Explicit expression for the transition density of the solution of a stochastic diffusion equation with piecewise-constant drift coefficient

The explicit form of the transition density is determined for the solution (t) of the stochastic diffusion equation d(t)=a((t))dt+dw(t), where a(z)= for x [a, b] and a(x)=0 for x [a, b], w(t) is a Wiener process.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 99–105, 1987.     

Sequential estimation of the parameters of diffusion processes

For the parameter of a diffusion process(t), satisfying the stochastic differential equation d(t)=f (t,)dt+dw(l), we propose an effective sequential estimation plan with an unbiased and normally distributed estimate. The proposed sequential plan is discussed in detail for the example of a process (t) having a linear stochastic differential.Translated from Matematicheskie Zametki, Vol. 12, No. 5, pp. 627–638, November, 1972.In conclusion the author wishes to express his deep gratitude to A. N. Shiryaev for formulating the problem and for useful observations     

Spline polynomials with a prescribed sequence of extrema

In the present note a theorem about strong suitability of the space of algebraic polynomials of degree n in C[a,b] (Theorem A in [1]) is generalized to the space of spline polynomials _{[a, b} ^{]n, k} (n2, 0) in C[a, b]. Namely, it is shown that the following theorem is valid: for arbitrary numbers ₀, ₁, ..., _n+k, satisfying the conditions (_i–_i–1) (_{i+1{ i}< 0(i=1, ..., n +k–1), there is a unique polynomials _n,k (t) _{[a, b} ^]/n,k and pointsa=₀,<₁<...< _n+k– 1< _n+k = b (₁1 <_n, ..., _kk<_n+k–1), such that s_n,k(_i) = _i(i=0, ..., n + k), s_n,k(_i)=0 (i=1, ..., n + k–1).Translated from Matematicheskii Zametki, Vol. 11, No. 3, pp. 251–258, March, 1972.     

On the structure of stationary flat processes. II

Summary The present paper continues the work by Davidson, Krickeberg, Papangelou, and the author on proving, under weakest possible assumptions, that a stationary random measure or a simple point process on the space of k-flats in R ^d is a.s. invariant or a Cox process respectively. The problems for and are related by the fact that is Cox whenever the Papangelou conditional intensity measure of (a thinning of) is a.s. invariant. In particular, is shown to be a.s. invariant, whenever it is absolutely continuous with respect to some fixed measure and has no (so called) outer degeneracies. When k=d–22, no absolute continuity is needed, provided that the first moments exist and that has no inner degeneracies either. Under a certain regularity condition on , it is further shown that and are simultaneously non-degenerate in either sense.     

On minimax filtration of vector processes

We study the problem of optimal linear estimation of the transformation of a stationary random process (t) with values in a Hilbert space by observations of the process (t) + (t) fort0. We obtain relations for computing the error and the spectral characteristic of the optimal linear estimate of the transformationA for given spectral densities of the processes (t) and (t). The minimax spectral characteristics and the least favorable spectral densities are obtained for various classes of densities.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 389–397, March, 1993.     

Distribution of the supremum of sums of independent variables with negative drift

Let {_n} be a sequence of identically distributed independent random variables,M₁=<0,M ₁ ² <;S ₀=0,S _n =₁+₂,+...+ _{n, n}1;¯ S=sup {S _n n=0.} The asymptotic behavior ofP(¯ St) as t is studied. If _t ^P (₁x dx=0((t)), thenP(¯ St)– 1/¦¦ _t P (₁x dx=0((t)) (t) is a positive function, having regular behavior at infinity.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 763–770, November, 1977.The author thanks B. A. Rogozin for the formulation of the problem and valuable remarks.     

Mathematical expectations of functions of sums of a random number of independent terms

Conditions are found which must be imposed on a function g(x) in order that M g(₁+₂+ + _v < if M g(_i) < and M g(v) < ,, ₁, ₂, , _n, ... being non-negative and independent, being integral, and {_i} being identically distributed. The result is applied to the theory of branching processes.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 387–394, April, 1968.     

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 ().     

Estimate of a random field based on the trajectory of a point moving in it

We consider the stochastic differential equationd _t =( _t )dt+ _t ( _t )dw _t in Euclidean space, where (x) is a Gaussian random field andw _t is a standard Wiener process. Let f _t ={ _s ,st}. Equations are obtained for the conditional meansm _t (x)=f _t } andB _t (x, y)=M{(x)(y)|f _t }.Translated fromTeariya Sluchaínykh Protsessov, Vol. 14, pp. 7–9, 1986.     

