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1.
It is studied the first-passage time (FPT) of a time homogeneous one-dimensional diffusion, driven by the stochastic differential equation dX(t) = μ(X(t))dt + σ(X(t)) dB t , X(0) = x 0, through b + Y(t), where b > x 0 and Y(t) is a compound Poisson process with rate λ > 0 starting at 0, which is independent of the Brownian motion B t . In particular, the FPT density is investigated, generalizing a previous result, already known in the case when X(t) = μt + B t , for which the FPT density is the solution of a certain integral equation. A numerical method is shown to calculate approximately the FPT density; some examples and numerical results are also reported.  相似文献   

2.
Extremes of independent Gaussian processes   总被引:1,自引:0,他引:1  
Zakhar Kabluchko 《Extremes》2011,14(3):285-310
For every n ∈ ℕ, let X 1n ,..., X nn be independent copies of a zero-mean Gaussian process X n  = {X n (t), t ∈ T}. We describe all processes which can be obtained as limits, as n→ ∞, of the process a n (M n  − b n ), where M n (t) =  max i = 1,...,n X in (t), and a n , b n are normalizing constants. We also provide an analogous characterization for the limits of the process a n L n , where L n (t) =  min i = 1,...,n |X in (t)|.  相似文献   

3.
We specify a function b 0(t) in terms of the Lévy triplet such that lim sup  t→0 X t /b 0(t)∈[1,1.8] a.s. iff ò01[` \varPi ](+)(b0(t)) dt < ¥\int_{0}^{1}\overline{ \varPi }^{(+)}(b_{0}(t))\,dt<\infty for any Lévy process X with unbounded variation and a Brownian component σ=0. We show with an example that there are cases where lim sup  t→0 X t /b(t)=1 a.s. but b(t) is not asymptotically equivalent to b 0(t) as t tends to 0. We achieve this by introducing an integral criterion which checks whether lim sup  t→0 X t /b(t) is 0, infinity, or a finite positive value for b(t) satisfying very mild conditions and any Lévy process.  相似文献   

4.
Let K⊂ℝ d (d≥ 1) be a compact convex set and Λ a countable Abelian group. We study a stochastic process X in K Λ, equipped with the product topology, where each coordinate solves a SDE of the form dX i (t) = ∑ j a(ji) (X j (t) −X i (t))dt + σ (X i (t))dB i (t). Here a(·) is the kernel of a continuous-time random walk on Λ and σ is a continuous root of a diffusion matrix w on K. If X(t) converges in distribution to a limit X(∞) and the symmetrized random walk with kernel a S (i) = a(i) + a(−i) is recurrent, then each component X i (∞) is concentrated on {xK : σ(x) = 0 and the coordinates agree, i.e., the system clusters. Both these statements fail if a S is transient. Under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of K, we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of the components. This distribution turns out to be universal in all recurrent kernels a S on Abelian groups Λ. Received: 10 May 1999 / Revised version: 18 April 2000 / Published online: 22 November 2000  相似文献   

5.
The paper deals with the infinite-dimensional stochastic equation dX= B(t, X) dt + dW driven by a Wiener process which may also cover stochastic partial differential equations. We study a certain finite dimensional approximation of B(t, X) and give a qualitative bound for its rate of convergence to be high enough to ensure the weak uniqueness for solutions of our equation. Examples are given demonstrating the force of the new condition. Received: 6 November 1999 / Revised version: 21 August 2000 / Published online: 6 April 2001  相似文献   

6.
Moderate Deviations for Random Sums of Heavy-Tailed Random Variables   总被引:2,自引:0,他引:2  
Let {Xn;n≥ 1} be a sequence of independent non-negative random variables with common distribution function F having extended regularly varying tail and finite mean μ = E(X1) and let {N(t); t ≥0} be a random process taking non-negative integer values with finite mean λ(t) = E(N(t)) and independent of {Xn; n ≥1}. In this paper, asymptotic expressions of P((X1 +… +XN(t)) -λ(t)μ 〉 x) uniformly for x ∈[γb(t), ∞) are obtained, where γ〉 0 and b(t) can be taken to be a positive function with limt→∞ b(t)/λ(t) = 0.  相似文献   

7.
Suppose that S is a subordinator with a nonzero drift and W is an independent 1-dimensional Brownian motion. We study the subordinate Brownian motion X defined by X t  = W(S t ). We give sharp bounds for the Green function of the process X killed upon exiting a bounded open interval and prove a boundary Harnack principle. In the case when S is a stable subordinator with a positive drift, we prove sharp bounds for the Green function of X in (0, ∞ ), and sharp bounds for the Poisson kernel of X in a bounded open interval.  相似文献   

