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Functional integral representations for solutions of the motion equations for wall-bounded incompressible viscous flows, expressed (implicitly) in terms of distributions of solutions to stochastic differential equations of McKean-Vlasov type, are established by using a perturbation technique. These representations are used to obtain exact random vortex dynamics for wall-bounded viscous flows. Numerical schemes therefore are proposed and the convergence of the numerical schemes for random vortex dynamics with an additional force term is established. Several numerical experiments are carried out for demonstrating the motion of a viscous flow within a thin layer next to the fluid boundary., Comment: 45 pages, 12 figures

In this paper, we consider the stochastic averaging principle and stability for multi-valued McKean-Vlasov stochastic differential equations with jumps. First, under certain averaging conditions, we are able to show that the solutions of the equations concerned can be approximated by solutions of the associated averaged multi-valued McKean-Vlasov stochastic differential equations with jumps in the sense of the mean square convergence. Second, we extend the classical It\^{o}'s formula from stochastic differential equations to multi-valued McKean-Vlasov stochastic differential equations with jumps. Last, as application of It\^{o}'s formula, we present the exponential stability of second moments, the exponentially 2-ultimate boundedness and the almost surely asymptotic stability for their solutions in terms of a Lyapunov function., Comment: 31 pages. arXiv admin note: text overlap with arXiv:2106.12080 by other authors

Distribution dependent stochastic differential equations have been a very hot subject with extensive studies. On the other hand, under the $G$-expectation framework, stochastic differential equations driven by $G$-Brownian motion (in short form, $G$-SDEs) have received increasing attentions, and the existence and uniqueness of solutions to $G$-SDEs under Lipschitz and non-Lipschitz conditions have been obtained. Based on these studies, it is very natural and also important to investigate the $G$-SDEs which are also distribution dependent. In this paper, we are concerned with the well-posedness of the distribution dependent $G$-SDEs. To this end, we first introduce a proper distance of the involved distribution functions and propose a new formulation of the distribution dependent $G$-SDEs. Then, by utilising fix point argument, we establish existence and uniqueness of the solutions of distributed dependent $G$-SDEs under Lipschitz condition. Finally, we derive certain estimates for the solutions of the distribution dependent $G$-SDEs.

In this paper we are concerned with distribution dependent backward stochastic differential equations (DDBSDEs) driven by Gaussian processes. We first show the existence and uniqueness of solutions to this type of equations. This is done by formulating a transfer principle to transfer the well-posedness problem to an auxiliary DDBSDE driven by Brownian motion. Then, we establish a comparison theorem under Lipschitz condition and boundedness of Lions derivative imposed on the generator. Furthermore, we get a new representation for DDBSDEs driven by Gaussian processes, this representation is even new for the case of the equations driven by Brownian motion. The new obtained representation enables us to prove a converse comparison theorem. Finally, we derive transportation inequalities and Logarithmic-Sobolev inequalities via the stability of the Wasserstein distance and the relative entropy of measures under the homeomorphism condition., Comment: 29 pages

In this paper, we are concerned with stochastic Volterra equations with singular kernels and H\"older continuous coefficients. We first establish the well-posedness of these equations by utilising the Yamada-Watanabe approach. Then, we aim to characterise the path-independence for additive functionals of these equations. The main challenge here is that the solutions of stochastic Volterra equations are not semimartingales nor Markov processes, thus the existing techniques for obtaining the path-independence of usual, semimartingale type stochastic differential equations are no longer applicable. To overcome this difficulty, we link the concerned stochastic Volterra equations to mild formulation of certain parabolic type stochastic partial differential equations, and further apply our previous results on the path-independence for stochastic evolution equations to get the desired result. Finally, as an important application, we consider a class of stochastic Volterra equations whose kernels are related with fractional Brownian motions and derive the path-independence of additive functionals for them., Comment: 26 pages

In this paper, we develop low regularity theory for 3D Burgers equation perturbed by a linear multiplicative stochastic force. This method is new and essentially different from the deterministic partial differential equations(PDEs). Our results and method can be widely applied to other stochastic hydrodynamic equations and the deterministic PDEs. As a further study, we establish a random version of maximum principle for random 3D Burgers equations, which will be an important tool for the study of 3D stochastic Burgers equations. As we know establishing moment estimates for highly nonlinear stochastic hydrodynamic equations is difficult. But moment estimates are very important for us to study the probabilistic properties and long-time behavior for the stochastic systems. Here, the random maximum principle helps us to achieve some important moment estimates for 3D stochastic Burgers equations and lays a solid foundation for the further study of 3D stochastic Burgers equations., Comment: there are some typos, we need some times to revise it

