scholarly journals The Integral of the Supremum Process of Brownian Motion

2009 ◽  
Vol 46 (02) ◽  
pp. 593-600
Author(s):  
Svante Janson ◽  
Niclas Petersson
Keyword(s):  
Brownian Motion ◽  
Laplace Transform ◽  
Explicit Formula ◽  
Local Time ◽  
Time Interval ◽  
Standard Brownian Motion ◽  
Excursion Theory ◽  
Double Laplace Transform ◽  
The Laplace Transform ◽  
Supremum Process

In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area)(T) covered by the process in the time interval [0,T]. The Laplace transform of(T) follows as a consequence. The main proof involves a double Laplace transform of(T) and is based on excursion theory and local time for Brownian motion.

scholarly journals The Integral of the Supremum Process of Brownian Motion

2009 ◽  
Vol 46 (2) ◽  
pp. 593-600 ◽  
Author(s):  
Svante Janson ◽  
Niclas Petersson
Keyword(s):  
Brownian Motion ◽  
Laplace Transform ◽  
Explicit Formula ◽  
Local Time ◽  
Time Interval ◽  
Standard Brownian Motion ◽  
Excursion Theory ◽  
Double Laplace Transform ◽  
The Laplace Transform ◽  
Supremum Process

In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area)(T) covered by the process in the time interval [0,T]. The Laplace transform of(T) follows as a consequence. The main proof involves a double Laplace transform of(T) and is based on excursion theory and local time for Brownian motion.

The integral of geometric Brownian motion

2001 ◽  
Vol 33 (1) ◽  
pp. 223-241 ◽  
Author(s):  
Daniel Dufresne
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Brownian Motion ◽  
Laplace Transform ◽  
Finite Time ◽  
Geometric Brownian Motion ◽  
Time Interval ◽  
Finite Time Interval ◽  
Special Cases ◽  
The Laplace Transform

This paper is about the probability law of the integral of geometric Brownian motion over a finite time interval. A partial differential equation is derived for the Laplace transform of the law of the reciprocal integral, and is shown to yield an expression for the density of the distribution. This expression has some advantages over the ones obtained previously, at least when the normalized drift of the Brownian motion is a non-negative integer. Bougerol's identity and a relationship between Brownian motions with opposite drifts may also be seen to be special cases of these results.

Excursions height- and length-related stopping times, and application to finance

2002 ◽  
Vol 34 (4) ◽  
pp. 846-868 ◽  
Author(s):  
Laurent Gauthier
Keyword(s):  
Brownian Motion ◽  
Laplace Transform ◽  
Stopping Time ◽  
Stopping Times ◽  
Excursion Theory ◽  
Delay Constraint ◽  
Investment Projects ◽  
The Laplace Transform

In this paper, we study the first instant when Brownian motion either spends consecutively more than a certain time above a certain level, or reaches another level. This stopping time generalizes the ‘Parisian’ stopping times that were introduced by Chesneyet al.(1997). Using excursion theory, we derive the Laplace transform of this stopping time. We apply this result to the valuation of investment projects with a delay constraint, but with an alternative: pay a higher cost and get the project started immediately

scholarly journals Tutte’s invariant approach for Brownian motion reflected in the quadrant

2017 ◽  
Vol 21 ◽  
pp. 220-234 ◽  
Author(s):  
S. Franceschi ◽  
Kilian Raschel
Keyword(s):  
Functional Equation ◽  
Brownian Motion ◽  
Laplace Transform ◽  
Explicit Formula ◽  
Chebyshev Polynomials ◽  
Quarter Plane ◽  
The Laplace Transform ◽  
Negative Drift ◽  
Invariant Approach ◽  
Continuous Setting

We consider a Brownian motion with negative drift in the quarter plane with orthogonal reflection on the axes. The Laplace transform of its stationary distribution satisfies a functional equation, which is reminiscent from equations arising in the enumeration of (discrete) quadrant walks. We develop a Tutte’s invariant approach to this continuous setting, and we obtain an explicit formula for the Laplace transform in terms of generalized Chebyshev polynomials.

Excursions height- and length-related stopping times, and application to finance

2002 ◽  
Vol 34 (04) ◽  
pp. 846-868 ◽  
Author(s):  
Laurent Gauthier
Keyword(s):  
Brownian Motion ◽  
Laplace Transform ◽  
Stopping Time ◽  
Stopping Times ◽  
Excursion Theory ◽  
Delay Constraint ◽  
Investment Projects ◽  
The Laplace Transform

In this paper, we study the first instant when Brownian motion either spends consecutively more than a certain time above a certain level, or reaches another level. This stopping time generalizes the ‘Parisian’ stopping times that were introduced by Chesney et al. (1997). Using excursion theory, we derive the Laplace transform of this stopping time. We apply this result to the valuation of investment projects with a delay constraint, but with an alternative: pay a higher cost and get the project started immediately

