Mireia Besalú, Carles Rovira
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.
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In this paper we consider stochastic differential equations with non-negativity constraints, driven by a fractional Brownian motion with Hurst parameter $H>\1/2$. We first study an ordinary integral equation where the integral is defined in the Young sense and then we apply this result pathwise to solve the stochastic problem.
On the negativity of the top Lyapunov exponent for stochastic differential equations driven by fractional Brownian motion
本文研究了由分数布朗运动驱动的随机微分方程的顶部李雅普诺夫指数的符号特性，通过分析随机动力系统并构造不变测度，得出噪声强度增加时该指数为负的结论。
In this work, we propose the balanced implicit method (BIM) to approximate the solution of the delay Cox-Ingersoll-Ross (CIR) model with jump which often gives rise to model an asset price and stochastic volatility . We show that this method preserves non-negativity property of the solution of this model with appropriate control functions. We prove the strong convergence and investigate the $p$th moment boundedness of the solution of BIM. Finally, we illustrate those results in the last section.
In this note we prove an existence and uniqueness result of solution for multidimensional delay differential equations with normal reflection and driven by a Hölder continuous function of order $β\in (\frac13,\frac12)$. We also obtain a bound for the supremum norm of this solution. As an application, we get these results for stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H $\in (\frac13,\frac12)$.
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 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 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.
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.
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.
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 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).
Anticipated BSDEs driven by fractional Brownian motion with time-delayed generator
本文研究由分数布朗运动驱动且生成子含时滞的预期倒向随机微分方程，利用不动点定理证明了解的存在唯一性，并建立了解的比较定理。
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…
On the positivity of the density of stochastic delay differential equations driven by a fractional Brownian motion
本文在系数光滑性与扩散项椭圆性条件下，证明分数布朗运动驱动的常延迟随机微分方程解的密度在其支撑集上严格为正，并通过结合 Nourdin-Viens 与 Kohatsu-Higa 方法给出高斯型下界。
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…
Convergence rate of nonlinear delayed neutral McKean-Vlasov SDEs driven by fractional Brownian motions
本文研究了由分数布朗运动驱动的超线性延迟中立型McKean-Vlasov随机微分方程的强收敛性，证明了解的唯一存在性、矩有界性与混沌传播，并给出了Euler-Maruyama格式的强收敛结果及数值验证。
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…
Stationary solution for a fractional stochastic delay differential equations with multiple delays
本文研究分数布朗运动驱动的线性随机延迟微分方程，通过基本解和路径wise积分获得显式解，证明在稳定性条件下解收敛于平稳高斯过程，并构造了唯一平稳解，为含延迟和分数记忆的平稳系统建模提供了框架。
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…
Singular stochastic delay equations driven by fractional Brownian motion: Dynamics, longtime behaviour, and pathwise stability
本文研究由分数布朗运动驱动的奇异随机延迟方程，证明其适定性，建立Lyapunov谱与局部流形，并给出路径局部指数稳定性条件。
Numerical scheme for delay-type stochastic McKean-Vlasov equations driven by fractional Brownian motion
本文针对分数布朗运动驱动的延迟型随机McKean-Vlasov方程，证明了解的存在唯一性及混沌传播，并指出经典Euler-Maruyama格式在多项式延迟条件下仍有效，给出了收敛速率，并通过意见动力学模型验证了理论。
We consider the sum of two self-similar centred Gaussian processes with different self-similarity indices. Under non-negativity assumptions of covariance functions and some further minor conditions, we show that the asymptotic behaviour of the persistence probability of the sum is the same as for the single process with the greater self-similarity index. In particular, this covers the mixed fractional Brownian motion introduced in Cheridito (2001) and shows that the corresponding persistence probability decays asymptotically polynomially with persistence exponent $1-\max(1/2,H),$ where $H$ is …
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.
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.
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.
