In this note we prove the existence of a density for the law of the solution for 1-dimensional stochastic delay differential equations with normal reflection. The equations are driven by a fractional Brownian motion with Hurst parameter $H > 1/2$. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann-Stieltjes integral.
In the setting of finite reflection groups, we prove that the projection of a Brownian motion onto a closed Weyl chamber is another Brownian motion normally reflected on the walls of the chamber. Our proof is probabilistic and the decomposition we obtain may be seen as a multidimensional extension of Tanaka's formula for linear Brownian motion. The paper is closed with a description of the boundary process through the local times at zero of the distances from the initial process to the facets.
In this paper, we study the reflected backward stochastic differential equations driven by G-Brownian motion with two reflecting obstacles, which means that the solution lies between two prescribed processes. A new kind of approximate Skorohod condition is proposed to derive the uniqueness and existence of the solutions. The uniqueness can be proved by a priori estimates and the existence is obtained via a penalization method.
In this note we prove an existence and uniqueness result of solution for stochastic differential delay equations with hereditary drift driven by a fractional Brownian motion with Hurst parameter $H > 1/2$. Then, we show that, when the delay goes to zero, the solutions to these equations converge, almost surely and in $L^p$, to the solution for the equation without delay. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann-Stieltjes integral.
We investigate the problem of nonparametric estimation of the trend for stochastic differential equations with delay and driven by a fractional Brownian motion through the method of kernel-type estimation for the estimation of a probability density function.
In this paper, we consider a class of stochastic delay fractional evolution equations driven by fractional Brownian motion in a Hilbert space. Sufficient conditions for the existence and uniqueness of mild solutions are obtained. An application to the stochastic fractional heat equation is presented to illustrate the theory.
In this work, we will show the existence and uniqueness of the solution to the semi linear stochastic differential equations driven by weighted fractional Brownian motion with delay. We also prove smoothness of the density of the solution with respect to Lebesgue's measure on R^d for $d \geq 1$.
In this paper we study the controllability results of impulsive neutral stochastic functional differential equations with infinite delay driven by fractional Brownian motion in a real separable Hilbert space. The controllability results are obtained using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result.
Let B(t), X(t) and Y(t) be independent standard 1d Borwnian motions. Define X^+(t) and Y^-(t) as the trajectories of the processes X(t) and Y(t) pushed upwards and, respectively, downwards by B(t), according to Skorohod-reflection. In a recent paper, Jon Warren proves inter alia that Z(t):= X^+(t)-Y^-(t) is a three-dimensional Bessel-process. In this note, we present an alternative, elementary proof of this fact.
In this paper we study the controllability of fractional neutral stochastic functional differential equations with infinite delay driven by fractional Brownian motion in a real separable Hilbert space. The controllability results are obtained by using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result.
On a multi-assets Black-Scholes economy, we introduce a class of barrier options. In this model we apply a generalized reflection principle in a context of the finite reflection group acting on a Euclidean space to give a valuation formula and the semi-static hedge.
The aim of this paper is to present the analysis for the solutions of nonlinear stochastic functional differential equation driven by G-Brownian motion with infinite delay (G-SFDEwID). Under some useful assumptions, we have proved that the G-SFDEwID admits a unique local solution. The mentioned theory has been further generalized to show that G-SFDEwID admits a unique strong global solution. The asymptotic properties, mean square boundedness and convergence of solutions with different initial data have been derived. We have assessed that the solution map $X_t$ is mean square bounded and two so…
In this article we study the connection of fractional Brownian motion, representation theory and reflection positivity in quantum physics. We introduce and study reflection positivity for affine isometric actions of a Lie group on a Hilbert space E and show in particular that fractional Brownian motion for Hurst index 0<H\le 1/2 is reflection positive and leads via reflection positivity to an infinite dimensional Hilbert space if 0<H <1/2. We also study projective invariance of fractional Brownian motion and relate this to the complementary series representations of GL(2,R). We relate this to …
We investigate the "hot--spots" property for the survival time probability of Brownian motion with killing and reflection in planar convex domains whose boundary consists of two curves, one of which is an arc of a circle, intersecting at acute angles. This leads to the "hot--spots" property for the mixed Dirichlet--Neumann eigenvalue problem in the domain with Neumann conditions on one of the curves and Dirichlet conditions on the other
A geometric Brownian motion with delay is the solution of a stochastic differential equation where the drift and diffusion coefficient depend linearly on the past of the solution, i.e. a linear stochastic functional differential equation. In this work the asymptotic behavior in mean square of a geometric Brownian motion with delay is completely characterized by a sufficient and necessary condition in terms of the drift and diffusion coefficients.
