On the inverse of the first hitting time problem for bidimensional processes

1997 ◽  
Vol 34 (3) ◽  
pp. 610-622 ◽  
Author(s):  
Mario Lefebvre
Keyword(s):  
Generating Function ◽  
Complex Function ◽  
Sufficient Conditions ◽  
Moment Generating Function ◽  
Hitting Time ◽  
First Hitting Time ◽  
Time Problem ◽  
The Moment ◽  
Image Position ◽  
Necessary And Sufficient

Bidimensional processes defined by dx(t) = ρ (x, y)dt and dy(t) = m(x, y)dt + [2v(x, y)]1/2dW(t), where W(t) is a Wiener process, are considered. Let T(x, y) be the first time the process (x(t), y(t)), starting from (x, y), hits the boundary of a given region in . A theorem is proved that gives necessary and sufficient conditions for a given complex function to be considered as the moment generating function of T(x, y) for some bidimensional diffusion process. Examples are given where the theorem is used to construct explicit solutions to first hitting time problems and to compute the infinitesimal moments that correspond to the chosen moment generating function.

On the inverse of the first hitting time problem for bidimensional processes

1997 ◽  
Vol 34 (03) ◽  
pp. 610-622
Author(s):  
Mario Lefebvre
Keyword(s):  
Generating Function ◽  
Complex Function ◽  
Sufficient Conditions ◽  
Moment Generating Function ◽  
Hitting Time ◽  
First Hitting Time ◽  
Time Problem ◽  
The Moment ◽  
Image Position ◽  
Necessary And Sufficient

Bidimensional processes defined by dx(t) = ρ (x, y)dt and dy(t) = m(x, y)dt + [2v(x, y)]1/2dW(t), where W(t) is a Wiener process, are considered. Let T(x, y) be the first time the process (x(t), y(t)), starting from (x, y), hits the boundary of a given region in . A theorem is proved that gives necessary and sufficient conditions for a given complex function to be considered as the moment generating function of T(x, y) for some bidimensional diffusion process. Examples are given where the theorem is used to construct explicit solutions to first hitting time problems and to compute the infinitesimal moments that correspond to the chosen moment generating function.

Note on the derivation of fluctuation formulae for statistical assemblies

1937 ◽  
Vol 33 (3) ◽  
pp. 390-393 ◽  
Author(s):  
M. S. Bartlett
Keyword(s):  
Generating Function ◽  
Simple Formula ◽  
Moment Generating Function ◽  
Statistical Distributions ◽  
The Moment ◽  
Image Position

1. A familiar device in the study of statistical distributions is to form the moment-generating functionwhere the bar denotes averaging over all values of the statistical variate x. The moments μr of x are the coefficients of αr/r!, and the derived coefficients in the expansion of K ≡ log M are termed the semi-invariants kr. In particular,and, for the normal (Gaussian) law,we have the simple formula

scholarly journals Randomization in the first hitting time problem

2009 ◽  
Vol 79 (23) ◽  
pp. 2422-2428 ◽  
Author(s):  
Ken Jackson ◽  
Alexander Kreinin ◽  
Wanhe Zhang
Keyword(s):  
Hitting Time ◽  
First Hitting Time ◽  
Time Problem

scholarly journals Topological entropy in totally disconnected locally compact groups

2016 ◽  
Vol 37 (7) ◽  
pp. 2163-2186 ◽  
Author(s):  
ANNA GIORDANO BRUNO ◽  
SIMONE VIRILI
Keyword(s):  
Topological Entropy ◽  
Sufficient Conditions ◽  
Invariant Subgroup ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Totally Disconnected ◽  
Dynamical Interpretation

Let $G$ be a topological group, let $\unicode[STIX]{x1D719}$ be a continuous endomorphism of $G$ and let $H$ be a closed $\unicode[STIX]{x1D719}$-invariant subgroup of $G$. We study whether the topological entropy is an additive invariant, that is, $$\begin{eqnarray}h_{\text{top}}(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719}\restriction _{H})+h_{\text{top}}(\bar{\unicode[STIX]{x1D719}}),\end{eqnarray}$$ where $\bar{\unicode[STIX]{x1D719}}:G/H\rightarrow G/H$ is the map induced by $\unicode[STIX]{x1D719}$. We concentrate on the case when $G$ is totally disconnected locally compact and $H$ is either compact or normal. Under these hypotheses, we show that the above additivity property holds true whenever $\unicode[STIX]{x1D719}H=H$ and $\ker (\unicode[STIX]{x1D719})\leq H$. As an application, we give a dynamical interpretation of the scale $s(\unicode[STIX]{x1D719})$ by showing that $\log s(\unicode[STIX]{x1D719})$ is the topological entropy of a suitable map induced by $\unicode[STIX]{x1D719}$. Finally, we give necessary and sufficient conditions for the equality $\log s(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719})$ to hold.

