scholarly journals On a first hit distribution of the running maximum of Brownian motion

2022 ◽  
Author(s):  
Julien Randon-Furling ◽  
Paavo Salminen ◽  
Pierre Vallois
Keyword(s):  
Brownian Motion ◽  
Running Maximum

Optimal Stopping of the Maximum Process

2014 ◽  
Vol 51 (03) ◽  
pp. 818-836 ◽  
Author(s):  
Luis H. R. Alvarez ◽  
Pekka Matomäki
Keyword(s):  
Brownian Motion ◽  
Free Boundary ◽  
Optimal Stopping ◽  
Original Problem ◽  
Alternative Route ◽  
Linear Diffusion ◽  
Maximum Process ◽  
Running Maximum ◽  
Parameterized Family ◽  
Optimal Stopping Problems

We consider a class of optimal stopping problems involving both the running maximum as well as the prevailing state of a linear diffusion. Instead of tackling the problem directly via the standard free boundary approach, we take an alternative route and present a parameterized family of standard stopping problems of the underlying diffusion. We apply this family to delineate circumstances under which the original problem admits a unique, well-defined solution. We then develop a discretized approach resulting in a numerical algorithm for solving the considered class of stopping problems. We illustrate the use of the algorithm in both a geometric Brownian motion and a mean reverting diffusion setting.

scholarly journals Occupation Times, Drawdowns, and Drawups for One-Dimensional Regular Diffusions

2015 ◽  
Vol 47 (1) ◽  
pp. 210-230 ◽  
Author(s):  
Hongzhong Zhang
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Time Horizon ◽  
Three Dimensional ◽  
Bessel Processes ◽  
Occupation Times ◽  
Exponential Time ◽  
One Dimensional ◽  
Brownian Motion With Drift ◽  
Running Maximum

The drawdown process of a one-dimensional regular diffusion process X is given by X reflected at its running maximum. The drawup process is given by X reflected at its running minimum. We calculate the probability that a drawdown precedes a drawup in an exponential time-horizon. We then study the law of the occupation times of the drawdown process and the drawup process. These results are applied to address problems in risk analysis and for option pricing of the drawdown process. Finally, we present examples of Brownian motion with drift and three-dimensional Bessel processes, where we prove an identity in law.

Optimal Stopping of the Maximum Process

2014 ◽  
Vol 51 (03) ◽  
pp. 818-836 ◽  
Author(s):  
Luis H. R. Alvarez ◽  
Pekka Matomäki
Keyword(s):  
Brownian Motion ◽  
Free Boundary ◽  
Optimal Stopping ◽  
Original Problem ◽  
Alternative Route ◽  
Linear Diffusion ◽  
Maximum Process ◽  
Running Maximum ◽  
Parameterized Family ◽  
Optimal Stopping Problems

We consider a class of optimal stopping problems involving both the running maximum as well as the prevailing state of a linear diffusion. Instead of tackling the problem directly via the standard free boundary approach, we take an alternative route and present a parameterized family of standard stopping problems of the underlying diffusion. We apply this family to delineate circumstances under which the original problem admits a unique, well-defined solution. We then develop a discretized approach resulting in a numerical algorithm for solving the considered class of stopping problems. We illustrate the use of the algorithm in both a geometric Brownian motion and a mean reverting diffusion setting.

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.

scholarly journals Minimal Expected Time in Drawdown through Investment for an Insurance Diffusion Model

Risks ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 17
Author(s):  
Leonie Violetta Brinker
Keyword(s):  
Brownian Motion ◽  
Optimal Investment ◽  
Investment Strategy ◽  
Insurance Company ◽  
Piecewise Monotone ◽  
Expected Time ◽  
Optimal Investment Strategy ◽  
Running Maximum ◽  
Hamilton Jacobi Bellman ◽  
The Value Function

Consider an insurance company whose surplus is modelled by an arithmetic Brownian motion of not necessarily positive drift. Additionally, the insurer has the possibility to invest in a stock modelled by a geometric Brownian motion independent of the surplus. Our key variable is the (absolute) drawdown Δ of the surplus X, defined as the distance to its running maximum X¯. Large, long-lasting drawdowns are unfavourable for the insurance company. We consider the stochastic optimisation problem of minimising the expected time that the drawdown is larger than a positive critical value (weighted by a discounting factor) under investment. A fixed-point argument is used to show that the value function is the unique solution to the Hamilton–Jacobi–Bellman equation related to the problem. It turns out that the optimal investment strategy is given by a piecewise monotone and continuously differentiable function of the current drawdown. Several numerical examples illustrate our findings.

scholarly journals On the Frequency of Drawdowns for Brownian Motion Processes

2015 ◽  
Vol 52 (01) ◽  
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.

scholarly journals Optimal Stopping of the Maximum Process

2014 ◽  
Vol 51 (3) ◽  
pp. 818-836 ◽  
Author(s):  
Luis H. R. Alvarez ◽  
Pekka Matomäki
Keyword(s):  
Brownian Motion ◽  
Free Boundary ◽  
Optimal Stopping ◽  
Original Problem ◽  
Alternative Route ◽  
Linear Diffusion ◽  
Maximum Process ◽  
Running Maximum ◽  
Parameterized Family ◽  
Optimal Stopping Problems

We consider a class of optimal stopping problems involving both the running maximum as well as the prevailing state of a linear diffusion. Instead of tackling the problem directly via the standard free boundary approach, we take an alternative route and present a parameterized family of standard stopping problems of the underlying diffusion. We apply this family to delineate circumstances under which the original problem admits a unique, well-defined solution. We then develop a discretized approach resulting in a numerical algorithm for solving the considered class of stopping problems. We illustrate the use of the algorithm in both a geometric Brownian motion and a mean reverting diffusion setting.

scholarly journals Performance Analysis of Production Systems with Correlated Demand via Diffusion Approximations

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Yingdong Lu
Keyword(s):  
Brownian Motion ◽  
Production Systems ◽  
Extreme Points ◽  
Joint Probability ◽  
Diffusion Approximations ◽  
Reflected Brownian Motion ◽  
Performance Metric ◽  
Quarter Plane ◽  
Running Maximum ◽  
Joint Probability Density

We investigate the performance of a production system with correlated demand through diffusion approximation. The key performance metric under consideration is the extreme points that this system can reach. This problem is mapped to a problem of characterizing the joint probability density of a two-dimensional Brownian motion and its coordinate running maximum. To achieve this goal, we obtain the stationary distribution of a reflected Brownian motion within the positive quarter-plane, which is of independent interest, through investigating a solution of an extended Helmhotz equation.

scholarly journals Occupation Times, Drawdowns, and Drawups for One-Dimensional Regular Diffusions

2015 ◽  
Vol 47 (01) ◽  
pp. 210-230 ◽  
Author(s):  
Hongzhong Zhang
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Time Horizon ◽  
Three Dimensional ◽  
Bessel Processes ◽  
Occupation Times ◽  
Exponential Time ◽  
One Dimensional ◽  
Brownian Motion With Drift ◽  
Running Maximum

The drawdown process of a one-dimensional regular diffusion processXis given byXreflected at its running maximum. The drawup process is given byXreflected at its running minimum. We calculate the probability that a drawdown precedes a drawup in an exponential time-horizon. We then study the law of the occupation times of the drawdown process and the drawup process. These results are applied to address problems in risk analysis and for option pricing of the drawdown process. Finally, we present examples of Brownian motion with drift and three-dimensional Bessel processes, where we prove an identity in law.

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