In this article we investigate the controllability for neutral stochastic functional integro-differential equations with finite delay, driven by a fractional Brownian motion with Hurst parameter lesser than $1/2$ in a Hilbert space. We employ the theory of resolvent operators combined with the Banach fixed point theorem to establish sufficient conditions to prove the desired result
In this paper we consider the controllability of certain class of non-autonomous neutral evolution stochastic functional differential equations, with time varying delays, driven by a fractional Brownian motion in a separable real Hilbert space. Sufficient conditions for controllability are obtained by employing a fixed point approach. A practical example is provided to illustrate the viability of the abstract result of this work.
In this paper we consider a class of time-dependent neutral stochastic functional differential equations with finite delay driven by a fractional Brownian motion in a Hilbert space. We prove an existence and uniqueness result for the mild solution by means of the Banach fixed point principle. A practical example is provided to illustrate the viability of the abstract result of this work.
In this paper, we study the existence and uniqueness of mild solution for a stochastic neutral partial functional integro-differential equation with delay in a Hilbert space driven by a fractional Brownian motion and with non-deterministic diffusion coefficient. We suppose that the linear part has a resolvent operator. We also establish a sufficient condition for the existence of the density of a function of the solution. An example is provided to illustrate the results of this work
Stability equivalence for stochastic differential equations, stochastic differential delay equations and their corresponding Euler-Maruyama methods in $G$-framework
该研究证明在 $G$-框架下，当延迟或数值步长足够小时，$G$-SDDE、辅助 $G$-SDE 及其 EM 方法在 $p$ 阶矩实际指数稳定性上彼此等价，从而支持通过模拟检验稳定性。
In this note we consider a class of neutral stochastic functional differential equations with finite delay driven simultaneously by a fractional Brownian motion and a Poisson point processes in a Hilbert space. We prove an existence and uniqueness result and we establish some conditions ensuring the exponential decay to zero in mean square for the mild solution by means of the Banach fixed point principle.
In this paper, we establish a large deviation principle for stochastic differential delay equations driven by both Brownian motions and Poisson random measures. The weak convergence method plays an important role.
In this paper we study the existence of a unique solution to a general class of Young delay differential equations driven by a Hölder continuous function with parameter greater that 1/2 via the Young integration setting. Then some estimates of the solution are obtained, which allow to show that the solution of a delay differential equation driven by a fractional Brownian motion (fBm) with Hurst parameter H>1/2 has a smooth density. To this purpose, we use Malliavin calculus based on the Frechet differentiability in the directions of the reproducing kernel Hilbert space associated with fBm.
The paper is concerned with a mixed stochastic delay differential equation involving both a Wiener process and a $γ$-Hölder continuous process with $γ>1/2$ (e.g. a fractional Brownian motion with Hurst parameter greater than $1/2$). It is shown that its solution depends continuously on the coefficients and the initial data. Two applications of this result are given: the convergence of solutions to equations with vanishing delay to the solution of equation without delay and the convergence of Euler approximations for mixed stochastic differential equations. As a side result of independent inter…
Aiming for more realistic optimal dividend policies, we consider a stochastic control problem with linearly bounded control rates using a performance function given by the expected present value of dividend payments made up to ruin. In a Brownian model, we prove the optimality of a member of a new family of control strategies called delayed linear control strategies, for which the controlled process is a refracted diffusion process. For some parameters specifications, we retrieve the strategy initially proposed by Avanzi & Wong (2012) to regularize dividend payments, which is more consistent w…
In this paper, we are concerned with convergence rate of Euler-Maruyama (EM) scheme for stochastic differential delay equations (SDDEs) of neutral type, where the neutral term, the drift term and the diffusion term are allowed to be of polynomial growth. More precisely, for SDDEs of neutral type driven by Brownian motions, we reveal that the convergence rate of the corresponding EM scheme is one half; Whereas for SDDEs of neutral type driven by jump processes, we show that the best convergence rate of the associated EM scheme is close to one half.
