Solutions Of SDEs Research Articles

Characteristics and Itô's formula for weak Dirichlet processes: an equivalence result

The main objective consists in generalizing a well-known Itô formula of J. Jacod and A. Shiryaev: given a càdlàg process S, there is an equivalence between the fact that S is a semimartingale with given characteristics ( B k , C , ν ) and a Itô formula type expansion of F ( S ) , where F is a bounded function of class C 2 . This result connects weak solutions of path-dependent SDEs and related martingale problems. We extend this to the case when S is a weak Dirichlet process. A second aspect of the paper consists of discussing some untreated features of stochastic calculus for finite quadratic variation processes.

Comparison of Statistical Approaches for Reconstructing Random Coefficients in the Problem of Stochastic Modeling of Air–Sea Heat Flux Increments

This paper compares two statistical methods for parameter reconstruction (random drift and diffusion coefficients of the Itô stochastic differential equation, SDE) in the problem of stochastic modeling of air–sea heat flux increment evolution. The first method relates to a nonparametric estimation of the transition probabilities (wherein consistency is proven). The second approach is a semiparametric reconstruction based on the approximation of the SDE solution (in terms of distributions) by finite normal mixtures using the maximum likelihood estimates of the unknown parameters. This approach does not require any additional assumptions for the coefficients, with the exception of those guaranteeing the existence of the solution to the SDE itself. It is demonstrated that the corresponding conditions hold for the analyzed data. The comparison is carried out on the simulated samples, modeling the case where the SDE random coefficients are represented in trigonometric form, which is related to common climatic models, as well as on the ERA5 reanalysis data of the sensible and latent heat fluxes in the North Atlantic for 1979–2022. It is shown that the results of these two methods are close to each other in a quantitative sense, but differ somewhat in temporal variability and spatial localization. The differences during the observed period are analyzed, and their geophysical interpretations are presented. The semiparametric approach seems promising for physics-informed machine learning models.

Well-posedness of the deterministic transport equation with singular velocity field perturbed along fractional Brownian paths

In this article we prove path-by-path uniqueness in the sense of Davie [25] and Shaposhnikov [46] for SDE's driven by a fractional Brownian motion with a sufficiently small Hurst parameter H∈(0,12), uniformly in the initial conditions, where the drift vector field is allowed to be merely bounded and measurable.Using this result, we construct weak unique regular solutions in Wlock,p([0,1]×Rd), p>d of the classical transport and continuity equations with singular velocity fields perturbed along fractional Brownian paths.The latter results provide a systematic way of producing examples of singular velocity fields, which cannot be treated by the regularity theory of DiPerna-Lions [28], Ambrosio [2] or Crippa-De Lellis [23].Our approach is based on a priori estimates at the level of flows generated by a sequence of mollified vector fields, converging to the original vector field, and which are uniform with respect to the mollification parameter. In addition, we use a compactness criterion based on Malliavin calculus from [24] as well as a supremum estimate in time of moments of the derivative of the flow of SDE solutions.

Weak variable step-size schemes for stochastic differential equations based on controlling conditional moments

We address the weak numerical solution of stochastic differential equations driven by independent Brownian motions (SDEs for short). This paper develops a new methodology to design adaptive strategies for determining automatically the step-sizes of the numerical schemes that compute the mean values of smooth functions of the solutions of SDEs. First, we introduce a general method for constructing variable step-size weak schemes for SDEs, which is based on controlling the match between the first conditional moments of the increments of the numerical integrator and the ones corresponding to an additional weak approximation. To this end, we use certain local discrepancy functions that do not involve sampling random variables. Precise directions for designing suitable discrepancy functions and for selecting starting step-sizes are given. Second, we introduce a variable step-size Euler scheme, together with a variable step-size second order weak scheme via extrapolation. Finally, numerical simulations are presented to show the potential of the introduced variable step-size strategy and the adaptive scheme to overcome known instability problems of the conventional fixed step-size schemes in the computation of diffusion functional expectations.

On SDEs with Lipschitz coefficients, driven by continuous, model-free martingales

We prove the existence and uniqueness of solutions of SDEs with Lipschitz coefficients, driven by continuous, model-free martingales. The main tool in our reasoning is Picard’s iterative procedure and a model-free version of the Burkholder-Davis-Gundy inequality for integrals driven by model-free, continuous martingales. We work with a new outer measure which assigns zero value exactly to those properties which are instantly blockable.

Limit theorems for local times and applications to SDEs with jumps

Consider a stochastic process X, regenerative at a state x which is instantaneous and regular. Let L be a regenerative local time for X at x. Suppose furthermore that X can be approximated by discrete time regenerative processes Xn for which x is accessible. We give conditions on X and Xn so that the naturally defined local time of Xn (which counts the quantity of visits to x) converges weakly to L. This limit theorem generalizes previous invariance principles that have appeared in the literature. Furthermore, it allows one to prove novel invariance principles for local times of regenerative processes converging to diffusions or to solutions of SDEs with jumps.

