Transformation of Wiener measure under anticipative flows

Summary LetT()=+F() be a transformation from the Wiener space to itself with the range ofF() assumed to be in the Cameron-Martin space. The absolute continuity and the density function associated withT is considered;T is assumed to be embedded in or defined through a parameterizationT _t =+F _t () andF _t is assumed to be differentiable int. The paper deals first with the case where the range of thet-derivative ofF _t () is also in the Cameron-Martin space and new representations for the Radon-Nikodym derivative and the Carleman-Fredholm determinant are derived. The case where thet-derivative ofF _t is not in the Cameron-Martin space is considered next and results on the absolute continuity and the density function, under conditions which are considerably weaker than previously known conditions, are presented.The work of the second author was supported by the fund for promotion of research at the Technion     

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.     

Study of a filtration expanded to include an honest time

Summary Let (,, P) be a measurable space, and { _t} be a filtration on (,). Then, given a fixed honest timeL a new filtrationG _t} is defined, the smallest containing { _t} and for whichL is a stopping time, and the martingales, semimartingales and stopping times of this new filtration are characterised.     

On the properties of ε(≥0) optimal policies in discounted unbounded return model

This paper investigates the properties of (0) optimal policies in the model of [2]. It is shown that, if * = ( ₀ ^* , ₁ ^* ,..., _n ^* , _n ^+1/* , ...) is a-discounted optimal policy, then ( ₀ ^* , ₁ ^* , ..., _n ^* ) for alln0 is also a-discounted optimal policy. Under some condition we prove that stochastic stationary policy _n ^* corresponding to the decision rule _n ^* is also optimal for the same discounting factor. We have also shown that for each-optimal stochastic stationary policy ₀ ^* , ₀ ^* can be decomposed into several decision rules to which the corresponding stationary policies are also-optimal separately; and conversely, a proper convex combination of these decision rules is identified with the former ₀ ^* . We have further proved that for any (,)-optimal policy, say *=( ₀ ^* , ₁ ^* , ..., _n ^* , _n ^+1/* , ...), _n–1 ^* ) is ((1– ⁿ )^–1 e, ) optimal forn>0. At the end of this paper we mention that the results about convex combinations and decompositions of optimal policies of § 4 in [1] can be extended to our case.Project supported by the Science Fund of the Chinese Academy of Sciences.     

On equivalence of rearrangements of the Haar system in dyadic Hardy and BMO spaces

H={h ₁,I } — , . : , I ¦(I)¦=¦I¦, ¦I¦ — I. H H ={h _(I),I} . , , . L ^p .

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Dedicated to Professor B. Szökefalvi-Nagy on his 75th birthday

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This research was supported in part by MTA-NSF Grants INT-8400708 and 8620153.     

Reducibility of nonlinear almost periodic systems of difference equations on an infinite-dimensional torus

We establish sufficient conditions for the reducibility of a nonlinear system of difference equationsx(t+1)=x(t)++P(x(t),t+ to a system y(t+1)= y(t)+, wherex, , m and the infinite-dimensional vector function P(x(t),t) is 2p-periodic inx _i i=1,2,...) and almost periodic int with a frequency basis.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1216–1223, September, 1994.     

Compact sets in the spaceL p (O,T; B)

Summary A characterization of compact sets in L^p (0, T; B) is given, where 1P and B is a Banach space. For the existence of solutions in nonlinear boundary value problems by the compactness method, the point is to obtain compactness in a space L^p (0,T; B) from estimates with values in some spaces X, Y or B where XBY with compact imbedding XB. Using the present characterization for this kind of situations, sufficient conditions for compactness are given with optimal parameters. As an example, it is proved that if {f_n} is bounded in L^q(0,T; B) and in L _loc ¹ (0, T; X) and if {f_n/t} is bounded in L _loc ¹ (0, T; Y) then {f_n} is relatively compact in L^p(0,T; B), p相似文献   

Excursions of a Markov process induced by continuous additive functionals

Summary Let be a continuous additive functional with supportC of a Hunt processX={X _t;t0}. LetS={S _t;t0} be the inverse of and put . For each time of discontinuityu ofS, letZ _u be the corresponding excursion ofX outside ofC. The conditional structure of the excursion process {Z _u;u0} given the paths ofY={Y _t;t0} is studied. It is shown that conditionally, givenY, the excursion process is a Poisson random measure.Support from the Office of Naval Research (Contract Number N-00014-67-A-0112-0011) and the National Science Foundation (Grant Number ENG 75-02026) is gratefully acknowledged     

Continuous dependence of stochastic control models on the noise distribution

Given a discrete-time stochastic control systemx _t+1 =F(x _t ,a _t , _t ),t=0, 1,.., N (N), where the noise process { _t } is a sequence of i.i.d. random elements with distribution, let ^N (x) be the optimal reward function when the initial state isx and the planning horizon isN. We give conditions under whichv ^N is a continuous function in for several reward criteria. The applicability of these results to nonparametric adaptive control of stochastic systems is briefly discussed.This research was supported in part by the Consejo Nacional de Ciencia y Tecnologia (CONACYT) under Grants PCCBBNA-020630 and PCEXCNA-040640.     

