The exact Hausdorff measures for the graph and image of a multidimensional iterated Brownian motion

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.     

Averaging of directional derivatives in vertices of nonobtuse regular triangulations

For a shape-regular triangulation ${\mathcal{T}_h}For a shape-regular triangulation _h{\mathcal{T}_h} without obtuse angles of a bounded polygonal domain W ì ?²{\Omega\subset\Re^2} , let L_h{\mathcal L_h} be the space of continuous functions linear on the triangles from T_h{\mathcal{T}_h} and Π _h the interpolation operator from C([`(W)]){C(\overline\Omega)} to L<sub>h</sub>{\mathcal L_h} . This paper is devoted to the following classical problem: Find a second-order approximation of the derivative ?u/?z(a){\partial u/\partial z(a)} in a direction z of a function u ? C<sup>3</sup>([`(W)]){u\in C^3(\overline\Omega)} in a vertex a in the form of a linear combination of the constant directional derivatives ?P_h(u)/?z{\partial \Pi_h(u)/\partial z} on the triangles surrounding a. An effective procedure for such an approximation is presented, its error is proved to be of the size O(h ²), an operator W_h: L_h?L_h×L_h{\mbox{W}_h: \mathcal L_h\longrightarrow\mathcal L_h\times\mathcal L_h} relating a second-order approximation W _h [Π _h (u)] of ?u{\nabla u} to every u ? C³([`(W)]){u\in C^3(\overline\Omega)} is constructed and shown to be a so-called recovery operator. The accuracy of the presented approximation is compared with the accuracies of the local approximations by other known techniques numerically.     

On the Vector Process Obtained by Iterated Integration of the Telegraph Signal

We analyse the vector process (X ₀(t), X ₁(t),...,X _n(t), t > 0) where , and X ₀(t) is the o two-valued telegraph process.In particular, the hyperbolic equations governing the joint distributions of the process are derived and analysed.Special care is given to the case of the process (X ₀(t), X ₁(t), X ₂(t), t > 0) representing a randomly accelerated motion where some explicit results on the probability distribution are derived.     

Nonautonomous Stochastic Search in Global Optimization

We present a general method how to prove convergence of a sequence of random variables generated by a nonautonomous scheme of the form X _t =T _t (X _t−1,Y _t ), where Y _t represents randomness, used as an approximation of the set of solutions of the global optimization problem with a continuous cost function. We show some of its applications.     

Some first passage problems related to cusum procedures

Let $\text{X}{}_1\text{,}\text{X}{}_2\text{,…}$ be independent random variables, and set W_n = max(0,W_n-1 + X_n), W₀ = 0, n ? 1. The so-called cusum (cumulative sum) procedure uses the first passage time T(h) = inf{n ? 1: W_n?h}for detecting changes in the mean μ of the process. It is shown that lim_h→∞ μET(h)/h = 1 if μ > 0. Also, a cusum procedure for detecting changes in the normal mean is derived when the variance is unknown. An asymptotic approximation to the average run length is given.     

Bose-Mesner Algebras Associated with Four-Weight Spin Models

It is known that for each matrix W _i and it's transpose ^t W _i in any four-weight spin model (X, W ₁, W ₂, W ₃, W ₄; D), there is attached the Bose-Mesner algebra of an association scheme, which we call Nomura algebra. They are denoted by N(W _i ) and N( ^t W _i ) = N′(W _i ) respectively. H. Guo and T. Huang showed that some of them coincide with a self-dual Bose-Mesner algebra, that is, N(W ₁) = N′(W ₁) = N(W ₃) = N′(W ₃) holds. In this paper we show that all of them coincide, that is, N(W _i ), N′(W _i ), i=1, 2, 3, 4, are the same self-dual Bose-Mesner algebra. Received: June 17, 1999 Final version received: Januray 17, 2000     

Canonical State/Signal Shift Realizations of Passive Continuous Time Behaviors

This work is devoted to the construction of canonical passive and conservative state/signal shift realizations of arbitrary passive continuous time behaviors. By definition, a passive future continuous time behavior is a maximal nonnegative right-shift invariant subspace of the Kreĭn space L²([0,￥);W){L^2([0,\infty);\mathcal W)}, where W{\mathcal W} is a Kreĭn space, and the inner product in L²([0,￥);W){L^2([0,\infty);\mathcal W)} is the one inherited from W{\mathcal W}. A state/signal system S = (V;X,W){\Sigma=(V;\mathcal X,\mathcal W)}, with a Hilbert state space X{\mathcal X} and a Kreĭn signal space W{\mathcal W}, is a dynamical system whose classical trajectories (x, w) on [0, ∞) satisfy x ? C¹([0,￥);X){x\in C^1([0,\infty);\mathcal X)}, w ? C([0,￥);W){w \in C([0,\infty);\mathcal W)}, and

Expansion of the global error for numerical schemes solving stochastic differential equations

Given the solution (X_t ) of a Stochastic Differential System, two situat,ions are considered: computat,ion of Ef(X_t ) by a Monte–Carlo method and, in the ergodic case, integration of a function f w.r.t. the invariant probability law of (X_t ) by simulating a simple t,rajectory.