In the paper, we consider a type of stochastic differential equations driven by G-L\'evy processes. We prove that a kind of their additive functionals has path independence and extend some known results., Comment: 13 pages

In this article, the path independent property of additive functionals of McKean-Vlasov stochastic differential equations with jumps is characterised by nonlinear partial integro-differential equations involving $L$-derivatives with respect to probability measures introduced by P.-L. Lions. Our result extends the recent work [16] by Ren and Wang where their concerned McKean-Vlasov stochastic differential equations are driven by Brownian motions., Comment: 16 pages

The flow of nematic liquid crystals can be described by a highly nonlinear stochastic hydrodynamical model, thus is often influenced by random fluctuations, such as uncertainty in specifying initial conditions and boundary conditions. In this article, we consider the $2$-D stochastic nematic liquid crystals with the velocity field perturbed by affine-linear multiplicative white noise, with random initial data and random boundary conditions. Our main objective is to establish the global well-posedness of the stochastic equations under certain sufficient Malliavin regularity of the initial conditions and the boundary conditions. The Malliavin calculus techniques play important roles in proving the global existence of the solutions to the stochastic nematic liquid crystal models with random initial and random boundary conditions. It should be pointed out that the global well-posedness is also true when the stochastic system is perturbed by the noise on the boundary.

In the paper, we are concerned with degenerate stochastic differential equations with jumps. Firstly, we establish two support theorems for the solutions of the degenerate stochastic equations, under different (sufficient) conditions. Secondly, we apply one of our support theorems to a class of degenerate stochastic evolution equations (i.e., infinite-dimensional stochastic differential equations) with jumps to get a characterisation of path-independence for the densities of their Girsanov transformations., Comment: 13 pages

In this paper, we established the Freidlin-Wentzell type large deviation principles for first-order scalar conservation laws perturbed by small multiplicative noise. Due to the lack of the viscous terms in the stochastic equations, the kinetic solution to the Cauchy problem for these first-order conservation laws is studied. Then, based on the well-posedness of the kinetic solutions, we show that the large deviations holds by utilising the weak convergence approach.

The aim of this paper is twofold. Firstly, we derive upper and lower non-Gaussian bounds for the densities of the marginal laws of the solutions to backward stochastic differential equations (BSDEs) driven by fractional Brownian motions. Our arguments consist of utilising a relationship between fractional BSDEs and quasilinear partial differential equations of mixed type, together with the profound Nourdin-Viens formula. In the linear case, upper and lower Gaussian bounds for the densities and the tail probabilities of solutions are obtained with simple arguments by their explicit expressions in terms of the quasi-conditional expectation. Secondly, we are concerned with Gaussian estimates for the densities of a BSDE driven by a Gaussian process in the manner that the solution can be established via an auxiliary BSDE driven by a Brownian motion. Using the transfer theorem we succeed in deriving Gaussian estimates for the solutions.

In the present paper, the effect of noise intensity on stochastic parabolic equations is discussed. We focus on the effect of noise on the energy solutions of the stochastic parabolic equations. By utilising It\^o's formula and the energy estimate method, we obtain the excitation indices of the energy solutions $u$ at any finite time $t$. Furthermore, we improve certain existing results in the literature by presenting a comparably simple method to show those existing results., Comment: 11 pages

We study a least squares estimator for an unknown parameter in the drift coefficient of a path- distribution dependent stochastic differential equation involving a small dispersion parameter epsilon greater than zero. The estimator, based on n discrete time observations of the stochastic differential equation, is shown to be convergent weakly to the true value as epsilon goes to zero and n goes to infinity. This indicates that the least squares estimator obtained is consistent with the true value. Moreover, we obtain the rate of convergence and derive the asymptotic distribution of least squares estimator., Comment: 24 pages

In this paper, we are concerned with regularity of nonlocal stochastic partial differential equations of parabolic type. By using Companato estimates and Sobolev embedding theorem, we first show the H\"{o}lder continuity (locally in the whole state space $\mathbb{R}^d$) for mild solutions of stochastic nonlocal diffusion equations in the sense that the solutions $u$ belong to the space $C^{\gamma}(D_T;L^p(\Omega))$ with the optimal H\"{o}lder continuity index $\gamma$ (which is given explicitly), where $D_T:=[0,T]\times D$ for $T>0$, and $D\subset\mathbb{R}^d$ being a bounded domain. Then, by utilising tail estimates, we are able to obtain the estimates of mild solutions in $L^p(\Omega;C^{\gamma^*}(D_T))$. What's more, we give an explicit formula between the two index $\gamma$ and $\gamma^*$. Moreover, we prove H\"{o}lder continuity for mild solutions on bounded domains. Finally, we present a new criteria to justify H\"{o}lder continuity for the solutions on bounded domains. The novelty of this paper is that our method are suitable to the case of time-space white noise., Comment: 23 pages

