scholarly journals Quantized noncommutative Riemann manifolds and stochastic processes: The theoretical foundations of the square root of Brownian motion

2021 ◽  
pp. 126037
Author(s):  
Marco Frasca ◽  
Alfonso Farina ◽  
Moawia Alghalith
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Square Root ◽  
Riemann Manifolds ◽  
Theoretical Foundations

On wald-type optimal stopping for Brownian motion

1997 ◽  
Vol 34 (1) ◽  
pp. 66-73 ◽  
Author(s):  
S. E. Graversen ◽  
G. Peškir
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Maximal Inequality ◽  
Stopping Times ◽  
Standard Brownian Motion ◽  
Square Root ◽  
Real Analysis ◽  
Optimal Stopping Problems ◽  
Wald's Identity

The solution is presented to all optimal stopping problems of the form supτE(G(|Β τ |) – cτ), where is standard Brownian motion and the supremum is taken over all stopping times τ for B with finite expectation, while the map G : ℝ+ → ℝ satisfies for some being given and fixed. The optimal stopping time is shown to be the hitting time by the reflecting Brownian motion of the set of all (approximate) maximum points of the map . The method of proof relies upon Wald's identity for Brownian motion and simple real analysis arguments. A simple proof of the Dubins–Jacka–Schwarz–Shepp–Shiryaev (square root of two) maximal inequality for randomly stopped Brownian motion is given as an application.

scholarly journals Stochastic Processes with a Particular Type of Variograms

2007 ◽  
Vol 2007 ◽  
pp. 1-5 ◽  
Author(s):  
Chunsheng Ma
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Stochastic Processes ◽  
Series Expansion ◽  
Real Line ◽  
Random Fields ◽  
The Other ◽  
Simple Construction ◽  
The Real ◽  
Line Form

This paper is concerned with a class of stochastic processes or random fields with second-order increments, whose variograms have a particular form, among which stochastic processes having orthogonal increments on the real line form an important subclass. A natural issue, how big this subclass is, has not been explicitly addressed in the literature. As a solution, this paper characterizes a stochastic process having orthogonal increments on the real line in terms of its variogram or its construction. Our findings are a little bit surprising: this subclass is big in terms of the variogram, and on the other hand, it is relatively “small” according to a simple construction. In particular, every such process with Gaussian increments can be simply constructed from Brownian motion. Using the characterizations we obtain a series expansion of the stochastic process with orthogonal increments.

Approximations of boundary crossing probabilities for a Brownian motion

1999 ◽  
Vol 36 (4) ◽  
pp. 1019-1030 ◽  
Author(s):  
Alex Novikov ◽  
Volf Frishling ◽  
Nino Kordzakhia
Keyword(s):  
Brownian Motion ◽  
Power Series ◽  
Numerical Integration ◽  
Piecewise Linear ◽  
Boundary Crossing ◽  
Square Root ◽  
Girsanov Transformation ◽  
Piecewise Linear Approximations ◽  
Infinite Power ◽  
Crossing Probabilities

Using the Girsanov transformation we derive estimates for the accuracy of piecewise approximations for one-sided and two-sided boundary crossing probabilities. We demonstrate that piecewise linear approximations can be calculated using repeated numerical integration. As an illustrative example we consider the case of one-sided and two-sided square-root boundaries for which we also present analytical representations in a form of infinite power series.

scholarly journals Dimensions of Fractional Brownian Images

2021 ◽  
Author(s):  
Stuart A. Burrell
Keyword(s):  
Brownian Motion ◽  
Hausdorff Dimension ◽  
Stochastic Processes ◽  
Fractional Brownian Motion ◽  
Box Dimension ◽  
General Strategy ◽  
Borel Sets ◽  
Exceptional Sets ◽  
Explicit Condition ◽  
Box Dimensions

AbstractThis paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Potential-theoretic methods are used to produce dimension bounds for images of sets under Hölder maps and certain stochastic processes. We apply this to compute the almost-sure value of the dimension of Borel sets under index-$$\alpha $$ α fractional Brownian motion in terms of dimension profiles defined using capacities. As a corollary, this establishes continuity of the profiles for Borel sets and allows us to obtain an explicit condition showing how the Hausdorff dimension of a set may influence the typical box dimension of Hölder images such as projections. The methods used propose a general strategy for related problems; dimensional information about a set may be learned from analysing particular fractional Brownian images of that set. To conclude, we obtain bounds on the Hausdorff dimension of exceptional sets, with respect to intermediate dimensions, in the setting of projections.

Weak Convergence to Stochastic Integrals

2021 ◽  
pp. 724-756
Author(s):  
James Davidson
Keyword(s):  
Brownian Motion ◽  
Weak Convergence ◽  
Stochastic Processes ◽  
Final Section ◽  
Stochastic Integrals ◽  
Martingale Difference ◽  
Mixing Processes ◽  
Linear Processes ◽  
Special Cases

The main object of this chapter is to prove the convergence of the covariances of stochastic processes with their increments to stochastic integrals with respect to Brownian motion. Some preliminary theory is given relating to random functionals on C, stochastic integrals, and the important Itô isometry. The main result is first proved for the tractable special cases of martingale difference increments and linear processes. The final section is devoted to proving the more difficult general case, of NED functions of mixing processes.

Introductory remarks

1990 ◽  
Vol 332 (1627) ◽  
pp. 417-418
Keyword(s):  
Probability Theory ◽  
Brownian Motion ◽  
Random Walk ◽  
Stochastic Processes ◽  
Statistical Physics ◽  
Late 19Th Century ◽  
Fields Of Study ◽  
Wide Range ◽  
Influential Paper ◽  
Telephone Traffic

Stochastic processes are systems that evolve in time probabilistically; their study is the ‘dynamics’ of probability theory as contrasted with rather more traditional ‘static’ problems. The analysis of stochastic processes has as one of its main origins late 19th century statistical physics leading in particular to studies of random walk and brownian motion (Rayleigh 1880; Einstein 1906) and via them to the very influential paper of Chandrasekhar (1943). Other strands emerge from the work of Erlang (1909) on congestion in telephone traffic and from the investigations of the early mathematical epidemiologists and actuarial scientists. There is by now a massive general theory and a wide range of special processes arising from applications in many fields of study, including those mentioned above. A relatively small part of the above work concerns techniques for the analysis of empirical data arising from such systems.

scholarly journals Estimation of the integral of a stochastic process

1978 ◽  
Vol 18 (1) ◽  
pp. 83-93 ◽  
Author(s):  
Noel Cressie
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Stochastic Processes ◽  
Poisson Process ◽  
Covariance Structure ◽  
Regression Equation ◽  
Linear Drift ◽  
Independent Increments ◽  
Sampling Points ◽  
Optimum Sampling

Consider the class of stochastic processes with stationary independent increments and finite variances; notable members are brownian motion, and the Poisson process. Now for Xt any member of this class of processes, we wish to find the optimum sampling points of Xt, for predicting . This design question is shown to be directly related to finding sampling points of Yt for estimating β in the regression equation, Yt = β + Xt. Since processes with stationary independent increments have linear drift, the regression equation for Yt is the first type of departure we might look for; namely quadratic drift, and unchanged covariance structure.

Sign in / Sign up

Export Citation Format

Share Document