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1

Hitting distributions of geometric Brownian motion

100%

Byczkowski T., Ryznar M.

Studia Mathematica

|

2006

|

tom 173

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nr 1

19-38

EN

Let τ be the first hitting time of the point 1 by the geometric Brownian motion X(t) = x exp(B(t) - 2μt) with drift μ ≥ 0 starting from x > 1. Here B(t) is the Brownian motion starting from 0 with EB²(t) = 2t. We provide an integral formula for the density function of the stopped exponential functional $A(τ) = ∫_0^τ X²(t)dt$ and determine its asymptotic behaviour at infinity. Although we basically rely on methods developed in [BGS], the present paper covers the case of arbitrary drifts μ ≥ 0 and provides a significant unification and extension of the results of the above-mentioned paper. As a corollary we provide an integral formula and give the asymptotic behaviour at infinity of the Poisson kernel for half-spaces for Brownian motion with drift in real hyperbolic spaces of arbitrary dimension.

2

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Equivalence theorems for infinite products of independent random elements on metric semigroups

80%

Byczkowski T., Woś J.

Colloquium Mathematicum

|

1977

|

tom 37

|

nr 2

287-293

3

Zero-one law for subgroups of paths of group-valued stochastic processes

70%

Byczkowska H., Byczkowski T.

Studia Mathematica

|

1988

|

tom 89

|

nr 1

65-73

4

On infinite products of independent random elements on metric semigroups

60%

Byczkowski T., Woś J.

Colloquium Mathematicum

|

1977

|

tom 37

|

nr 2

271-285

5

The invariance principle for group-valued random variables

60%

Byczkowski T.

Studia Mathematica

|

1976

|

tom 56

|

nr 2

187-198

6

Zero-one laws for Gaussian measures on metric abelian groups

50%

Byczkowski T.

Studia Mathematica

|

1981

|

tom 69

|

nr 2

159-189

7

On the density of log concave seminorms on vector spaces

50%

Byczkowski T.

Studia Mathematica

|

1991

|

tom 99

|

nr 2

101-108

8

Gaussian measures on $L_{p}$ spaces, 0 ≤ p < ∞

50%

Byczkowski T.

Studia Mathematica

|

1976-1977

|

tom 59

|

nr 3

249-261

9

Norm convergent expansion for $L_{Φ}$-valued Gaussian random elements

50%

Byczkowski T.

Studia Mathematica

|

1979

|

tom 64

|

nr 2

87-95

Ograniczanie wyników

7 Studia Mathematica

2 Colloquium Mathematicum

9 Byczkowski T.

2 Woś J.

1 Byczkowska H.

1 Ryznar M.

1 2006

1 1991

1 1988

1 1981

1 1979

3 1977

1 1976

Hitting distributions of geometric Brownian motion

Equivalence theorems for infinite products of independent random elements on metric semigroups

Zero-one law for subgroups of paths of group-valued stochastic processes

On infinite products of independent random elements on metric semigroups

The invariance principle for group-valued random variables

Zero-one laws for Gaussian measures on metric abelian groups

On the density of log concave seminorms on vector spaces

Gaussian measures on $L_{p}$ spaces, 0 ≤ p < ∞

Norm convergent expansion for $L_{Φ}$-valued Gaussian random elements