The problem of controlling a Wiener process when the number of switchings is bounded

We are considering the problem of controlling a one-dimensional Wiener process (t) (0)=0,E=0,D= ²t.Translated fromProblemy Ustoichivosti Stokhasticheskikh Modelei. Trudy Seminara, 1988, pp. 53–55.     

Estimation of density functions by order statistics

The properties of the empirical density function,f _n(x) = k/n( _j ⁺ – _j-1 ⁺ ) if _j-1 ⁺ < x ⁺ where _j-1 ⁺ and _j ⁺ are sample elements and there are exactlyk – 1 sample elements between them, are studied in that practical point of view how to choose a suitablek for a good estimation. A bound is given for the expected value of the absolute value of difference between the empirical and theoretical density functions.     

Über sphärische Blätterungen und die Vollständigkeit ihrer Blätter

Let M be a Riemannian manifold, a differentiable form on M with values in a bundle Hom (TM, ), and let G be an open subset of M such that Ker forms a vector bundle of constant fibre dimension k>0 over G. We prove: If satisfies some analytical conditions, then is completely integrable, the integral manifolds of are spherically bent in M, and in some interesting cases they are complete.     

A characterization of Gaussian measures on locally compact Abelian groups

Let and be independent random variables having equal variance. In order that + and – be independent, it is necessary and sufficient that and have normal distributions. This result of Bernshtein [1] is carried over in [7] to the case when and take values in a locally compact Abelian group. In the present note, a characterization of Gaussian measures on locally compact Abelian groups is given in which in place of + and –, functions of and are considered which satisfy the associativity equation.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 759–762, November, 1977.     

On a class of 3-dimensional contact metric manifolds

Let M³ be a 3-dimensional contact metric manifold with contact structure (, , , g), such that and =R(.,)) commute. Such a manifold is called 3--manifold. We prove that every 3--manifold with -parallel Weyl tensor is either flat or a Sasakian manifold with constant curvature 1.     

Extrapolation of transformations of random processes perturbed by white noise

We consider the problem of linear mean square optimal estimation of transformation of a stationary random process (t) in observations of process (t) + n(t) for t < – 0, where (t) is white noise uncorrelated with (t). We find least favorable spectral densities f₀() D and minimax (robust) spectral characteristics of an optimal estimator of transformation A for various classesD of densities.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 216–223, February, 1991.     

The regularity of Lagrangiansf(x, ξ)=254-01254-01254-01with Hölder exponents α(x)

In this paper the regularity of the Lagrangiansf(x, )=||^(x)(1< ₁(x)₂< +) is studied. Our main result: If(x) is Holder continuous, then the Lagrangianf(x, )=f(x, )=||^(x) is regular. This result gives a negative answer to a conjecture of V. Zhikov.Supported by the National Natural Science Foundation of China.     

An explicit solution of the linear filtering problem for certain classes of nonrational spectral densities

We present an explicit solution of the problem of optimal linear filtering: the recovery of the useful signal(s) at the instantt+, (>0,<0, or=0) from known values of the received signal(s)=(s)+(s) in the past, i.e., at the instantt–s, s0. In doing so we assume the random processes(s) and /gr(s) are stationary and jointly stationary, while the stationary process of noise (s) with zero mean is assumed to be mutually correlated and jointly stationary with the process(s) under the assumption that there exists a common spectral densityf() for these processes.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 83–91, 1986.     

Hitting probability of a Gaussian vector with values in a Hilbert space in a sphere of small radius

Let be a Gaussian random vector with values in a Hilbert space H. We denote by(a, z) the ball in H with center ata and of radius z. Some asymptotic formulas for I (a, z)=P (a, z), z 0, are presented in the paper.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 85, pp. 75–93, 1979.     

Surgery on the Novikov complex

Let M be a closed connected smooth manifold with dim M=n6, and : ₁(M) Z be an epimorphism. Denote by the group ring of ₁(M) and let ^– be its Novikov completion. Let D _* be a free-based finitely generated chain complex over ^– . Assume that D _ii=0 for i1 and in–1 and that D _* has the same simple homotopy type as the Novikov-completed simplicial chain complex of the universal covering M. Let N be an integer. We prove that D _* can be realized, up to the terms of ^– of degree N as the Novikov complex of a Morse map : M S ¹, belonging to . Applications to Arnold's conjectures and to the theory of fibering of M over S ¹ are given.