Last Exit Time and Harmonic Measure for Brownian Motion in Rd

In this paper, distributional questions which arise in certain mathematical finance models are studied: the distribution of the integral over a fixed time interval [0, T] of the exponential of Brownian motion with drift is computed explicitly, with the help of computations previously made by the author for Bessel processes. The moments of this integral are obtained independently and take a particularly simple form. A subordination result involving this integral and previously obtained by Bougerol is recovered and related to an important identity for Bessel functions. When the fixed time T is replaced by an independent exponential time, the distribution of the integral is shown to be related to last-exit-time distributions and the fixed time case is recovered by inverting Laplace transforms.

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On some exponential functionals of Brownian motion

Advances in Applied Probability ◽

10.2307/1427477 ◽

1992 ◽

Vol 24 (3) ◽

pp. 509-531 ◽

Cited By ~ 177

Author(s):

Marc Yor

Keyword(s):

Brownian Motion ◽

Bessel Functions ◽

Exit Time ◽

Fixed Time ◽

Mathematical Finance ◽

Time Interval ◽

Bessel Processes ◽

Fixed Time Interval ◽

Last Exit Time ◽

Subordination Result

In this paper, distributional questions which arise in certain mathematical finance models are studied: the distribution of the integral over a fixed time interval [0, T] of the exponential of Brownian motion with drift is computed explicitly, with the help of computations previously made by the author for Bessel processes. The moments of this integral are obtained independently and take a particularly simple form. A subordination result involving this integral and previously obtained by Bougerol is recovered and related to an important identity for Bessel functions. When the fixed time T is replaced by an independent exponential time, the distribution of the integral is shown to be related to last-exit-time distributions and the fixed time case is recovered by inverting Laplace transforms.

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On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary

Advances in Applied Probability ◽

10.2307/1427397 ◽

1988 ◽

Vol 20 (2) ◽

pp. 411-426 ◽

Cited By ~ 48

Author(s):

Paavo Salminen

Keyword(s):

Brownian Motion ◽

Smooth Function ◽

Moving Boundary ◽

Exit Time ◽

Expected Value ◽

Explicit Solutions ◽

First Hitting Time ◽

Standard Brownian Motion ◽

Square Root ◽

Last Exit Time

Let t → h(t) be a smooth function on ℝ+, and B = {Bs; s ≥ 0} a standard Brownian motion. In this paper we derive expressions for the distributions of the variables Th: = inf {S; Bs = h(s)} and λth: = sup {s ≦ t; Bs = h(s)}, where t> 0 is given. Our formulas contain an expected value of a Brownian functional. It is seen that this can be computed, principally, using Feynman–Kac&s formula. Further, we discuss in our framework the familiar examples with linear and square root boundaries. Moreover our approach provides in some extent explicit solutions for the second-order boundaries.

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On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary

Advances in Applied Probability ◽

10.1017/s0001867800017043 ◽

1988 ◽

Vol 20 (02) ◽

pp. 411-426 ◽

Cited By ~ 11

Author(s):

Paavo Salminen

Keyword(s):

Brownian Motion ◽

Smooth Function ◽

Moving Boundary ◽

Exit Time ◽

Expected Value ◽

Explicit Solutions ◽

First Hitting Time ◽

Standard Brownian Motion ◽

Square Root ◽

Last Exit Time

Lett → h(t) be a smooth function on ℝ+, andB= {Bs;s≥ 0} a standard Brownian motion. In this paper we derive expressions for the distributions of the variablesTh: = inf {S;Bs=h(s)} and λth: = sup {s≦t; Bs= h(s)}, wheret>0 is given. Our formulas contain an expected value of a Brownian functional. It is seen that this can be computed, principally, using Feynman–Kac&s formula. Further, we discuss in our framework the familiar examples with linear and square root boundaries. Moreover our approach provides in some extent explicit solutions for the second-order boundaries.

Download Full-text