We study the asymptotic behaviour of the time-changed stochastic process $\vphantom{X}^f\!X(t)=B(\vphantom{S}^f\!S (t))$, where $B$ is a standard one-dimensional Brownian motion and $\vphantom{S}^f\!S$ is the (generalized) inverse of a subordinator, i.e. the first-passage time process corresponding to an increasing Lévy process with Laplace exponent $f$. This type of processes plays an important role in statistical physics in the modeling of anomalous subdiffusive dynamics. The main result of the paper is the proof of the mixing property for the sequence of stationary increments of a subdiffus…
We investigate solutions of backward stochastic differential equations (BSDE) with time delayed generators driven by Brownian motions and Poisson random measures, that constitute the two components of a Levy process. In this new type of equations, the generator can depend on the past values of a solution, by feeding them back into the dynamics with a time lag. For such time delayed BSDE, we prove existence and uniqueness of solutions provided we restrict on a sufficiently small time horizon or the generator possesses a sufficiently small Lipschitz constant. We study differentiability in the va…
Protein translocation in cells has been modelled by \emph{Brownian ratchets}. In such models, the protein diffuses through a nanopore. On one side of the pore, ratcheting molecules bind to the protein and hinder it to diffuse out of the pore. We study a Brownian ratchet by means of a reflected Brownian motion $(X_t)_{t\geq 0}$ with a changing reflection point $(R_t)_{t\geq 0}$. The rate of change of $R_t$ is $γ(X_t-R_t)$ and the new reflection boundary is distributed uniformly between $R_{t-}$ and $X_t$. The asymptotic speed of the ratchet scales with $γ^{1/3}$ and the asymptotic variance is i…
We prove that reflected Brownian motion with normal reflections in a convex domain satisfies a dimension free Talagrand type transportation cost-information inequality. The result is generalized to other reflected diffusions with suitable drifts and diffusions. We apply this to get such an inequality for interacting Brownian particles with rank-based drift and diffusion coefficients such as the infinite Atlas model. This is an improvement over earlier dimension-dependent results.
In this paper, we study the tail behavior of $\max_{i\leq N}\sup_{s>0}\left(W_i(s)+W_A(s)-βs\right)$ as $N\to\infty$, with $(W_i,i\leq N)$ i.i.d. Brownian motions and $W_A$ an independent Brownian motion. This random variable can be seen as the maximum of $N$ mutually dependent Brownian queues, which in turn can be interpreted as the backlog in a Brownian fork-join queue. In previous work, we have shown that this random variable centers around $\frac{σ^2}{2β}\log N$. Here, we analyze the rare-event that this random variable reaches the value $(\frac{σ^2}{2β}+a)\log N$, with $a>0$. It turns out…
In this paper, we consider a class of multi-dimensional stochastic delay differential equations with jump reflection. Based on existence and uniqueness of the strong solution to the equation, we prove that the Markov semigroup generated by the segment process corresponding to the solution admits a unique invariant measure on the Skorohod space when the coefficients of equation satisfy a class of monotone conditions. Finally, we establish a relationship between the regulator and the local time of the solution and discuss a local time property at large time under the stationary setting.
Based on the classical probability, the stability criteria for stochastic differential delay equations (SDDEs) where their coefficients are either linear or nonlinear but bounded by linear functions have been investigated intensively. Moreover, the dependent stability of the highly nonlinear hybrid stochastic differential equations is recently studied. In this paper, by using the nonlinear expectation theory, we explore the dependent stability of a class of highly nonlinear hybrid stochastic differential delay equations driven by $G$-Brownian motion ($G$-SDDEs). Firstly, we give preliminaries …
We construct obliquely reflected Brownian motions in all bounded simply connected planar domains, including non-smooth domains, with general reflection vector fields on the boundary. Conformal mappings and excursion theory are our main technical tools. A key intermediate step, which may be of independent interest, is an alternative characterization of reflected Brownian motions in smooth bounded planar domains with a given field of angles of oblique reflection on the boundary in terms of a pair of quantities, namely an integrable positive harmonic function, which represents the stationary dist…
We prove strong existence and uniqueness for a reflection process $X$ in a smooth, bounded domain $D$ that behaves like obliquely-reflected-Brownian-motion, except that the direction of reflection depends on a (spin) parameter $S$, which only changes when $X$ is on the boundary of $D$ according to a physical rule. The process $(X,S)$ is a degenerate diffusion. We show uniqueness of the stationary distribution by using techniques based on excursions of $X$ from $\partial D$, and an associated exit system. We also show that the process admits a submartingale formulation and use related results t…
The well-known reflection coupling gives a maximal coupling of two one-dimensional Brownian motions with different starting points. Nevertheless, the reflection coupling does not generalize to more than two Brownian motions. In this paper, we construct a coupling of all Brownian motions with all possible starting points (i.e., a grand coupling), such that the coupling for any pair of the coupled processes is close to being maximal, that is, the distribution of the coupling time of the pair approaches that of the maximal coupling as the time tends to $0$ or $\infty$, and the coupling time of th…
Brownian motion in R 2 + with covariance matrix $Σ$ and drift $μ$ in the interior and reflection matrix R from the axes is considered. The asymptotic expansion of the stationary distribution density along all paths in R 2 + is found and its main term is identified depending on parameters ($Σ$, $μ$, R). For this purpose the analytic approach of Fayolle, Iasnogorodski and Malyshev in [12] and [36], restricted essentially up to now to discrete random walks in Z 2 + with jumps to the nearest-neighbors in the interior is developed in this article for diffusion processes on R 2 + with reflections on…
(i) Uncountably many synchronized reflected Brownian motions can hit the boundary of a $C^2$ domain at the same time. (ii) Measures associated to local times of two synchronized reflected Brownian motions are mutually singular until the time when the normal vectors at the reflection locations become identical. (iii) Mirror coupled reflected Brownian motions can simultaneously hit opposite sides of a wedge at different distances from the origin.