Some expansion formulae for Fox'sH-function of two variables involving associated Legendre functions

H- .     

Coupling of the GB set property for ergodic averages

Let (Y,,,T) be an ergodic dynamical system. LetA be an nonempty subset ofL ²() such that , whereA=sup{||sȒt||₂ ,s, tA} andN(A, u) is the smallest number ofL ²()-open balls of radiusu, centered inA, enough to coverA. Let . We prove as a consequence of a more general result, thatC(A) is aGB subset ofL ²().     

Finite extinction time for some perturbed Hamilton-Jacobi equations

In this paper we study initial value problems likeu _t–R¦u¦^m+u^q=0 in ⁿ× ₊, u(·,0⁺)=u_o(·) in ^N, whereR > 0, 0 <q < 1,m 1, andu _o is a positive uniformly continuous function verifying –R¦u _o¦^m+u ₀ ^q 0 in ^N . We show the existence of the minimum nonnegative continuous viscosity solutionu, as well as the existence of the function t(·) defined byu(x, t) > 0 if 0<t<t (x) andu(x, t)=0 ift t (x). Regularity, extinction rate, and asymptotic behavior of t(x) are also studied. Moreover, form=1 we obtain the representation formulau(x, t)=max{([(u _o(x – t))^1–q –(1–q)t]₊)^1/(1–q): ¦¦R}, (x, t) ₊ ^N+1 .Partially supported by the DGICYT No. 86/0405 project.     

Feller Processes Generated by Pseudo-Differential Operators: On the Hausdorff Dimension of Their Sample Paths

Let {X _t} _t0 be a Feller process generated by a pseudo-differential operator whose symbol satisfiesÇⁿ|q(Ç,)|c(1=)()) for some fixed continuous negative definite function (). The Hausdorff dimension of the set {X _t:tE}, E [0, 1] is any analytic set, is a.s. bounded above by dim E. is the Blumenthal–Getoor upper index of the Levy Process associated with ().     

Tail behavior for semigroups of measures on groups

Let (_t)_t>0 be a convolution semigroup of probability measures on a measurable group (G, ). In this paper, we provide precise information about the asymptotic behavior of _t{q>s, whereq is a measurable seminorm and (_t)_t>0 isq-continuous.     

Best approximation and best unilateral approximation of the kernel of a biharmonic equation and optimal renewal of the values of operators

For the classB _p , 0 < 1, 1p , of 2-periodic functions of the form f(t)=u(,t), whereu (,t) is a biharmonic function in the unit disk, we obtain the exact values of the best approximation and best unilateral approximation of the kernel K(t) of the convolution f= K *g, gl, with respect to the metric of L₁. We also consider the problem of renewal of the values of the convolution operator by using the information about the values of the boundary functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 11, pp. 1549–1557, November, 1995.     

The monotone follower problem in stochastic decision theory

We consider the problem of optimally tracking the random demandx+w _t, w. Brownian motion, by a nondecreasing process. adapted to the Brownian past, so as to minimize the expected lossE ₀ ^T (x+w_t–_t)dt. The decision problem is reduced to a free boundary one, and the latter is studied and solved for a large class of cost functions().This research was supported in part by the Air Force Office of Scientific Research, under AF-AFOSR 77-3063.     

A note on the global asymptotic stability of the unreacting state in a simple model for quadratic autocatalysis with linear decay

In this note we consider a reaction-diffusion problem which describes a simple model chemical reaction scheme for quadratic autocatalysis with linear decay. We show that withk>1 (wherek is a parameter measuring the relative strength of the decay step to the autocatalytic step) the dimensionless unreacting state 1, 0 is globally asymptotically stable, with 1+0(t ^–1/2) and 0(t ^–1/2 e ^–(k–1)t) ast-. Here and are the concentrations of the reactant and the autocatalyst respectively, andt is time. The casek<1 has been considered in detail by Merkin et al. [1].     

Large Deviations of Local Times of Lévy Processes

For X(t) a real-valued symmetric Lévy process, its characteristic function is E(e ^iX(t))=exp(–t()). Assume that is regularly varying at infinity with index 1<2. Let L ^x _t denote the local time of X(t) and L* _t =sup _xR L ^x _t . Estimates are obtained for P(L ⁰ _t y) and P(L* _t y) as y and t fixed.     

Central Limit Theorem for a Class of Nonhomogeneous Random Walks

A spatially nonhomogeneous random walk _t on the grid =^m X ⁿ is considered. Let _t ⁰ be a random walk homogeneous in time and space, and let _t be obtained from it by changing transition probabilities on the set A= X ⁿ, || < , so that the walk remains homogeneous only with respect to the subgroup ⁿ of the group . It is shown that if >m 2 or the drift is distinct from zero, then the central limit theorem holds for _t.     

Asymptotics of the partial sums of a set of integral transforms

In this paper, we explore the asymptotic distribution of the zeros of the partial sums of the family of entire functions of order 1 and type 1, defined by G(,,z)=₀ ¹(t)t ^–1×(1–t)^–1e ^zt dt, where Re,Re>0, is Riemann-integrable on [0,1], continuous at t=0, 1 and satisfies (0)(1)0.