Moving Weighted Averages

1993 ◽  
Vol 45 (3) ◽  
pp. 449-469 ◽  
Author(s):  
M. A. Akcoglu ◽  
Y. Déniel
Keyword(s):  
Weight Function ◽  
Sufficient Conditions ◽  
Nonnegative Function ◽  
Finite Measure ◽  
Weight Functions ◽  
Ergodic Averages ◽  
Fixed Weight ◽  
Finite Measure Space ◽  
Image Position ◽  
Necessary And Sufficient

AbstractLet ℝ denote the real line. Let {Tt}tєℝ be a measure preserving ergodic flow on a non atomic finite measure space (X, ℱ, μ). A nonnegative function φ on ℝ is called a weight function if ∫ℝ φ(t)dt = 1. Consider the weighted ergodic averagesof a function f X —> ℝ, where {θk} is a sequence of weight functions. Some sufficient and some necessary and sufficient conditions are given for the a.e. convergence of Akf, in particular for a special case in whichwhere φ is a fixed weight function and {(ak, rk)} is a sequence of pairs of real numbers such that rk > 0 for all k. These conditions are obtained by a combination of the methods of Bellow-Jones-Rosenblatt, developed to deal with moving ergodic averages, and the methods of Broise-Déniel-Derriennic, developed to deal with unbounded weight functions.

scholarly journals Sufficient conditions for matchings

1972 ◽  
Vol 18 (2) ◽  
pp. 129-136 ◽  
Author(s):  
Ian Anderson
Keyword(s):  
Simple Proof ◽  
Perfect Matching ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Number Of Components ◽  
Image Position ◽  
Necessary And Sufficient

A graph G is said to possess a perfect matching if there is a subgraph of G consisting of disjoint edges which together cover all the vertices of G. Clearly G must then have an even number of vertices. A necessary and sufficient condition for G to possess a perfect matching was obtained by Tutte (3). If S is any set of vertices of G, let p(S) denote the number of components of the graph G – S with an odd number of vertices. Then the conditionis both necessary and sufficient for the existence of a perfect matching. A simple proof of this result is given in (1).

On the linear exponential family

1968 ◽  
Vol 64 (2) ◽  
pp. 481-483 ◽  
Author(s):  
J. K. Wani
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Scalar Product ◽  
Exponential Family ◽  
Characterization Theorem ◽  
Random Variable ◽  
Moment Generating Function ◽  
The Moment ◽  
Image Position

In this paper we give a characterization theorem for a subclass of the exponential family whose probability density function is given bywhere a(x) ≥ 0, f(ω) = ∫a(x) exp (ωx) dx and ωx is to be interpreted as a scalar product. The random variable X may be an s-vector. In that case ω will also be an s-vector. For obvious reasons we will call (1) as the linear exponential family. It is easy to verify that the moment generating function (m.g.f.) of (1) is given by

A NOTE ON INEQUALITY CONSTRAINTS IN THE GARCH MODEL

2008 ◽  
Vol 24 (3) ◽  
pp. 823-828 ◽  
Author(s):  
Henghsiu Tsai ◽  
Kung-Sik Chan
Keyword(s):  
Generating Function ◽  
Sufficient Conditions ◽  
Inequality Constraints ◽  
Conditional Variance ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Economic Statistics ◽  
Absolute Monotonicity ◽  
Necessary And Sufficient ◽  
The Garch Model

We consider the parameter restrictions that need to be imposed to ensure that the conditional variance process of a GARCH(p,q) model remains nonnegative. Previously, Nelson and Cao (1992, Journal of Business ’ Economic Statistics 10, 229–235) provided a set of necessary and sufficient conditions for the aforementioned nonnegativity property for GARCH(p,q) models with p ≤ 2 and derived a sufficient condition for the general case of GARCH(p,q) models with p ≥ 3. In this paper, we show that the sufficient condition of Nelson and Cao (1992) for p ≥ 3 actually is also a necessary condition. In addition, we point out the linkage between the absolute monotonicity of the generalized autoregressive conditional heteroskedastic (GARCH) generating function and the nonnegativity of the GARCH kernel, and we use it to provide examples of sufficient conditions for this nonnegativity property to hold.

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