Transition Semigroup Research Articles

Strong Feller Property of Sticky Reflected Distorted Brownian Motion

Using Girsanov transformations we construct from sticky reflected Brownian motion on $$[0,\infty )$$ a conservative diffusion on $$E:=[0,\infty )^n$$ , $$n \in \mathbb {N}$$ , and prove that its transition semigroup possesses the strong Feller property for a specified general class of drift functions. By identifying the Dirichlet form of the constructed process we characterize it as sticky reflected distorted Brownian motion. In particular, the relations of the underlying analytic Dirichlet form methods to the probabilistic methods of random time changes and Girsanov transformations are presented. Our studies of the mathematical model are motivated by its applications to the dynamical wetting model with $$\delta $$ -pinning and repulsion.

Stochastic Optimal Control with Delay in the Control I: Solving the HJB Equation through Partial Smoothing

Stochastic optimal control problems governed by delay equations with delay in the control are usually more difficult to study than those in which the delay appears only in the state. This is particularly true when we look at the associated Hamilton--Jacobi--Bellman (HJB) equation. Indeed, even in the simplified setting (introduced first by Vinter and Kwong [SIAM J. Control Optim., 19 (1981), pp. 139--153] for the deterministic case) the HJB equation is an infinite-dimensional second order semilinear partial differential equation that does not satisfy the so-called structure condition, which substantially means that the control can act on the system, modifying its dynamics at most along the same directions upon which the noise acts. The absence of such a condition---together with the lack of smoothing properties, a common feature of problems with delay---prevents the use of known techniques (based on backward stochastic differential equations or on the smoothing properties of the linear part) to prove the existence of regular solutions of this HJB equation, and so no results in this direction have been proved till now. In this paper we provide a result on existence of regular solutions of HJB equations of this type. This opens the door to proving existence of optimal feedback controls, a task that will be accomplished in the companion paper [F. Gozzi and F. Masiero, SIAM J. Control Optim., 55 (2017), pp. 3013--3038]. The main tool used is a partial smoothing property that we prove for the transition semigroup associated to the uncontrolled problem. Such results hold for a specific class of equations and data which arise naturally in many applied problems.

The support projection of state of QMS with unbounded generator and the classical theorem for Markov chains with analytic transition semi-group

We characterize the support projection of a state evolving under the action of a with unbounded generator represented in the generalized GKSL form and we give another proof of the classical Lévy-Austin-Ornstein theorem.

Approximating a diffusion by a finite-state hidden Markov model

For a wide class of continuous-time Markov processes evolving on an open, connected subset of Rd, the following are shown to be equivalent: (i)The process satisfies (a slightly weaker version of) the classical Donsker–Varadhan conditions;(ii)The transition semigroup of the process can be approximated by a finite-state hidden Markov model, in a strong sense in terms of an associated operator norm;(iii)The resolvent kernel of the process is ‘v-separable’, that is, it can be approximated arbitrarily well in operator norm by finite-rank kernels. Under any (hence all) of the above conditions, the Markov process is shown to have a purely discrete spectrum on a naturally associated weighted L∞ space.

Mild solutions of semilinear elliptic equations in Hilbert spaces

This paper extends the theory of regular solutions (C1 in a suitable sense) for a class of semilinear elliptic equations in Hilbert spaces. The notion of regularity is based on the concept of G-derivative, which is introduced and discussed. A result of existence and uniqueness of solutions is stated and proved under the assumption that the transition semigroup associated to the linear part of the equation has a smoothing property, that is, it maps continuous functions into G-differentiable ones. The validity of this smoothing assumption is fully discussed for the case of the Ornstein–Uhlenbeck transition semigroup and for the case of invertible diffusion coefficient covering cases not previously addressed by the literature. It is shown that the results apply to Hamilton–Jacobi–Bellman (HJB) equations associated to infinite horizon optimal stochastic control problems in infinite dimension and that, in particular, they cover examples of optimal boundary control of the heat equation that were not treatable with the approaches developed in the literature up to now.

TRANSITIVE HYPERIDENTITY IN SEMIGROUPS

In this paper we characterize all semigroups in which the hyperidentity of transitivity $X(X(x, y), X(y, z)) = X(x, z) $ is polynomially satisfied. In particular, we show that every transitive semigroup (that is a semigroup with the identity $xy^2z = xz$ ) is also hypertransitive.

