Narrow Results

Let E be a second countable locally compact Hausdorff space and let L be a linear operator with domain D(L) and range R(L) in C₀(E). Suppose that D(L) is dense in E and that the operator L possesses the Korovkin property or, more generally, the Stone property with respect to some large collection of continuous functions (cf. Definition 3.8). For every x∈E the martingale problem is well-posed if and only if there exists a unique extension of L that generates a Feller semigroup in C₀(E). Next let L₀ be the generator of a Feller semigroup in C₀(E) and let L₁ and T be linear operators with the following properties: the operator I–T has range D(L₁), the domain of L₁, L₁ verifies the maximum principle, the vector sum of the spaces R(I–T) and R(L₁(I–T)) is dense in C₀(E), and . Then there exists at most one linear extension L of the operator L₁ for which LT is bounded and that generates a Feller semigroup. Similarly, if the martingale problem is solvable for L₁, then it is uniquely solvable for L₁, provided that the operator is a bounded linear map in C₀(E). Here (ℙ_x, X(t)) is a solution to the martingale problem for L₁.

Some related results for dissipative operators in a Banach space are presented as well.

We derive deterministic criteria for the existence and nonexistence of equivalent (local) martingale measures for financial markets driven by multi-dimensional time-inhomogeneous diffusions. Our conditions can be used to construct financial markets in which the no unbounded profit with bounded risk condition holds, while the classical no free lunch with vanishing risk condition fails.

In this note we study the 2D stochastic quasi-geostrophic equation in 𝕋² for general parameter α ∈ (0, 1) and multiplicative noise. We prove the existence of martingale solutions and pathwise uniqueness under some condition in the general case, i.e. for all α ∈ (0, 1). In the subcritical case α > 1/2, we prove existence and uniqueness of (probabilistically) strong solutions and construct a Markov family of solutions. In particular, it is uniquely ergodic for α > 2/3 provided the noise is non-degenerate. In this case, the convergence to the (unique) invariant measure is exponentially fast. In the general case, we prove the existence of Markov selections.

In this paper we prove the existence of martingale solutions for the 2D stochastic fractional vorticity Navier–Stokes equation driven by space-time white noise for α ∈ (½, 1] and the 2D stochastic quasi-geostrophic equation on 𝕋² for α ∈ (0, 1] driven by non-trace class noise. We also cover the case driven by non-trace class multiplicative noise for all α ∈ (0, 1].

The purpose of this work is to develop a unified approach to study the dynamics of a single-degree-of-freedom system excited by both periodic and random perturbations. As a prototype, we consider the noisy Duffing–van der Pol–Mathieu equation and achieve a reduction by developing rigorous methods to replace, in a limiting regime, the original complicated system by a simpler, constructive, and rational approximation — a lower-dimensional model of the dynamical system. To this end, we study the equations as a random perturbation of a two-dimensional weakly dissipative time-periodic Hamiltonian system. We achieve the model-reduction through stochastic averaging and the reduced Markov process takes its values on a graph with certain glueing conditions at the vertex of the graph. Examination of the reduced Markov process on the graph yields many important results, namely, mean exit times, probability density functions, and stochastic bifurcations.

We consider a two-dimensional weakly dissipative dynamical system with time-periodic drift and diffusion coefficients. The average of the drift is governed by a degenerate Hamiltonian whose set of critical points has an interior. The dynamics of the system is studied in the presence of three time scales. Using the martingale problem approach and separating the time scales, we average the system to show convergence to a Markov process on a stratified space. The averaging combines the deterministic time averaging of periodic coefficients, and the stochastic averaging of the resulting system. The corresponding strata of the reduced space are a two-sphere, a point and a line segment. Special attention is given to the description of the domain of the limiting generator, including the analysis of the gluing conditions at the point where the strata meet. These gluing conditions, resulting from the effects of the hierarchy of time scales, are similar to the conditions on the domain of skew Brownian motion and are related to the description of spider martingales.