For each case it is proved the expansion of the global approximat,ion error—for a class of discret,isat,ion schemes and of funct,ions f—in powers of the discretisation step size, extending in the fist case a result of Gragg for deterministic O.D.E.

Some nn~nerical examples are shown to illust,rate the applicat,ion of extrapolation methods, justified by the foregoing expansion, in order to improve the approximation accuracy     

The supercritical multitype age-dependent branching process

Let X(t) = (X₁(t),…, X_p(t)) be a p-dimensional supercritical age-dependent branching process. For an appropriate α > 0, necessary and sufficient conditions are found for X(t) e^?αt to converge to a nondegenerate random vector W. Several properties of W are also determined.     

Some ratio inequalities for iterated stochastic integrals

Let X = (X_t, ?_t) be a continuous local martingale with quadratic variation 〈X〉 and X₀ = 0. Define iterated stochastic integrals I_n(X) = (I_n(t, X), ?_t), n ≥ 0, inductively by $$ I_{n} (t, X) = \int ^{t} _{0} I_{n-1} (s, X)dX_{s} $$ with I₀(t, X) = 1 and I₁(t, X) = X_t. Let (??^x_t(X)) be the local time of a continuous local martingale X at x ∈ ?. Denote ??*t(X) = sup_x∈? ??^x_t(X) and X* = sup_t≥0 |X_t|. In this paper, we shall establish various ratio inequalities for I_n(X). In particular, we show that the inequalities $$ c_{n,p} \, \left\Vert (G ( \langle X \rangle _{\infty} )) ^{n/2} \right\Vert _{p} \; \le \; \left\Vert {\mathop \sup \limits _{t \ge 0}} \; {\left\vert I_{n} (t, X) \right\vert \over {(1+ \langle X \rangle _{t} ) ^{n/2}}} \right\Vert _{p} \; \le C_{n, p} \, \left\Vert (G ( \langle X \rangle _{\infty} )) ^{n/2} \right\Vert _{p} $$ hold for 0 < p < ∞ with some positive constants c_n,p and C_n,p depending only on n and p, where G(t) = log(1+ log(1+ t)). Furthermore, we also show that for some γ ≥ 0 the inequality $$ E \left[ U ^{p}_{n} \exp \left( \gamma {U ^{1/n} _{n} \over {V}} \right) \right] \le C_{n, p, \gamma} E [V ^{n, p}] \quad (0 < p < \infty ) $$ holds with some positive constant C_n,p,γ depending only on n, p and γ, where U_n is one of 〈I_n(X)〉^1/2_∞ and I*_n(X), and V one of the three random variables X*, 〈X〉^1/2_∞ and ??*_∞(X). (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Topological characterizations of ordered groups with quasi-divisor theory

For an order embedding of a partly ordered group G into an l-group a topology is introduced on which is defined by a family of valuations W on G. Some density properties of sets h(G), h(X _t ) and (X _t being t-ideals in G) in the topological space are then investigated, each of them being equivalent to the statement that h is a strong theory of quasi-divisors.     

Integration par parties dans l'espace de Wiener et approximation du temps local

Summary Let be a real-valued stochastic process having a continuous local timeL(u,t),u —, 0tT andX (t) = ( *X)(t),t 0, the regularization ofX by means of the convolution with the approximation of unity . The main theorem in this paper (Theorem 3.5) is a generalization of various results about the approximation (for fixedu) of the local timeL(u, ) by means of a convenient normalization of the numberN ^X (u;) of crossings of the processX with the levelu. Especially, this Theorem extends to a class of not necessarily Markovian continuous martingales, a result of this type for one-dimensional diffusions due to Azais [A2]). The methods of proof combine some estimations of the moments of the number of crossings with a level of a regular stochastic processes with stochastic analysis techniques based upon integration by parts in the Wiener space.

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Support partill du CICYT, N^o PB86-0238     

On the Potential Theory of One-dimensional Subordinate Brownian Motions with Continuous Components

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.     