We extend Krylov and R\"{o}ckner's result \cite{KR} to the drift coefficients in critical Lebesgue space, and prove the existence and uniqueness of weak solutions for a class of SDEs. To be more precise, let $b: [0,T]\times{\mathbb R}^d\rightarrow{\mathbb R}^d$ be Borel measurable, where $T>0$ is arbitrarily fixed. Consider $$X_t=x+\int_0^tb(s,X_s)ds+W_t,\quad t\in[0,T], \, x\in{\mathbb R}^d,$$ where $\{W_t\}_{t\in[0,T]}$ is a $d$-dimensional standard Wiener process. If $b=b_1+b_2$ such that $b_1(T-\cdot)\in\mathcal{C}_q^0((0,T];L^p({\mathbb R}^d))$ with $2/q+d/p=1$ for $p,q\ge1$ and $\|b_1(T-\cdot)\|_{\mathcal{C}_q((0,T];L^p({\mathbb R}^d))}$ is sufficiently small, and that $b_2$ is bounded and Borel measurable, then there exits a unique weak solution to the above equation. Furthermore, we obtain the strong Feller property of the semi-group and existence of density associated with above SDE. Besides, we extend the classical partial differential equations (PDEs) results for $L^q(0,T;L^p({\mathbb R}^d))$ coefficients to $L^\infty_q(0,T;L^p({\mathbb R}^d))$ ones, and derive the Lipschitz regularity for solutions of second order parabolic PDEs (see Lemma 2.1).

In this paper, we consider the extended stochastic Navier-Stokes equations with Caputo derivative driven by fractional Brownian motion. We firstly derive the pathwise spatial and temporal regularity of the generalized Ornstein-Uhlenbeck process. Then we discuss the existence, uniqueness, and H\"{o}lder regularity of mild solutions to the given problem under certain sufficient conditions, which depend on the fractional order $\alpha$ and Hurst parameter $H$. The results obtained in this study improve some results in existing literature.

In this paper, motivated by modelling currency exchange markets with matrix-valued stochastic processes, matrix-valued stochastic differential equations (SDEs) are formulated. This is done based on the matrix trace, as for the purpose of modelling currency exchange markets. To be more precise, we set up a Hilbert space structure for $n\times n$ square matrices via the trace of the Hadamard product of two matrices. With the help of this framework, one can then define stochastic integral of It\^o type and It\^o SDEs. Two types of sufficient conditions are discussed for the existence and uniqueness of solutions to the matrix-valued SDEs., Comment: 28 pages, 1 figure

The purpose of this paper is to derive a critical link of parameters for the self-similar trajectories of jump-diffusions which are described as solutions of stochastic differential equations driven by $\alpha$-stable noise. This is done by a multivariate Lagrange interpolation approach. To this end, we utilise computer simulation algorithm in MATLAB to visualise the trajectories of the jump-diffusions for various combinations of parameters arising in the stochastic differential equations., Comment: 27 pages, 10 figures

In this paper, we introduce a matrix-valued time series model for foreign exchange market. We then formulate trading matrices, foreign exchange options and return options (matrices), as well as on-line portfolio strategies. Moreover, we attempt to predict returns of portfolios by developing a cross rate method. This leads us to construct an on-line portfolio selection algorithm for this model. At the end, we prove the profitability and the universality of our algorithm., Comment: 34 pages, 1 figure

In this paper, we are concerned with the estimates for the moments of stochastic convolution integrals. We first deal with the stochastic singular integral operators and we aim to derive the Morrey-Campanato estimates for the $p$-moments (for $p\ge1$). Then, by utilising the embedding theory between the Campanato space and H\"older space, we establish the norm of $C^{\theta,\theta/2}(\bar D)$, where $\theta\ge0, \bar D=\bar G\times[0,T]$ for arbitrarily fixed $T\in(0,\infty)$ and $G\subset\mathbb{R}^d$. As an application, we consider the following stochastic (fractional) heat equations with additive noises \bess du_t(x)=\Delta^\alpha u_t(x)dt+g(t,x)d\eta_t,\ \ \ u_0=0,\ 0\leq t\leq T, x\in G, \eess where $\Delta^\alpha=-(-\Delta)^\alpha$ with $0<\alpha\leq1$ (the fractional Laplacian), $g:[0,T]\times G\times\Omega\to\mathbb{R}$ is a joint measurable coefficient, and $\eta_t, t\in[0,T]$, is either the Brownian motion or a L\'evy process on a given filtered probability space $(\Omega,\mathcal{F},P;\{\mathcal{F}_t\}_{t\in[0,T]})$. The Schauder estimate for the $p$-moments of the solution of the above equation is obtained. The novelty of the present paper is that we obtain the Schauder estimate for parabolic stochastic partial differential equations with L\'evy noise., Comment: 19 pages