8.
The evolution of the growth of an individual in a random environment can be described through stochastic differential equations of the form dY t  = β(α − Y t )dt + σdW t , where Y t  = h(X t ), X t is the size of the individual at age t, h is a strictly increasing continuously differentiable function, α = h(A), where A is the average asymptotic size, and β represents the rate of approach to maturity. The parameter σ measures the intensity of the effect of random fluctuations on growth and W t is the standard Wiener process. We have previously applied this monophasic model, in which there is only one functional form describing the average dynamics of the complete growth curve, and studied the estimation issues. Here, we present the generalization of the above stochastic model to the multiphasic case, in which we consider that the growth coefficient β assumes different values for different phases of the animal’s life. For simplicity, we consider two phases with growth coefficients β 1 and β 2. Results and methods are illustrated using bovine growth data.  相似文献   

9.
We study equidistribution properties of nil-orbits (b n x) n∈ℕ when the parameter n is restricted to the range of some sparse sequence that is not necessarily polynomial. For example, we show that if X = G/Γ is a nilmanifold, bG is an ergodic nilrotation, and c ∈ ℝ \ ℤ is positive, then the sequence $ (b^{[n^c ]} x)_{n \in \mathbb{N}} $ (b^{[n^c ]} x)_{n \in \mathbb{N}} is equidistributed in X for every xX. This is also the case when n c is replaced with a(n), where a(t) is a function that belongs to some Hardy field, has polynomial growth, and stays logarithmically away from polynomials, and when it is replaced with a random sequence of integers with sub-exponential growth. Similar results have been established by Boshernitzan when X is the circle.  相似文献   

10.
For a given bi-continuous semigroup (T(t)) t⩾0 on a Banach space X we define its adjoint on an appropriate closed subspace X° of the norm dual X′. Under some abstract conditions this adjoint semigroup is again bi-continuous with respect to the weak topology σ(X°,X). We give the following application: For Ω a Polish space we consider operator semigroups on the space Cb(Ω) of bounded, continuous functions (endowed with the compact-open topology) and on the space M(Ω) of bounded Baire measures (endowed with the weak*-topology). We show that bi-continuous semigroups on M(Ω) are precisely those that are adjoints of bi-continuous semigroups on Cb(Ω). We also prove that the class of bi-continuous semigroups on Cb(ω) with respect to the compact-open topology coincides with the class of equicontinuous semigroups with respect to the strict topology. In general, if is not a Polish space this is not the case.  相似文献   

11.
We survey recent results related to uniqueness problems for parabolic equations for measures. We consider equations of the form ∂ t μ = L * μ for bounded Borel measures on ℝ d  × (0, T), where L is a second order elliptic operator, for example, Lu = Dxu + ( b,?xu ) Lu = {\Delta_x}u + \left( {b,{\nabla_x}u} \right) , and the equation is understood as the identity
ò( ?tu + Lu )dm = 0 \int \left( {{\partial_t}u + Lu} \right)d\mu = 0  相似文献   

12.
Let {X(t): t [a, b]} be a Gaussian process with mean μ L2[a, b] and continuous covariance K(s, t). When estimating μ under the loss ∫ab ( (t)−μ(t))2 dt the natural estimator X is admissible if K is unknown. If K is known, X is minimax with risk ∫ab K(t, t) dt and admissible if and only if the three by three matrix whose entries are K(ti, tj) has a determinant which vanishes identically in ti [a, b], i = 1, 2, 3.  相似文献   

13.
Let X 1 , X 2 , . . . be a sequence of negatively dependent and identically distributed random variables, and let N be a counting random variable independent of X i ’s. In this paper, we study the asymptotics for the tail probability of the random sum SN = ?k = 1N Xk {S_N} = \sum\nolimits_{k = 1}^N {{X_k}} in the presence of heavy tails. We consider the following three cases: (i) P(N > x) = o(P(X 1> x)), and the distribution function (d.f.) of X 1 is dominatedly varying; (ii) P(X 1> x) = o(P(N > x)), and the d.f. of N is dominatedly varying; (iii) the tails of X 1 and N are asymptotically comparable and dominatedly varying.  相似文献   

14.
We study in this paper an M/M/1 queue whose server rate depends upon the state of an independent Ornstein–Uhlenbeck diffusion process (X(t)) so that its value at time t is μ φ(X(t)), where φ(x) is some bounded function and μ>0. We first establish the differential system for the conditional probability density functions of the couple (L(t),X(t)) in the stationary regime, where L(t) is the number of customers in the system at time t. By assuming that φ(x) is defined by φ(x)=1−ε((x a/ε)(−b/ε)) for some positive real numbers a, b and ε, we show that the above differential system has a unique solution under some condition on a and b. We then show that this solution is close, in some appropriate sense, to the solution to the differential system obtained when φ is replaced with Φ(x)=1−ε x for sufficiently small ε. We finally perform a perturbation analysis of this latter solution for small ε. This allows us to check at the first order the validity of the so-called reduced service rate approximation, stating that everything happens as if the server rate were constant and equal to .   相似文献   