Delay rough evolution equations
该研究证明了一类延迟粗糙偏微分方程解的存在性与稳定性，验证了延迟趋于零时其解向粗糙偏微分方程解的收敛性，并将结果应用于分数布朗运动驱动的延迟随机偏微分方程研究。
Consider the motion of a Brownian particle in two or more dimensions, whose coordinate processes are standard Brownian motions with zero drift initially, and then at some random/unobservable time, one of the coordinate processes gets a (known) non-zero drift permanently. Given that the position of the Brownian particle is being observed in real time, the problem is to detect the time at which a coordinate process gets the drift as accurately as possible. We solve this problem in the most uncertain scenario when the random/unobservable time is (i) exponentially distributed and (ii) independent …
In this paper we investigate the argmin process of Brownian motion $B$ defined by $α_t:=\sup\left\{s \in [0,1]: B_{t+s}=\min_{u \in [0,1]}B_{t+u} \right\}$ for $t \geq 0$. The argmin process $α$ is stationary,with invariant measure which is arcsine distributed. We prove that $(α_t; t \geq 0)$ is a Markov process with the Feller property, and provide its transition kernel $Q_t(x,\cdot)$ for $t>0$ and $x \in [0,1]$. Similar results for the argmin process of random walks and Lévy processes are derived. We also consider Brownian extrema of a given length. We prove that these extrema form a delayed…
We consider a stationary fluid queue with fractional Brownian motion input. Conditional on the workload at time zero being greater than a large value $b$, we provide the limiting distribution for the amount of time that the workload process spends above level $b$ over the busy cycle straddling the origin, as $b\to\infty$. Our results can be interpreted as showing that long delays occur in large clumps of size of order $b^{2-1/H}$. The conditional limit result involves a finer scaling of the queueing process than fluid analysis, thereby departing from previous related literature.
Strong convergence of tamed theta scheme for superlinearly growing McKean-Vlasov NSDDEs driven by fractional Brownian motions
针对分数布朗运动驱动的超线性增长McKean-Vlasov中立型随机微分延迟方程，提出驯化theta Euler-Maruyama格式，证明解的存在唯一性，并建立数值格式的收敛速率。
In this paper, employing the weak convergence method, based on a variational representation for expected values of positive functionals of a Brownian motion, we investigate moderate deviation %(CLT for abbreviation) for a class of stochastic differential delay equations with small noises, where the coefficients are allowed to be highly nonlinear growth with respect to the variables. Moreover, we obtain the central limit theorem for stochastic differential delay equations which the coefficients are polynomial growth with respect to the delay variables.
Non-local heat equations with moving boundary
本文研究时间递增抛物集上的非局部（时间）热方程，其边界由适当曲线确定，通过延迟布朗运动随机表示证明解的存在性、唯一性和连续性，并发现均方位移的异常扩散行为。
We introduce a new class of stochastic partial differential equations (SPDEs) with seed bank modeling the spread of a beneficial allele in a spatial population where individuals may switch between an active and a dormant state. Incorporating dormancy and the resulting seed bank leads to a two-type coupled system of equations with migration between both states. We first discuss existence and uniqueness of seed bank SPDEs and provide an equivalent delay representation that allows a clear interpretation of the age structure in the seed bank component. The delay representation will also be crucial…
In this paper, we are interested in path-dependent stochastic differential equations (SDEs) which are controlled by Brownian motion and its delays. Within this non-Markovian context, we give a H örmander-type criterion for the regularity of solutions. Indeed, our criterion is expressed as a spanning condition with brackets. A novelty in the case of delays is that noise can "flow from the past" and give additional smoothness thanks to semi-brackets. The proof follows the general lines of Malliavin's probabilistic proof, in the Markovian case. Nevertheless, in order to handle the non-Markovian a…
On/off super-Brownian motion: Characterization, construction and long-term behaviour
该研究引入并构造了作为临界开/关分支布朗运动测度值标度极限的开/关超布朗运动，揭示其与经典超布朗运动的性质差异，证明活跃亚种群会以正概率在有限时间内归零，并提出未来研究方向。
This paper is concerned with strong convergence and almost sure convergence for neutral stochastic differential delay equations under non-globally Lipschitz continuous coefficients. Convergence rates of $θ$-EM schemes are given for these equations driven by Brownian motion and pure jumps respectively, where the drift terms satisfy locally one-sided Lipschitz conditions, and diffusion coefficients obey locally Lipschitz conditions, and the corresponding coefficients are highly nonlinear with respect to the delay terms.