In this paper, we consider the stochastic singular integral operators and obtain the BMO estimates. As an application, we consider the fractional Laplacian equation with additive noises \bess du_t(x)=\Delta^{\frac{\alpha}{2}}u_t(x)dt+\sum_{k=1}^\infty\int_{\mathbb{R}^m}g^k(t,x)z\tilde N_k(dz,dt),\ \ \ u_0=0,\ 0\leq t\leq T, \eess where $\Delta^{\frac{\alpha}{2}}=-(-\Delta)^{\frac{\alpha}{2}}$, and $\int_{\mathbb{R}^m}z\tilde N_k(t,dz)=:Y_t^k$ are independent $m$-dimensional pure jump L\'{e}vy processes with L\'{e}vy measure of $\nu^k$. Following the idea of \cite{Kim}, we obtain the $q$-th order BMO quasi-norm of the $\frac{\alpha}{q_0}$-order derivative of $u$ is controlled by the norm of $g$., Comment: 22 pages

In this paper, we derive a characterization theorem for the path-independent property of the density of the Girsanov transformation for {\it degenerated} stochastic differential equations (SDEs), extending the characterization theorem of \cite{twwy} for the non-degenerated SDEs. We further extends our consideration to non-Lipschitz SDEs with jumps and with degenerated diffusion coefficients, which generalizes the corresponding characterization theorem established in \cite{hqwu}.

As a class of L\'evy type Markov generators, nonlocal Waldenfels operators appear naturally in the context of investigating stochastic dynamics under L\'evy fluctuations and constructing Markov processes with boundary conditions (in particular the construction with jumps). This work is devoted to prove the weak and strong maximum principles for `parabolic' equations with nonlocal Waldenfels operators. Applications in stochastic differential equations with $\alpha$-stable L\'evy processes are presented to illustrate the maximum principles., Comment: 38 pages, 3 figures

In this paper, we investigate stochastic continuity (with respect to the initial value), irreducibility and non confluence property of the solutions of stochastic differential equations with jumps. The conditions we posed are weaker than those relevant conditions existing in the literature. We also provide an example to support our new conditions., Comment: 16 pages

In this paper, we investigate new sufficient conditions to ensure the existence of a unique global strong solution of stochastic differential equations with jumps. By using Euler approximation and by utilising a new test function $\varphi_\delta$ (see the following definition (\ref{pntas1})), we prove that there is a unique global strong solution for the initial value problem of the equation. The condition we posed is even weaker than the local Lipschitzian continuity of the coefficients. We also present an example to show that our conditions are indeed weaker than those relevant conditions existing in the literature., Comment: This paper has been withdrawn by the author due to a crucial error in the proof

The object of the present paper is to find new sufficient conditions for the existence of unique strong solutions to a class of (time-inhomogeneous) stochastic differential equations with random, non-Lipschitzian coefficients. We give an example to show that our conditions are indeed weaker than those relevant conditions existing in the literature. We also derive moment estimations for the maximum process of the solution. Finally, we present a sufficient condition to ensure the non confluence property of the solution of time-homogeneous SDE which, in one dimension, is nothing but stochastic monotone property of the solution., Comment: 21 pages

This paper is concerned with the following Markovian stochastic differential equation of mean-reversion type \[ dR_t= (\theta +\sigma \alpha(R_t, t))R_t dt +\sigma R_t dB_t \] with an initial value $R_0=r_0\in\mathbb{R}$, where $\theta\in\mathbb{R}$ and $\sigma>0$ are constants, and the mean correction function $\alpha:\mathbb{R}\times[0,\infty)\to \alpha(x,t)\in\mathbb{R}$ is twice continuously differentiable in $x$ and continuously differentiable in $t$. We first derive that under the assumption of path independence of the density process of Girsanov transformation for the above stochastic differential equation, the mean correction function $\alpha$ satisfies a non-linear partial differential equation which is known as the viscous Burgers equation. We then develop a Galerkin type approximation scheme for the function $\alpha$ by utilizing truncation of discretised Fourier transformation to the viscous Burgers equation.