Distribution of the supremum of the two-parameter Yeh-Wiener process on the boundary

Let D = [0, S] × [0, T] be a rectangle in E2 and X(s, t), (s, t)∈D, be a two parameter Yeh-Wiener process. This paper finds the probability distribution of the supremum of X(s, t) on the boundary of D by taking the limit of the probability distribution of the supremum of X(s, t) along certain paths as these paths approach the boundary of D. The probability distribution of the supremum of X(s, t) on the boundary of D gives a nice lower bound for the probability distribution of the supremum of X(s, t) on D, which is unknown.

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Some lower bounds for the distribution of the supremum of the Yeh-Wiener process over a rectangular region

Journal of Applied Probability ◽

10.1017/s0021900200048798 ◽

1975 ◽

Vol 12 (04) ◽

pp. 824-830

Author(s):

Arthur H. C. Chan

Keyword(s):

Lower Bound ◽

Gaussian Process ◽

Wiener Process ◽

Lower Bounds ◽

Upper Bound ◽

Exact Distribution ◽

Rectangular Region ◽

Process With Independent Increments ◽

Independent Increments ◽

Two Parameter

Let W (s, t), s, t ≧ 0, be the two-parameter Yeh–Wiener process defined on the first quadrant of the plane, that is, a Gaussian process with independent increments in both directions. In this paper, a lower bound for the distribution of the supremum of W (s, t) over a rectangular region [0, S]×[0, T], for S, T > 0, is given. An upper bound for the same was known earlier, while its exact distribution is still unknown.

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Some lower bounds for the distribution of the supremum of the Yeh-Wiener process over a rectangular region

Journal of Applied Probability ◽

10.2307/3212734 ◽

1975 ◽

Vol 12 (4) ◽

pp. 824-830 ◽

Cited By ~ 4

Author(s):

Arthur H. C. Chan

Keyword(s):

Lower Bound ◽

Gaussian Process ◽

Wiener Process ◽

Lower Bounds ◽

Upper Bound ◽

Exact Distribution ◽

Rectangular Region ◽

Process With Independent Increments ◽

Independent Increments ◽

Two Parameter

Let W (s, t), s, t ≧ 0, be the two-parameter Yeh–Wiener process defined on the first quadrant of the plane, that is, a Gaussian process with independent increments in both directions. In this paper, a lower bound for the distribution of the supremum of W (s, t) over a rectangular region [0, S]×[0, T], for S, T > 0, is given. An upper bound for the same was known earlier, while its exact distribution is still unknown.

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Two-parameter harnesses and the Wiener process

Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete ◽

10.1007/bf00531429 ◽

1981 ◽

Vol 56 (4) ◽

pp. 507-514 ◽

Cited By ~ 4

Author(s):

M. Dozzi

Keyword(s):

Wiener Process ◽

Two Parameter

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Stochastic inclusions and set-valued stochastic equations driven by a two-parameter Wiener process

Stochastics and Dynamics ◽

10.1142/s0219493718500478 ◽

2018 ◽

Vol 18 (06) ◽

pp. 1850047 ◽

Cited By ~ 1

Author(s):

Mariusz Michta ◽

Kamil Łukasz Świa̧tek

Keyword(s):

Wiener Process ◽

Stochastic Equation ◽

Stochastic Equations ◽

Continuous Selection ◽

Attainable Sets ◽

Multivalued Solution ◽

Stochastic Differential Inclusions ◽

Multivalued Solutions ◽

Two Parameter ◽

Selection Of

In the paper we study properties of solutions to stochastic differential inclusions and set-valued stochastic differential equations driven by a two-parameter Wiener process. We establish new connections between their solutions. We prove that attainable sets of solutions to such inclusions are subsets of values of multivalued solutions of associated set-valued stochastic equations. Next we show that every solution to stochastic inclusion is a continuous selection of a multivalued solution of an associated set-valued stochastic equation. Additionally we establish other properties of such solutions. The results obtained in the paper extends results dealing with this topic known both in deterministic and stochastic cases.

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New Characteristic Relations in Type-II and III Intermittency

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127497000613 ◽

1997 ◽

Vol 07 (04) ◽

pp. 831-836 ◽

Cited By ~ 5

Author(s):

M. O. Kim ◽

Hoyun Lee ◽

Chil-Min Kim ◽

Hyun-Soo Pang ◽

Eok-Kyun Lee ◽

...

Keyword(s):

Numerical Simulation ◽

Lower Bound ◽

Probability Distribution ◽

Critical Exponent ◽

Simulation Study ◽

Critical Exponents ◽

Control Parameter ◽

The Other ◽

Square Root ◽

Type Ii

We obtained new characteristic relations in Type-II and III intermittencies according to the reinjection probability distribution. When the reinjection probability distribution is fixed at the lower bound of reinjection, the critical exponents are -1, as is well known. However when the reinjection probability distribution is uniform, the critical exponent is -1/2, and when it is of form [Formula: see text], -3/4. On the other hand, if the square root of Δ, which represents the lower bound of reinjection, is much smaller than the control parameter ∊, i.e. ∊ ≫ Δ1/2, critical exponent is always -1, independent of the reinjection probability distribution. Those critical exponents are confirmed by numerical simulation study.

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Distribution Estimates for Functionals of the Two-Parameter Wiener Process

The Annals of Probability ◽

10.1214/aop/1176995940 ◽

1976 ◽

Vol 4 (6) ◽

pp. 977-982 ◽

Cited By ~ 17

Author(s):

Victor Goodman

Keyword(s):

Wiener Process ◽

Two Parameter

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Embedding submartingales in wiener processes with drift, with applications to sequential analysis

Journal of Applied Probability ◽

10.1017/s0021900200026668 ◽

1969 ◽

Vol 6 (03) ◽

pp. 612-632 ◽

Cited By ~ 2

Author(s):

W. J. Hall

Keyword(s):

Lower Bound ◽

Random Walks ◽

Sample Size ◽

Wiener Process ◽

Likelihood Ratio ◽

Sequential Analysis ◽

Stopping Times ◽

Wiener Processes ◽

Size Function ◽

Log Likelihood

Summary Skorokhod (1961) demonstrated how the study of martingale sequences (and zero-mean random walks) can be reduced to the study of the Wiener process (without drift) at a sequence of random stopping times. We show how the study of certain submartingale sequences, including certain random walks with drift and log likelihood ratio sequences, can be reduced to the study of the Wiener process with drift at a sequence of stopping times (Theorem 4.1). Applications to absorption problems are given. Specifically, we present new derivations of a number of the basic approximations and inequalities of classical sequential analysis, and some variations on them — including an improvement on Wald's lower bound for the expected sample size function (Corollary 7.5).

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