Relaxed Optimal Control Problem for a Finite Horizon G-SDE with Delay and Its Application in Economics
本文研究有限时域下G-布朗运动驱动的随机时滞微分方程的松弛最优控制存在性，关联其HJB方程与解耦G-正倒向随机时滞微分方程，通过仿真得到最优策略与成本，并给出经济学模型应用。
Path-dependent processes from signatures
本文利用时间增广布朗运动的路径签名，为随机线性 Volterra 和延迟方程等路径依赖积分方程提供了显式级数展开，揭示了其无限维马尔可夫结构并提出了简单的数值近似方案。
This note provides a simple sufficient condition ensuring that solutions of stochastic delay differential equations (SDDEs) driven by subordinators are non-negative. While, to the best of our knowledge, no simple non-negativity conditions are available in the context of SDDEs, we compare our result to the literature within the subclass of invertible continuous-time ARMA (CARMA) processes. In particular, we analyze why our condition cannot be necessary for CARMA($p,q$) processes when $p=2$, and we show that there are various situations where our condition applies while existing results do not a…
We consider an Ito stochastic differential equation with delay, driven by brownian motion, whose solution, by an appropriate reformulation, defines a Markov process $X$ with values in a space of continuous functions $\mathbf C$, with generator $\mathcal L$. We then consider a backward stochastic differential equation depending on $X$, with unknown processes $(Y,Z)$, and we study properties of the resulting system, in particular we identify the process $Z$ as a deterministic functional of $X$. We next prove that the forward-backward system provides a suitable solution to a class of parabolic pa…
The Euler Scheme for Fractional Stochastic Delay Differential Equations with Additive Noise
本文研究分数阶布朗运动驱动的随机延迟微分方程的Euler-Maruyama格式，证明了一致性并分析了归一化误差过程的收敛速度，发现在延迟平滑且$H>1/2$时可达$H+1/2$阶次优收敛。
Many real time-series exhibit behavior adequate to long range dependent data. Additionally very often these time-series have constant time periods and also have characteristics similar to Gaussian processes although they are not Gaussian. Therefore there is need to consider new classes of systems to model these kind of empirical behavior. Motivated by this fact in this paper we analyze two processes which exhibit long range dependence property and have additional interesting characteristics which may be observed in real phenomena. Both of them are constructed as the superposition of fractional…
Quickest Real-Time Detection of Multiple Brownian Drifts
本文针对多维布朗运动在随机时刻部分坐标获得漂移的实时检测问题，在指数分布时间下，通过最小化错误检测概率和延迟的停止时间，首次精确解决了亚椭圆情形，并推广到任意漂移坐标数。
Sufficient and necessary conditions are presented for the order-preservation of stochastic functional differential equations on $\R^d$ with non-Lipschitzian coefficients driven by the Brownian motion and Poisson processes. The sufficiency of the conditions extends and improves some known comparison theorems derived recently for one-dimesional equations and multidimensional equations without delay, and the necessity is new even in these special situations.
We show that all the time-dependent statistical properties of the rightmost points of a branching Brownian motion can be extracted from the traveling wave solutions of the Fisher-KPP equation. We show that the distribution of all the distances between the rightmost points has a long time limit which can be understood as the delay of the Fisher-KPP traveling waves when the initial condition is modified. The limiting measure exhibits the surprising property of superposability: the statistical properties of the distances between the rightmost points of the union of two realizations of the branchi…
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