Some conditional crossing results of Brownian motion over a piecewise-linear boundary

Explicit formulae are found for the probability that the Brownian motion, Bt, up-crosses, in [0,T], a piecewise-linear function S(t), with the condition that the value of Bt is assigned at a future time u>T or at an intermediate time u

[1]  A. Novikov,et al.  Approximations of boundary crossing probabilities for a Brownian motion , 1999 .

[2]  Liqun Wang,et al.  Boundary crossing probability for Brownian motion and general boundaries , 1997, Journal of Applied Probability.

[3]  Thomas H. Scheike.  A boundary-crossing result for Brownian motion , 1992 .

[4]  A. G. Nobile,et al.  On the evaluation of first-passage-time probability densities via non-singular integral equations , 1989, Advances in Applied Probability.

[5]  Abundo.  On first-passage-times for one-dimensional jump-diffusion processes. , 2000 .

[6]  Mario Abundo,et al.  On first–crossing times of one–dimensional diffusions over two time–dependent boundaries , 2000 .

[7]  ON THE TWO-BOUNDARY FIRST-CROSSING-TIME PROBLEM FOR DIFFUSION PROCESSES , 1990 .

[8]  Anders Martin-Löf,et al.  The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier , 1998 .

[9]  J. Doob.  Heuristic Approach to the Kolmogorov-Smirnov Theorems , 1949 .

[10]  J. Durbin.  Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test , 1971, Journal of Applied Probability.

[11]  Ya. Yu. Nikitin,et al.  Limit distributions and comparative asymptotic efficiency of the Smirnov — Kolmogorov statistics with a random index , 1981 .

[12]  P. Groeneboom.  Brownian motion with a parabolic drift and airy functions , 1989 .

[13]  J. Durbin,et al.  The first-passage density of the Brownian motion process to a curved boundary , 1992, Journal of Applied Probability.

[14]  W. J. Hall.  The distribution of brownian motion on linear stopping boundaries , 1997 .

[15]  T. W. Anderson.  A MODIFICATION OF THE SEQUENTIAL PROBABILITY RATIO TEST TO REDUCE THE SAMPLE SIZE , 1960 .

[16]  A. Borodin,et al.  Handbook of Brownian Motion - Facts and Formulae , 1996 .

[17]  P. Levy.  Processus stochastiques et mouvement brownien , 1948 .

[18]  A. G. Nobile,et al.  ON A SYMMETRY-BASED CONSTRUCTIVE APPROACH TO PROBABILITY DENSITIES FOR TWO-DIMENSIONAL DIFFUSION PROCESSES , 1989 .

[19]  Luisa Beghin,et al.  On the maximum of the generalized Brownian bridge , 1999 .

[20]  P. Salminen.  On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary , 1988, Advances in Applied Probability.

[21]  R. Trappl,et al.  Cybernetics and Systems 2000 , 2000 .

[22]  J. Durbin.  THE FIRST-PASSAGE DENSITY OF A CONTINUOUS GAUSSIAN PROCESS , 1985 .

[23]  L. Breiman.  First exit times from a square root boundary , 1967 .

[24]  A. G. Nobile,et al.  A new integral equation for the evaluation of first-passage-time probability densities , 1987, Advances in Applied Probability.

[25]  H. R. Lerche.  Boundary Crossing of Brownian Motion , 1986 .