Exponential ergodicity of CIR interest rate model with random switching

In this paper, we consider ergodicity of Cox–Ingersoll–Ross (CIR) interest rate model with random switching. First, we show that the CIR model with switching has a unique stationary distribution. Next, we prove that the transition semigroup for the CIR model with switching converges to the stationary distribution at an exponential rate in the Wasserstein distance. Moreover, under two particular cases, the explicit expressions for stationary distributions are presented. Finally, the central limit theorem for the CIR model with random switching is established.

Ergodicity of large scale stochastic geophysical flows with degenerate Gaussian noise

This paper considers ergodicity of the stochastic quasigeostrophic equations driven by degenerate Gaussian noise. Uniqueness of invariant measures is shown by proving the asymptotically strong Feller property of the probability transition semigroups.

Estimate for $P_{t}D$ for the stochastic Burgers equation

Nous considèrons l’équation de Burgers stochastique sur $H=L^{2}(0,1)$ dirigée par un bruit blanc, de semi-groupe de transition $P_{t}$, et démontrons une nouvelle formule qui permet d’exprimer $P_{t}D_{x}\varphi$ en terme de $\varphi$ mais pas de sa différentielle. Celle-ci nous permet d’obtenir des estimations sur $D_{x}\varphi$ dans $L^{2}(H,\nu)$, où $\nu$ est la mesure invariante de $P_{t}$, dont découlent quelques conséquences telles que l’existence de dérivées de Fomin pour $\nu$ ou encore une formule d’intégration par partie qui généralise celle bien connue pour les mesures gaussiennes.

Finite automata and numbers

Abstract We study finite automata representations of numerical rings. Such representations correspond to the class of linear p-adic automata that compute homogeneous linear functions with rational coefficients in the ring of p-adic integers. Finite automata act both as ring elements and as operations. We also study properties of transition diagrams of automata that compute a function f(x)= cx of one variable. In particular we obtain precise values for the number of states of such automata and show that for c > 0 transition diagrams are self-dual (this property generalises self-duality of Boolean functions). We also obtain the criterion for an automaton computing a function f(x)= cx to be a permutation automaton, and fully describe groups that are transition semigroups of such automata.

HJB equations in infinite dimensions under weak regularizing properties

We solve in mild sense Hamilton Jacobi Bellman equations, both in an infinite dimensional Hilbert space and in a Banach space, with Lipschitz Hamiltonian and Lipschitz continuous final condition, and asking only a weak regularizing property on the transition semigroup of the corresponding state equation. The results are applied to solve stochastic optimal control problems; the models we can treat include a controlled stochastic heat equation in space dimension one and with control and noise on a subdomain.

Deep factorisation of the stable process

The Lamperti–Kiu transformation for real-valued self-similar Markov processes (rssMp) states that, associated to each rssMp via a space-time transformation, there is a Markov additive process (MAP). In the case that the rssMp is taken to be an $\alpha $-stable process with $\alpha \in (0,2)$, [16] and [24] have computed explicitly the characteristics of the matrix exponent of the semi-group of the embedded MAP, which we henceforth refer to as the Lamperti-stable MAP. Specifically, the matrix exponent of the Lamperti-stable MAP’s transition semi-group can be written in a compact form using only gamma functions. Just as with Levy processes, there exists a factorisation of the (matrix) exponents of MAPs, with each of the two factors uniquely characterising the ascending and descending ladder processes, which themselves are again MAPs. Although the case of MAPs with jumps in only one direction should be relatively straightforward to handle, to the author’s knowledge, not a single example of such a factorisation for two-sided jumping MAPs currently exists in the literature. In this article we provide a completely explicit Wiener–Hopf factorisation for the Lamperti-stable MAP. The main value and novelty of exploring the matrix Wiener–Hopf factorisation of the underlying MAP comes about through style of the computational approach. Understanding the fluctuation theory of the underlying MAP offers new insight into different ways of analysing stable processes. Indeed, we obtain new space-time invariance properties of stable processes, as well as demonstrating examples how new fluctuation identities for stable processes can be developed as a consequence of the reasoning in deriving the matrix Wiener–Hopf factors. The methodology in this paper has already lead to new applications in [27] and [28].

Invariant measures for stochastic nonlinear beam and wave equations

Existence of an invariant measure for a stochastic extensible beam equation and for a stochastic damped wave equation with polynomial nonlinearities is proved. It is shown first that the corresponding transition semigroups map the space of all bounded sequentially weakly continuous functions on the state space into itself and then by a Lyapunov functions approach solutions bounded in probability are found.