Markovian maximal couplings of Markov processes are characterized by an equality of total variation and a distance of Wasserstein type. If a Markovian maximal coupling is a Feller process, the generator can be calculated, e.g. for reflection coupled Brownian motion. Apart from processes with continuous paths also jump processes are treated for the first time. For subordinated Brownian motion a Markovian maximal coupling is constructed by subordinating reflection coupled Brownian motion. This coupling is the unique Markovian maximal coupling and its generator is determined by state-space depend…
The $h$-trimming of a tree is a natural regularization procedure which consists in pruning the small branches of a tree: given $h\geq0$, it is obtained by only keeping the vertices having at least one leaf above them at a distance greater or equal to $h$. The $h$-cut of a function $f$ is the function of minimal total variation uniformly approximating the increments of $f$ with accuracy $h$, and can be explicitly constructed via the two-sided Skorohod reflection of $f$ on the interval $[0,h]$. In this work, we show that the contour path of the $h$-trimming of a rooted real tree is given by the …
Using coupling by change of measure and an approximation technique, Wang's Harnack inequalities are established for a class of functional SDEs driven by subordinate Brownian motions. The results cover the corresponding ones in the case without delay.
We derive sufficient conditions for asymptotic and monotone exponential decay in mean square of solutions of the geometric Brownian motion with delay. The conditions are written in terms of the parameters and are explicit for the case of asymptotic decay. For exponential decay, they are easily resolvable numerically. The analytical method is based on construction of a Lyapunov functional (asymptotic decay) and forward-backward estimate for the square mean (exponential decay).
In this note we prove an existence and uniqueness result for the solution of multidimensional stochastic delay differential equations with normal reflection. The equations are driven by a fractional Brownian motion with Hurst parameter $H>1/2$. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann--Stieltjes integral.
We give necessary and sufficient conditions for the stationary density of semimartingale reflected Brownian motion in a wedge to be written as a finite sum of terms of exponential product form. Relying on geometric ideas reminiscent of the reflection principle, we give an explicit formula for the density in such cases.
In this paper, we study the reflected solutions of one-dimensional backward stochastic differential equations driven by G-Brownian motion (RGBSDE for short). The reflection keeps the solution above a given stochastic process. In order to derive the uniqueness of reflected GBSDEs, we apply a "martingale condition" instead of the Skorohod condition. Similar to the classical case, we prove the existence by approximation via penalization.
In this paper we study the stochastic differential equations driven by $G$-Brownian motion ($G$-SDEs for short). We extend the notion of conditional $G$-expectation from deterministic time to the more general optional time situation. Then, via this conditional expectation, we develop the strong Markov property for $G$-SDEs. In particular, we obtain the strong Markov property for $G$-Brownian motion. Some applications including the reflection principle for $G$-Brownian motion are also provided.
We consider two reflecting diffusion processes $(X_t)_{t \ge 0}$ with a moving reflection boundary given by a non-decreasing pure jump Markov process $(R_t)_{t \ge 0}$. Between the jumps of the reflection boundary the diffusion part behaves as a reflecting Brownian motion with negative drift or as a reflecting Ornstein-Uhlenbeck process. In both cases at rate $γ(X_t -R_t)$ for some $γ\ge 0$ the reflection boundary jumps to a new value chosen uniformly in $[R_{t-},X_t]$. Since after each jump of the reflection boundary the diffusions are reflected at a higher level we call the processes Brownia…
This paper focuses on controllability results of stochastic delay partial functional integro-differential equations perturbed by fractional Brownian motion. Sufficient conditions are established using the theory of resolvent operators combined with a fixed point approach for achieving the required result. An example is provided to illustrate the theory.
