter of a given domain, with a geometric character, which can be defined and finite for some unbounded domains. In the proof, Cabre used the Krylov–Safonov boundary weak Harnack inequality from Trudinger; thus, it is required that the first-order coefficients belong to a Lebesgue [Formula: see text]-integrable function space. Using a growth lemma from Safonov and an approximation method, we improve the result to Lebesgue [Formula: see text]-integrable first-order coefficients, which is optimal and coincides with the condition for the original ABP estimate.

Density and gradient estimates for non degenerate Brownian SDEs with unbounded measurable drift

We consider non degenerate Brownian SDEs with Hölder continuous in space diffusion coefficient and unbounded drift with linear growth. We derive two sided bounds for the associated density and pointwise controls of its derivatives up to order two under some additional spatial Hölder continuity assumptions on the drift. Importantly, the estimates reflect the transport of the initial condition by the unbounded drift through an auxiliary, possibly regularized, flow.

Penalisation techniques for one-dimensional reflected rough differential equations

In this paper we solve real-valued rough differential equations (RDEs) reflected on an irregular boundary. The solution $Y$ is constructed as the limit of a sequence $(Y^n)_{n\in\mathbb{N}}$ of solutions to RDEs with unbounded drifts $(\psi_n)_{n\in\mathbb{N}}$. The penalisation $\psi_n$ increases with $n$. Along the way, we thus also provide an existence theorem and a Doss-Sussmann representation for RDEs with a drift growing at most linearly. In addition, a speed of convergence of the sequence of penalised paths to the reflected solution is obtained. \\ We finally use the penalisation method to prove that the law at time $t>0$ of some reflected Gaussian RDE is absolutely contiuous with respect to the Lebesgue measure.

Well-posedness of some non-linear stable driven SDEs

<p style='text-indent:20px;'>We prove the well-posedness of some non-linear stochastic differential equations in the sense of McKean-Vlasov driven by non-degenerate symmetric <inline-formula><tex-math id="M1">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-stable Lévy processes with values in <inline-formula><tex-math id="M2">\begin{document}$ {{{\mathbb R}}}^d $\end{document}</tex-math></inline-formula> under some mild Hölder regularity assumptions on the drift and diffusion coefficients with respect to both space and measure variables. The methodology developed here allows to consider unbounded drift terms even in the so-called super-critical case, i.e. when the stability index <inline-formula><tex-math id="M3">\begin{document}$ \alpha \in (0,1) $\end{document}</tex-math></inline-formula>. New strong well-posedness results are also derived from the previous analysis.

Asymptotic behavior of the steady Navier-Stokes flow in the exterior domain

We consider an elliptic equation with unbounded drift in an exterior domain, and obtain quantitative uniqueness estimates at infinity, i.e. the non-trivial solution of −△u+W⋅∇u=0 decays in the form of exp⁡(−C|x|log2⁡|x|) at infinity provided ‖W‖L∞(R2∖B1)≲1, which is sharp with the help of some counterexamples. These results also generalize the decay theorem by Kenig-Wang [13] or Kenig-Silvestre-Wang [14] in the whole space. As an application, the asymptotic behavior of an incompressible fluid around a bounded obstacle is also considered. Specially for the two-dimensional case, we can improve the decay rate in [16] to exp⁡(−C|x|log2⁡|x|), where the minimal decaying rate of exp⁡(−C|x|32+) is obtained by Kow-Lin in a recent paper [16] by using appropriate Carleman estimates.

Existence and uniqueness of invariant measures of 3D stochastic MHD-α model driven by degenerate noise

ABSTRACT In this paper, we establish the existence and uniqueness of invariant measures of the 3D stochastic magnetohydrodynamic-α model (MHD-α) driven by degenerate additive noise. We firstly study the Feller property of solutions and establish the existence of invariant measures by utilizing the classical Krylov–Bogoliubov theorem. Then, we prove the uniqueness of invariant measures for the corresponding transition semigroup by utilizing the notion of asymptotic strong Feller proposed by Hairer and Mattingly [Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann Math (2). 2006;164(3):993–1032]. The proof not only requires the investigation of degenerate noise, but also the study of highly nonlinear, unbounded drifts.