15.
 Let a, b, m, and t be integers such that 1≤a<b and 1≤t≤⌉(bm+1)/a⌉. Suppose that G is a graph of order |G| and H is any subgraph of G with the size |E(H)|=m. Then we prove that G has an [a,b]-factor containing all the edges of H if the minimum degree is at least a, |G|>((a+b)(t(a+b−1)−1)+2m)/b, and |N G (x 1)∪⋯ ∪N G (x t )|≥(a|G|+2m)/(a+b) for every independent set {x 1,…,x t }⊆V(G). This result is best possible in some sense and it is an extension of the result of H. Matsuda (A neighborhood condition for graphs to have [a,b]-factors, Discrete Mathematics 224 (2000) 289–292). Received: October, 2001 Final version received: September 17, 2002 RID="*" ID="*" This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement of Young Scientists, 13740084, 2001  相似文献   

16.
The stochastic equation dX t =dS t +a(t,X t )dt, t≥0, is considered where S is a one-dimensional Levy process with the characteristic exponent ψ(ξ),ξ∈ℝ. We prove the existence of (weak) solutions for a bounded, measurable coefficient a and any initial value X 0=x 0∈ℝ when (ℛeψ(ξ))−1=o(|ξ|−1) as |ξ|→∞. These conditions coincide with those found by Tanaka, Tsuchiya and Watanabe (J. Math. Kyoto Univ. 14(1), 73–92, 1974) in the case of a(t,x)=a(x). Our approach is based on Krylov’s estimates for Levy processes with time-dependent drift. Some variants of those estimates are derived in this note.  相似文献   

17.
LetX be a Banach space and leta, b, q be real numbers such thata<b,q>0. Denote byD a locally closed subset ofX. A necessary and sufficient condition for the existence of a mild solutionu∈C([a−q, b 1],X),a<b 1<b, to the differential equationdu(t)/dt=Au(t)+f(t, u t), such thatu:[a,b 1]→D, u a=ϕ is given. The linear operatorA is the generator of aC 0 semigroupT(t), t≧0, withT(t) compact fort>0,f: [a, b)×C([−q,0],D λ)→X is continuous and ϕ∈C([−q,0],D λ) with ϕ(0)∈D. D λ is a neighbourhood ofD. Applications to parabolic partial differential equations with retarded argument are given.  相似文献   

18.
In this paper we prove a stochastic representation for solutions of the evolution equation
where L  ∗  is the formal adjoint of a second order elliptic differential operator L, with smooth coefficients, corresponding to the infinitesimal generator of a finite dimensional diffusion (X t ). Given ψ 0 = ψ, a distribution with compact support, this representation has the form ψ t  = E(Y t (ψ)) where the process (Y t (ψ)) is the solution of a stochastic partial differential equation connected with the stochastic differential equation for (X t ) via Ito’s formula.   相似文献   

19.
Let {W(t),t∈R}, {B(t),t∈R } be two independent Brownian motions on R with W(0) = B(0) = 0. In this paper, we shall consider the exact Hausdorff measures for the image and graph sets of the d-dimensional iterated Brownian motion X(t), where X(t) = (Xi(t),... ,Xd(t)) and X1(t),... ,Xd(t) are d independent copies of Y(t) = W(B(t)). In particular, for any Borel set Q (?) (0,∞), the exact Hausdorff measures of the image X(Q) = {X(t) : t∈Q} and the graph GrX(Q) = {(t, X(t)) :t∈Q}are established.  相似文献   

20.
Let X be a Banach space, A : D(A) X → X the generator of a compact C0- semigroup S(t) : X → X, t ≥ 0, D a locally closed subset in X, and f : (a, b) × X →X a function of Caratheodory type. The main result of this paper is that a necessary and sufficient condition in order to make D a viable domain of the semilinear differential equation of retarded type u'(t) = Au(t) + f(t, u(t - q)), t ∈ [to, to + T], with initial condition uto = φ ∈C([-q, 0]; X), is the tangency condition lim infh10 h^-1d(S(h)v(O)+hf(t, v(-q)); D) = 0 for almost every t ∈ (a, b) and every v ∈ C([-q, 0]; X) with v(0), v(-q)∈ D.  相似文献   

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