On weak solution of SDE driven by inhomogeneous singular Lévy noise

We study a time-inhomogeneous SDE in R d \mathbb {R}^d driven by a cylindrical Lévy process with independent coordinates which may have different scaling properties. Such a structure of the driving noise makes it strongly spatially inhomogeneous and complicates the analysis of the model significantly. We prove that the weak solution to the SDE is uniquely defined, is Markov, and has the strong Feller property. The heat kernel of the process is presented as a combination of an explicit ‘principal part’ and a ‘residual part’, subject to certain L ∞ ( d x ) ⊗ L 1 ( d y ) L^\infty (dx)\otimes L^1(dy) and L ∞ ( d x ) ⊗ L ∞ ( d y ) L^\infty (dx)\otimes L^\infty (dy) -estimates showing that this part is negligible in a short time, in a sense. The main tool of the construction is the analytic parametrix method, specially adapted to Lévy-type generators with strong spatial inhomogeneities.

Almost sure stability of stochastic theta methods with random variable stepsize for stochastic differential equations

In this paper, we use the non-negative discrete semimartingale convergence theorems to study the stochastic theta methods with random stepsizes to reproduce the almost sure stability of the exact solution of stochastic differential equations. Moreover, the choice of the stepsize in each step is based on the stochastic theta methods of random variable stepsize. In numerical experiments, we propose an algorithm that successfully use θ-Maruyama and θ-Milstein methods to simulate the numerical solutions of stochastic differential equations, reproduce the almost sure stability of exact solutions of SDEs and simulate the random variable stepsize in each timestep, and compared with constant stepsizes, random stepsize can speed up the decay process and reduce the iterations greatly.

Numerical solution of quadratic SDE with measurable drift

In this paper we are interested in solving numerically quadratic SDEs with non-necessary continuous drift of the from Xt = x + ? t 0 b(s,Xs)ds + ? t 0 f (Xs)?2(Xs)ds + ? t 0 ?(Xs)dWs, where, x is the initial data b and ? are given coefficients that are assumed to be Lipschitz and bounded and f is a measurable bounded and integrable function on the whole space R. Numerical simulations for this class of SDE of quadratic growth and measurable drift, induced by the singular term f (x)?2(x), is implemented and illustrated by some examples. The main idea is to use a phase space transformation to transform our initial SDEs to a standard SDE without the discontinuous and quadratic term. The Euler-Maruyama scheme will be used to discretize the new equation, thus numerical approximation of the original equation is given by taking the inverse of the space transformation. The rate of convergence are shown to be of order 1/2.

Path-by-path uniqueness of multidimensional SDE’s on the plane with nondecreasing coefficients

In this paper we study path-by-path uniqueness for multidimensional stochastic differential equations driven by the Brownian sheet. We assume that the drift coefficient is unbounded, verifies a spatial linear growth condition and is componentwise nondeacreasing. Our approach consists of showing the result for bounded and componentwise nondecreasing drift using both a local time-space representation and a law of iterated logarithm for Brownian sheets. The desired result follows using a Gronwall type lemma on the plane. As a by product, we obtain the existence of a unique strong solution of multidimensional SDEs driven by the Brownian sheet when the drift is non-decreasing and satisfies a spatial linear growth condition.

Existence of relaxed stochastic optimal control for G-SDEs with controlled jumps

In this article, we study the relaxed control problem where the admissible controls are measure-valued processes and the state variable is governed by a G-stochastic differential equation (SDEs) driven by a relaxed Poisson measure where the compensator is a product measure. The control variable appears in the drift and in the jump term. We prove that every solution of our SDE associated to a relaxed control can be written as a limit of a sequence of solutions of SDEs associated to strict controls (stability results). In the end, we show the existence of our relaxed control.

On Krylov’s estimates for optional semimartingales

Abstract The estimates of N. V. Krylov for distributions of stochastic integrals by means of the L d {L_{d}} -norm of a measurable function are well-known and are widely used in the theory of stochastic differential equations and controlled diffusion processes. We generalize estimates of this type for optional semimartingales, then apply these estimates to prove the change of variables formula for a general class of functions from the Sobolev space W d 2 {W^{2}_{d}} . We also show how to use these estimates for the investigation of L 2 {L^{2}} -convergence of solutions of optional SDE’s.

On multi-step estimation of delay for SDE

We consider the problem of delay estimation by the observations of the solutions of several SDEs. It is known that the MLEs for these models are consistent and asymptotically normal, but the likelihood ratio functions are not differentiable w.r.t. the parameter, and therefore the numerical calculation of the MLEs encounter certain difficulties. We propose One-step and Two-step MLEs, whose calculation has no such problems and provide an estimator asymptotically equivalent to the MLE. These constructions are realized in two or three steps. First, we construct preliminary estimators which are consistent and asymptotically normal, but not asymptotically efficient. Then we use these estimators and a modified Fisher-score device to obtain One-step and Two-step MLEs. We suppose that its numerical realization is much more simple. Stochastic Pantograph equation is introduced and related statistical problems are discussed.