SMOLYANOV–WEIZSÄCKER SURFACE MEASURES GENERATED BY DIFFUSIONS ON THE SET OF TRAJECTORIES IN RIEMANNIAN MANIFOLDS

2008 ◽  
Vol 11 (01) ◽  
pp. 21-31 ◽  
Author(s):  
ILYA V. TELYATNIKOV
Keyword(s):  
Brownian Motion ◽  
Euclidean Space ◽  
Diffusion Process ◽  
Riemannian Manifolds ◽  
Marginal Distribution ◽  
Diffusion Processes ◽  
Ambient Space ◽  
Time Interval ◽  
Standard Brownian Motion ◽  
Limit Surface

We consider surface measures on the set of trajectories in a smooth compact Riemannian submanifold of Euclidean space generated by diffusion processes in the ambient space. A construction of surface measures on the path space of a smooth compact Riemannian submanifold of Euclidean space was introduced by Smolyanov and Weizsäcker for the case of the standard Brownian motion. The result presented in this paper extends the result of Smolyanov and Weizsäcker to the case when we consider measures generated by diffusion processes in the ambient space with nonidentical correlation operators. For every partition of the time interval, we consider the marginal distribution of the diffusion process in the ambient space under the condition that it visits the manifold at all times of the partition, when the mesh of the partition tends to zero. We prove the existence of some limit surface measures and the equivalence of the above measures to the distribution of some diffusion process on the manifold.

scholarly journals The Regularity of the Laplace Transform

2019 ◽  
pp. 5-11
Author(s):  
Andrey Pavlov
Keyword(s):  
Laplace Transform ◽  
Double Laplace Transform ◽  
The Laplace Transform

The paper proves the regularity of the double Laplace transform in the neighborhood of zero. The class of the transform of Laplace from the transform of Fourier is considered from the functions without a regularity in null.

scholarly journals Double-Barrier Parisian Options

2011 ◽  
Vol 48 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Angelos Dassios ◽  
Shanle Wu
Keyword(s):  
Brownian Motion ◽  
Markov Model ◽  
Laplace Transform ◽  
Mathematical Finance ◽  
Important Application ◽  
Double Barrier ◽  
Parisian Options ◽  
Brownian Motion With Drift ◽  
The Laplace Transform ◽  
Excursion Time

In this paper we study the excursion time of a Brownian motion with drift outside a corridor by using a four-state semi-Markov model. In mathematical finance, these results have an important application in the valuation of double-barrier Parisian options. We subsequently obtain an explicit expression for the Laplace transform of its price.

A result on the Laplace transform associated with the sticky Brownian motion on an interval

2020 ◽  
pp. 2150031
Author(s):  
Shiyu Song
Keyword(s):  
Brownian Motion ◽  
Boundary Conditions ◽  
Laplace Transform ◽  
Laplace Transforms ◽  
Perturbation Approach ◽  
Integrated Process ◽  
Modified Bessel Function ◽  
Airy Functions ◽  
Occupation Times ◽  
The Laplace Transform

In this paper, we study the joint Laplace transform of the sticky Brownian motion on an interval, its occupation time at zero and its integrated process. The perturbation approach of Li and Zhou [The joint Laplace transforms for diffusion occupation times, Adv. Appl. Probab. 45 (2013) 1049–1067] is adopted to convert the problem into the computation of three Laplace transforms, which is essentially equivalent to solving the associated differential equations with boundary conditions. We obtain the explicit expression for the joint Laplace transform in terms of the modified Bessel function and Airy functions.

scholarly journals On the Frequency of Drawdowns for Brownian Motion Processes

2015 ◽  
Vol 52 (1) ◽  
pp. 191-208 ◽  
Author(s):  
David Landriault ◽  
Bin Li ◽  
Hongzhong Zhang
Keyword(s):  
Risk Management ◽  
Brownian Motion ◽  
Laplace Transform ◽  
Insurance Policies ◽  
Frequency Rate ◽  
The Laplace Transform ◽  
Running Maximum ◽  
Time Sequences ◽  
Quantities Of Interest

Drawdowns measuring the decline in value from the historical running maxima over a given period of time are considered as extremal events from the standpoint of risk management. To date, research on the topic has mainly focused on the side of severity by studying the first drawdown over a certain prespecified size. In this paper we extend the discussion by investigating the frequency of drawdowns and some of their inherent characteristics. We consider two types of drawdown time sequences depending on whether a historical running maximum is reset or not. For each type we study the frequency rate of drawdowns, the Laplace transform of the nth drawdown time, the distribution of the running maximum, and the value process at the nth drawdown time, as well as some other quantities of interest. Interesting relationships between these two drawdown time sequences are also established. Finally, insurance policies protecting against the risk of frequent drawdowns are also proposed and priced.

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