### Abstract

The purpose of this note is to identify an interesting and surprising duality between the equations governing the probability distribution and expected value functional of the stochastic process defined by A_{t} := ∫^{t}_{0} exp{Z_{s}} ds, t ≥ 0, where {Z_{s}: s ≥ 0} is a one-dimensional Brownian motion with drift coefficient μ and diffusion coefficient σ^{2}. In particular, both expected values of the form v(t, x) := Ef(x+A_{t}), f homogeneous, as well as the probability density a(t, y) dy := P(A_{t} ∈ dy) are shown to be governed by a pair of linear parabolic partial differential equations. Although the equations are not the backward/forward adjoint pairs one would naturally have in the general theory of Markov processes, unifying and remarkably simple derivations of these equations are provided.

Original language | English (US) |
---|---|

Pages (from-to) | 187-192 |

Number of pages | 6 |

Journal | Statistics and Probability Letters |

Volume | 55 |

Issue number | 2 |

DOIs | |

State | Published - Nov 15 2001 |

Externally published | Yes |

### Fingerprint

### Keywords

- Asian options
- Geometric Brownian motion
- Turbulence

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Statistics and Probability

### Cite this

*Statistics and Probability Letters*,

*55*(2), 187-192. https://doi.org/10.1016/S0167-7152(01)00117-1

**A note on the distribution of integrals of geometric Brownian motion.** / Bhattacharya, Rabindra N; Thomann, Enrique; Waymire, Edward.

Research output: Contribution to journal › Article

*Statistics and Probability Letters*, vol. 55, no. 2, pp. 187-192. https://doi.org/10.1016/S0167-7152(01)00117-1

}

TY - JOUR

T1 - A note on the distribution of integrals of geometric Brownian motion

AU - Bhattacharya, Rabindra N

AU - Thomann, Enrique

AU - Waymire, Edward

PY - 2001/11/15

Y1 - 2001/11/15

N2 - The purpose of this note is to identify an interesting and surprising duality between the equations governing the probability distribution and expected value functional of the stochastic process defined by At := ∫t0 exp{Zs} ds, t ≥ 0, where {Zs: s ≥ 0} is a one-dimensional Brownian motion with drift coefficient μ and diffusion coefficient σ2. In particular, both expected values of the form v(t, x) := Ef(x+At), f homogeneous, as well as the probability density a(t, y) dy := P(At ∈ dy) are shown to be governed by a pair of linear parabolic partial differential equations. Although the equations are not the backward/forward adjoint pairs one would naturally have in the general theory of Markov processes, unifying and remarkably simple derivations of these equations are provided.

AB - The purpose of this note is to identify an interesting and surprising duality between the equations governing the probability distribution and expected value functional of the stochastic process defined by At := ∫t0 exp{Zs} ds, t ≥ 0, where {Zs: s ≥ 0} is a one-dimensional Brownian motion with drift coefficient μ and diffusion coefficient σ2. In particular, both expected values of the form v(t, x) := Ef(x+At), f homogeneous, as well as the probability density a(t, y) dy := P(At ∈ dy) are shown to be governed by a pair of linear parabolic partial differential equations. Although the equations are not the backward/forward adjoint pairs one would naturally have in the general theory of Markov processes, unifying and remarkably simple derivations of these equations are provided.

KW - Asian options

KW - Geometric Brownian motion

KW - Turbulence

UR - http://www.scopus.com/inward/record.url?scp=0000860630&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000860630&partnerID=8YFLogxK

U2 - 10.1016/S0167-7152(01)00117-1

DO - 10.1016/S0167-7152(01)00117-1

M3 - Article

VL - 55

SP - 187

EP - 192

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

SN - 0167-7152

IS - 2

ER -