Existence and uniqueness of solutions of SDEs with discontinuous drift and finite activity jumps

In this letter we prove existence and uniqueness of strong solutions to multi-dimensional SDEs with discontinuous drift and finite activity jumps.

On bounded solutions of linear SDEs driven by convergent system matrix processes with Hurwitz limits

Linear time-varying stochastic differential equations with a.s. convergent continuous random system matrix processes are considered. It is shown that given the limit is known to be Hurwitz (i.e. asymptotically stable), the generated state solutions are a.s. bounded. This property is shown to hold by substantiating that, w.p.1, (i) no finite escape time exists and (ii) no divergence to infinity, as , may occur. An application to stochastic adaptive control is provided.

Wasserstein convergence rates for random bit approximations of continuous Markov processes

We determine the convergence speed of a numerical scheme for approximating one-dimensional continuous strong Markov processes. The scheme is based on the construction of certain Markov chains whose laws can be embedded into the process with a sequence of stopping times. Under a mild condition on the process' speed measure we prove that the approximating Markov chains converge at fixed times at the rate of 1/4 with respect to every p-th Wasserstein distance. For the convergence of paths, we prove any rate strictly smaller than 1/4. Our results apply, in particular, to processes with irregular behavior such as solutions of SDEs with irregular coefficients and processes with sticky points.

Semigroup properties of solutions of SDEs driven by Lévy processes with independent coordinates

We study the stochastic differential equation dXt=A(Xt−)dZt, X0=x, where Zt=(Zt(1),…,Zt(d))T and Zt(1),…,Zt(d) are independent one-dimensional Lévy processes with characteristic exponents ψ1,…,ψd. We assume that each ψi satisfies a weak lower scaling condition WLSC(α,0,C̲), a weak upper scaling condition WUSC(β,1,C¯) (where 0<α≤β<2) and some additional regularity properties. We consider two mutually exclusive assumptions: either (i) all ψ1,…,ψd are the same and α,β are arbitrary, or (ii) not all ψ1,…,ψd are the same and α>(2∕3)β. We also assume that the determinant of A(x)=(aij(x)) is bounded away from zero, and aij(x) are bounded and Lipschitz continuous. In both cases (i) and (ii) we prove that for any fixed γ∈(0,α)∩(0,1] the semigroup Pt of the process X satisfies |Ptf(x)−Ptf(y)|≤ct−γ∕α|x−y|γ||f||∞ for arbitrary bounded Borel function f. We also show the existence of a transition density of the process X.

Randomized derivative-free Milstein algorithm for efficient approximation of solutions of SDEs under noisy information

We deal with pointwise approximation of solutions of scalar stochastic differential equations in the presence of informational noise about underlying drift and diffusion coefficients. We define a randomized derivative-free version of Milstein algorithm Āndf−RM and investigate its error. We also study the lower bounds on the error of arbitrary algorithm. It turns out that in some case the scheme Āndf−RM is the optimal one. Finally, in order to test the algorithm Āndf−RM in practice, we report performed numerical experiments.

Probability density function of SDEs with unbounded and path-dependent drift coefficient

In this paper, we first prove that the existence of a solution of SDEs under the assumptions that the drift coefficient is of linear growth and path-dependent, and diffusion coefficient is bounded, uniformly elliptic and Hölder continuous. We apply Gaussian upper bound for a probability density function of a solution of SDE without drift coefficient and local Novikov condition, in order to use Maruyama–Girsanov transformation. The aim of this paper is to prove the existence with explicit representations (under linear/super-linear growth condition), Gaussian two-sided bound and Hölder continuity (under sub-linear growth condition) of a probability density function of a solution of SDEs with path-dependent drift coefficient. As an application of explicit representation, we provide the rate of convergence for an Euler–Maruyama (type) approximation, and an unbiased simulation scheme.

Transportation inequalities for non-globally dissipative SDEs with jumps via Malliavin calculus and coupling

By using the mirror coupling for solutions of SDEs driven by pure jump L\'evy processes, we extend some transportation and concentration inequalities, which were previously known only in the case where the coefficients in the equation satisfy a global dissipativity condition. Furthermore, by using the mirror coupling for the jump part and the coupling by reflection for the Brownian part, we extend analogous results for jump diffusions. To this end, we improve some previous results concerning such couplings and show how to combine the jump and the Brownian case. As a crucial step in our proof, we develop a novel method of bounding Malliavin derivatives of solutions of SDEs with both jump and Gaussian noise, which involves the coupling technique and which might be of independent interest. The bounds we obtain are new even in the case of diffusions without jumps.