Multivalued Stochastic Differential Equations: Convergence of a Numerical Scheme

In this paper we show the strong mean square convergence of a numerical scheme for a R ^d -multivalued stochastic differential equation: dX _t +A(X _t )dtb(t,X _t )dt+(t,X _t )dW _t and obtain the rate of convergence O(( log(1/)^1/2) when the diffusion coefficient is bounded. By introducing a discrete Skorokhod problem, we establish L ^p -estimates (p2) for the solutions and prove the convergence by using a deterministic result. Numerical experiments for the rate of convergence are presented.     

On stochastic integral representation of stable processes with sample paths in Banach spaces

Certain path properties of a symmetric α-stable process X(t) = ∫_Sh(t, s) dM(s), t T, are studied in terms of the kernel h. The existence of an appropriate modification of the kernel h enables one to use results from stable measures on Banach spaces in studying X. Bounds for the moments of the norm of sample paths of X are obtained. This yields definite bounds for the moments of a double α-stable integral. Also, necessary and sufficient conditions for the absolute continuity of sample paths of X are given. Along with the above stochastic integral representation of stable processes, the representation of stable random vectors due to[13], Ann. Probab.9, 624–632) is extensively used and the relationship between these two representations is discussed.     

Remark on the phase shift in the Kuzmak-Whitham ansatz

We consider one-phase (formal) asymptotic solutions in the Kuzmak-Whitham form for the nonlinear Klein-Gordon equation and for the Korteweg-de Vries equation. In this case, the leading asymptotic expansion term has the form X(S(x, t)/h+Φ(x, t), I(x, t), x, t) +O(h), where h ≪ 1 is a small parameter and the phase S}(x, t) and slowly changing parameters I(x, t) are to be found from the system of “averaged” Whitham equations. We obtain the equations for the phase shift Φ(x, t) by studying the second-order correction to the leading term. The corresponding procedure for finding the phase shift is then nonuniform with respect to the transition to a linear (and weakly nonlinear) case. Our observation, which essentially follows from papers by Haberman and collaborators, is that if we incorporate the phase shift Φ into the phase and adjust the parameter Ĩ by setting $ \tilde S $ \tilde S = S +hΦ+O(h ²),Ĩ = I + hI ₁ + O(h ²), then the functions $ \tilde S $ \tilde S (x, t, h) and Ĩ(x, t, h) become solutions of the Cauchy problem for the same Whitham system but with modified initial conditions. These functions completely determine the leading asymptotic term, which is X($ \tilde S $ \tilde S (x, t, h)/h, Ĩ(x, t, h), x, t) + O(h).     

Exponential-cosine operator-valued functional equation in the strong operator topology

LetX be a Banach Space and letB(X) denote the family of bounded linear operators onX. LetR ⁺ = [0, ). A one parameter family of operators {S(t);t R ⁺},S:R ⁺ B(X), is called exponential-cosine operator function ifS(O) =I andS(s +t) – 2S(s)S(t) = (S(2s) – 2S ²(s))S(t –s), for alls, t R ⁺,s t. Let ,fD(A), and ,fD(B). It is shown that for a strongly continuous exponential-cosine operator {S(t)},fD(A ²) implies ₀ ^t (t –u(S(u)fduD(B) and B ₀ ^t (t –u)S(u)fdu =S(t)f –f +tAf – 2A ₀ ^t S(u)fdu + 2A ² ₀ ^t (t –u)S(u)fdu.D(B) is seen to be dense inD(A ²). Some regularity properties ofS(t) have also been obtained.     

Queueing approximation of suprema of spectrally positive Lévy process

Let W=sup _0≤t<∞(X(t)−β t), where X is a spectrally positive Lévy process with expectation zero and 0<β<∞. One of the main results of the paper says that for such a process X, there exists a sequence of M/GI/1 queues for which stationary waiting times converge in distribution to W. The second result shows that condition (III) of Proposition 2 in the paper is not implied by all other conditions.     

Multiphasic Individual Growth Models in Random Environments

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.     

Limit distributions of conditionalU-statistics

Let{(X_n, Y_n)}_n1 be a sequence of i.i.d. bi-variate vectors. In this article, we study the possible limit distributions ofU _n ^h (t), the so-calledconditional U-statistics, introduced by Stute.⁽¹⁰⁾ They are estimators of functions of the formm ^h (t)=E{h(Y ₁,...,Y _k )|X ₁=t ₁,...,X _k =t _k },t=(t ₁,...,t _k ) ^k whereE |h|<. Heret is fixed. In caset ₁=...=t_k=t (say), we describe the limiting random variables asmultiple Wiener integrals with respect toP _t, the conditional distribution ofY, givenX=t. Whent _i, 1ik, are not all equal, we introduce and use a slightly generalized version of a multiple Wiener integral.Research supported by National Board for Higher Mathematics, Bombay, India.