Geometric Brownian motion | EPFL Graph Search

A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. It is an important example of stochastic processes satisfying a stochastic differential equation (SDE); in particular, it is used in mathematical finance to model stock prices in the Black–Scholes model. Technical definition: the SDE A stochastic process St is said to follow a GBM if it satisfies the following stochastic differential equation (SDE): : dS_t = \mu S_t,dt + \sigma S_t,dW_t where W_t is a Wiener process or Brownian motion, and \mu ('the percentage drift') and \sigma ('the percentage volatility') are constants. The former parameter is used to model deterministic trends, while the latter parameter models unpredictable events occurring during the motion. Sol

Optimal solution and asymptotic properties of a stochastic control problem arising in sailboat trajectory optimization

Carlo Ciccarella

We study the optimal strategy for a sailboat to reach an upwind island under the hypothesis that the wind direction fluctuates according to a Brownian motion and the wind speed is constant. The work is motivated by a concrete problem which typically arises during sailing regattas, namely finding the best tacking strategy to reach the upwind buoy as quickly as possible. We assume that there is no loss of time when tacking. We first guess an optimal strategy and then we establish its optimality by using the dynamic programming principle. The Hamilton Jacobi Bellmann equation obtained is a parabolic PDE with Neumann boundary conditions. Since it does not admit a closed form solution, the proof of optimality involves an intricate estimate of derivatives of the value function. We explicitly provide the asymptotic shape of the value function. In order to do so, we prove a result on large time behavior for solutions to time dependent parabolic PDE using a coupling argument. In particular, a boat far from the island approaches the island at

\frac{1}{2} + \frac{\sqrt{2}}{\pi} = 95.02\%

of the boat's speed.

EPFL2017

Interfaces of the one-dimensional voter models

Samir Brahim Belhaouari

We show that for a voter model on {0,1}Z corresponding to a random walk with kernel p(·) and starting from unanimity to the right and opposing unanimity to the left, a tight interface between 0's and 1's exists if p(·) has second moments but does not if p(·) fails to have α moment for some α < 2. We study the evolution of the interface for the one-dimensional voter model. We show that if the random walk kernel associated with the voter model has finite γth moment for some γ > 3, then the evolution of the interface boundaries converges weakly to a Brownian motion. This extends recent work of Newman, Ravishankar and Sun. Our result is optimal in the sense that a finite γth moment is necessary for this convergence for all γ ∈ (0, 3). We also obtain relatively sharp estimates for the tail distribution of the size of the equilibrium interface, extending earlier results of Cox and Durrett.

EPFL2006

The sources of sovereign risk: a calibration based on Levy stochastic processes

Sylvain Jean Pascal Carré, Daniel Cohen

Governments choose to issue risky or riskless debt depending on the nature of the stochastic process of output. We use Brownian motion and Poisson shocks a modeling method in the literature on corporate default known as Levy processes to approximate a decomposition of the output process into a smooth and a jump component. Using an Eaton and Gersovitz (1981) model of debt repudiation, we show that the Brownian part explains the counter-cyclical behavior of the current account, and the Poisson part explains the risk of default thus enabling our model to account for key stylized facts regarding sovereign risk. (C) 2019 Elsevier B.V. All rights reserved.

ELSEVIER SCIENCE BV2019

Optimal solution and asymptotic properties of a stochastic control problem arising in sailboat trajectory optimization

Carlo Ciccarella

\frac{1}{2} + \frac{\sqrt{2}}{\pi} = 95.02\%

of the boat's speed.

EPFL2017