Large Aperiodic Semigroups

We search for the largest syntactic semigroups of star-free languages having n left quotients; equivalently, we look for the largest transition semigroups of aperiodic finite automata with n states. We first introduce unitary semigroups generated by transformations that change only one state. In particular, we study unitary-complete semigroups which have a special structure, and show that each maximal unitary semigroup is unitary-complete. For [Formula: see text] we exhibit a unitary-complete semigroup that is larger than any aperiodic semigroup known to date. We then present even larger aperiodic semigroups, generated by transformations that map a non-empty subset of states to a single state; we call such transformations and semigroups semiconstant. We examine semiconstant tree semigroups which have a structure based on full binary trees. The semiconstant tree semigroups are at present the best candidates for largest aperiodic semigroups.

On some smoothening effects of the transition semigroup of a Lévy process

Let (Pt) be the transition semigroup of a Lévy process (Lt) taking values in a Hilbert space H. Let ν and N˜ respectively be the Lévy measure and compensated Poisson random measure of (Lt). It is shown that for any bounded and measurable function f,AqPtf(x)=1tE[f(Ltx)∫0t∫Hq(y)N˜(ds,dy)]for all t>0,x∈H, where Aq is some nonlocal operator. As a corollary,∫H|Ptf(x+y)−Ptf(x)|2ν(dy)≤1tPtf2(x)for all t>0,x∈H. As ν can be infinite this formula establishes some smoothening effect of the semigroup (Pt). In the paper some applications of the formula will be presented as well.

M-Dissipativity for Kolmogorov Operator of a Fractional Burgers Equation with Space-time White Noise

This paper is concerned with the essential m-dissipativity of the Kolmogorov operator associated with a fractional stochastic Burgers equation with space-time white noise. Some estimates on the solution and its moments with respect to the invariant measure are given. Moreover we also study the smoothing properties of the transition semigroup and the corresponding fractional Ornstein-Uhlenbeck operator by introducing an auxiliary semigroup and (generalized) Bismut-Elworthy formula. From these results, we prove that the Kolmogorov operator of the problem is m-dissipative and the domain of the infinitesimal generator of the fractional Ornstein-Uhlenbeck operator is a core.

Self-repelling diffusions via an infinite dimensional approach

In the present work we study self-interacting diffusions following an infinite dimensional approach. First we prove existence and uniqueness of a solution with Markov property. Then we study the corresponding transition semigroup and, more precisely, we prove that it has Feller property and we give an explicit form of an invariant probability of the system.

Transitive representations of inverse semigroups

While every group is isomorphic to a transitive group of permutations, the analogous property fails for inverse semigroups: not all inverse semigroups are isomorphic to transitive inverse semigroups of one-to-one partial transformations of a set. We describe those inverse semigroups that are.

Asymptotic properties of stochastic Cahn–Hilliard equation with singular nonlinearity and degenerate noise

We consider a stochastic partial differential equation with a logarithmic nonlinearity with singularities at 1 and −1 and a constraint of conservation of the space average. The equation, driven by a trace-class space–time noise, contains a bi-Laplacian in the drift. We obtain existence of solution for equation with polynomial approximation of the nonlinearity. Tightness of this sequence of approximations is proved, leading to a limit transition semi-group. We study the asymptotic properties of this semi-group, showing the existence and uniqueness of invariant measure, asymptotic strong Feller property and topological irreducibility.

An Efficient Algorithm Based on Eigenfunction Expansions for Some Optimal Timing Problems in Finance

This paper considers the optimal switching problem and the optimal multiple stopping problem for one-dimensional Markov processes in a finite horizon discrete time framework. We develop a dynamic programming procedure to solve these problems and provide easy-to-verify conditions to characterize connectedness of switching and exercise regions. When the transition or Feynman-Kac semigroup of the Markov process has discrete spectrum, we develop an efficient algorithm based on eigenfunction expansions that explicitly solves the dynamic programming problem. We also prove that the algorithm converges exponentially in the series truncation level. Our method is applicable to a rich family of Markov processes which are widely used in financial applications, including many diffusions as well as jump-diffusions and pure jump processes that are constructed from diffusion through time change. In particular, many of these processes are often used to model mean-reversion. We illustrate the versatility of our method by considering three applications: valuation of combination shipping carriers, interest-rate chooser flexible caps and commodity swing options. Numerical examples show that our method is highly efficient and has significant computational advantages over standard numerical PDE methods that are typically used to solve such problems.