We consider the stochastic Burgers equation

Given a controlled diffusion with jumps, it is shown under some conditions that there exists a Markov controlled diffusion which has the same cost under any criterion which only depends on the one-dimensional distributions. This result is then used to prove that the finite dimensional marginal distribution of a controlled diffusion with jumps at a prescribed set of time instants can also be attained by using a control from a much smaller class of controls called "nearly Markov controls", extending the work of Borkar [4] to the discontinuous setting.

The characteristic equation for a linear delay differential equation (DDE) has countably infinite roots on the complex plane. We deal with linear DDEs that are on the verge of instability, i.e. a pair of roots of the characteristic equation (eigenvalues) lie on the imaginary axis of the complex plane, and all other roots have negative real parts. We show that, when the system is perturbed by small noise, under an appropriate change of time scale, the law of the amplitude of projection onto the critical eigenspace is close to the law of a certain one-dimensional stochastic differential equation (SDE) without delay. Further, we show that the projection onto the stable eigenspace is small. These results allow us to give an approximate description of the delay-system using an SDE (without delay) of just one dimension. The proof is based on the martingale problem technique.

We introduce a generalized notion of semilinear elliptic partial differential equations where the corresponding second order partial differential operator $L$ has a generalized drift. We investigate existence and uniqueness of generalized solutions of class $C^1$. The generator $L$ is associated with a Markov process $X$ which is the solution of a stochastic differential equation with distributional drift. If the semilinear PDE admits boundary conditions, its solution is naturally associated with a backward stochastic differential equation (BSDE) with random terminal time, where the forward process is $X$. Since $X$ is a weak solution of the forward SDE, the BSDE appears naturally to be driven by a martingale. In the paper we also discuss the uniqueness of solutions of a BSDE with random terminal time when the driving process is a general càdlàg martingale.

Let $d\ge 2$. In this paper, we study weak solutions for the following type of stochastic differential equations: $d{X}_t=d{S}_t+b(s+t,X_t)dt,t\ge 0,X_0=x$, where $(s,x)\in {ℝ}_{+}\times {ℝ}^d$ is the starting point, $b:{ℝ}_{+}\times {ℝ}^d\to {ℝ}^d$ is measurable, and $S={(S_t)}_{t\ge 0}$ is a $d$-dimensional centered $\alpha$-stable process with index $\alpha \in (1,2)$. We show that if the centered $\alpha$-stable process $S$ is non-degenerate and $b\in L_{\mathrm{\text{loc}}}^{\infty }({ℝ}_{+};L^{\infty }({ℝ}^d))+L_{\mathrm{\text{loc}}}^q({ℝ}_{+};L^p({ℝ}^d))$ for some $p,q>0$ with $d/p+\alpha /q<\alpha -1$, then the above SDE has a unique weak solution for every starting point $(s,x)\in {ℝ}_{+}\times {ℝ}^d$.

We establish weak well-posedness for critical symmetric stable driven SDEs in ${ℝ}^d$ with additive noise $Z$, $d\ge 1$. Namely, we study the case where the stable index of the driving process $Z$ is $\alpha =1$ which exactly corresponds to the order of the drift term having the coefficient $b$ which is continuous and bounded. In particular, we cover the cylindrical case when $Z_t=(Z_t^1,\dots ,Z_t^d)$ and $Z^1,\dots ,Z^d$ are independent one-dimensional Cauchy processes. Our approach relies on $L^p$-estimates for stable operators and uses perturbative arguments.

We prove existence and uniqueness of martingale solutions to a (slightly) hyper-viscous stochastic Navier–Stokes equation in 2d with initial conditions absolutely continuous with respect to the Gibbs measure associated to the energy, getting the results both in the torus and in the whole space setting.

A finite-state Markov chain is introduced in the noise terms of the three-dimensional stochastic Navier–Stokes equations in order to allow for transitions between two types of multiplicative noises. We call such systems as stochastic Navier–Stokes equations with Markov switching. To solve such a system, a family of regularized stochastic systems is introduced. For each such regularized system, the existence of a unique strong solution (in the sense of stochastic analysis) is established by the method of martingale problems and pathwise uniqueness. The regularization is removed in the limit by obtaining a weakly convergent sequence from the family of regularized solutions, and identifying the limit as a solution of the three-dimensional stochastic Navier–Stokes equation with Markov switching.