We consider the problem of strong existence and uniqueness of a Brownian motion forced to stay in the quadrant by an electrostatic repulsion from the sides that works obliquely. The results are reminiscent of the study of a Brownian motion with oblique reflection in a wedge. Actually, the same skew symmetry condition is involved when looking for a stationary distribution in product form. the terms of the product are now gamma distributions in place of exponential ones. An associate purely deterministic problem is also considered.
Consider a sequence of n bi-infinite and stationary Brownian queues in tandem. Assume that the arrival process entering in the first queue is a zero mean ergodic process. We prove that the departure process from the n-th queue converges in distribution to a Brownian motion as n goes to infinity. In particular this implies that the Brownian motion is an attractive invariant measure for the Brownian queueing operator. Our proof exploits the relationship between the Brownian queues in tandem and the last-passage Brownian percolation model, developing a coupling technique in the second setting. Th…
A multidimensional Brownian motion with partial reflection on a hyperplane $S$ in the direction $qN+α$, where $N$ is the conormal vector to the hyperplane and $q\in [-1,1], α\in S$ are given parametres, is constructed and this construction is based on both analytic and probabilistic approaches. The joint distribution of $d-$dimensional analogy to skew Brownian motion and its local time on the hyperplane is obtained.
We consider a mean-field optimal control problem for stochastic differential equations with delay driven by fractional Brownian motion with Hurst parameter greater than one half. Stochastic optimal control problems driven by fractional Brownian motion can not be studied using classical methods, because the fractional Brownian motion is neither a Markov process nor a semi-martingale. However, using the fractional White noise calculus combined with some special tools related to the differentiation for functions of measures, we establish and prove necessary and sufficient stochastic maximum princ…
Semimartingale reflecting Brownian motions (SRBMs) living in the closures of domains with piecewise smooth boundaries are of interest in applied probability because of their role as heavy traffic approximations for some stochastic networks. In this paper, assuming certain conditions on the domains and directions of reflection, a perturbation result, or invariance principle, for SRBMs is proved. This provides sufficient conditions for a process that satisfies the definition of an SRBM, except for small random perturbations in the defining conditions, to be close in distribution to an SRBM. A cr…
For Brownian motion in a (two-dimensional) wedge with negative drift and oblique reflection on the axes, we derive an explicit formula for the Laplace transform of its stationary distribution (when it exists), in terms of Cauchy integrals and generalized Chebyshev polyno-mials. To that purpose we solve a Carleman-type boundary value problem on a hyperbola, satisfied by the Laplace transforms of the boundary stationary distribution.
We consider a Brownian motion with drift in the quarter plane with orthogonal reflection on the axes. The Laplace transform of its stationary distribution satisfies a functional equation, which is reminiscent from equations arising in the enumeration of (discrete) quadrant walks. We develop a Tutte's invariant approach to this continuous setting, and we obtain an explicit formula for the Laplace transform in terms of generalized Chebyshev polynomials.
We prove the existence and uniqueness of a strong solution of a stochastic differential equation with normal reflection representing the random motion of finitely many globules. Each globule is a sphere with time-dependent random radius and a center moving according to a diffusion process. The spheres are hard, hence non-intersecting, which induces in the equation a reflection term with a local (collision-)time. A smooth interaction is considered too and, in the particular case of a gradient system, the reversible measure of the dynamics is given. In the proofs, we analyze geometrical properti…
We consider the problem of conditioning the Brownian excursion to have a fixed time average over the interval [0,1] and we study an associated stochastic partial differential equation with reflection at 0 and with the constraint of conservation of the space average. The equation is driven by the derivative in space of a space-time white noise and contains a double Laplacian in the drift. Due to the lack of the maximum principle for the double Laplacian, the standard techniques based on the penalization method do not yield existence of a solution.
We construct optimal Markov couplings of Lévy processes, whose Lévy (jump) measure has an absolutely continuous component. The construction is based on properties of subordinate Brownian motions and the coupling of Brownian motions by reflection.
In this article, we illustrate the flexibility of the algebraic integration formalism introduced by M. Gubinelli (2004), by establishing an existence and uniqueness result for delay equations driven by rough paths. We then apply our results to the case where the driving path is a fractional Brownian motion with Hurst parameter H>1/3.
Stochastic variational inequalities provide a unified treatment for stochastic differential equations living in a closed domain with normal reflection and (or) singular repellent drift. When the domain is a polyhedron, we prove that the reflected-repelled Brownian motion does not hit the non-smooth part of the boundary. A sufficient condition for non-hitting a face of the polyhedron is derived from the one-dimensional case. A complete answer to the question of attainability of the walls of the Weyl chamber may be given for a radial Dunkl process.
