scholarly journals The ruin problem for a Wiener process with state-dependent jumps

2020 ◽  
Vol 16 (1) ◽  
pp. 13-23
Author(s):  
M. Lefebvre
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Diffusion Process ◽  
Wiener Process ◽  
Jump Diffusion ◽  
Moment Generating Function ◽  
Continuous Part ◽  
State Dependent ◽  
The Moment ◽  
First Time

AbstractLet X(t) be a jump-diffusion process whose continuous part is a Wiener process, and let T (x) be the first time it leaves the interval (0,b), where x = X(0). The jumps are negative and their sizes depend on the value of X(t). Moreover there can be a jump from X(t) to 0. We transform the integro-differential equation satisfied by the probability p(x) := P[X(T (x)) = 0] into an ordinary differential equation and we solve this equation explicitly in particular cases. We are also interested in the moment-generating function of T (x).

scholarly journals Uniform approximation of the Cox-Ingersoll-Ross process

2015 ◽  
Vol 47 (4) ◽  
pp. 1132-1156 ◽  
Author(s):  
Grigori N. Milstein ◽  
John Schoenmakers
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Wiener Process ◽  
Point Of View ◽  
First Passage ◽  
Cir Process ◽  
Quality Of Approximation ◽  
Time Grid ◽  
Passage Times

The Doss-Sussmann (DS) approach is used for uniform simulation of the Cox-Ingersoll-Ross (CIR) process. The DS formalism allows us to express trajectories of the CIR process through solutions of some ordinary differential equation (ODE) depending on realizations of a Wiener process involved. By simulating the first-passage times of the increments of the Wiener process to the boundary of an interval and solving the ODE, we uniformly approximate the trajectories of the CIR process. In this respect special attention is payed to simulation of trajectories near 0. From a conceptual point of view the proposed method gives a better quality of approximation (from a pathwise point of view) than standard, even exact, simulation of the stochastic differential equation at some deterministic time grid.

scholarly journals Joint distribution of a Lévy process and its running supremum

2018 ◽  
Vol 55 (2) ◽  
pp. 488-512 ◽  
Author(s):  
Laure Coutin ◽  
Monique Pontier ◽  
Waly Ngom
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Weak Solution ◽  
Diffusion Process ◽  
Lebesgue Measure ◽  
Joint Distribution ◽  
Random Variable ◽  
Jump Diffusion ◽  
Marginal Density ◽  
The Law

Abstract Let X be a jump-diffusion process and X* its running supremum. In this paper we first show that for any t > 0, the law of the pair (X*t, Xt) has a density with respect to the Lebesgue measure. This allows us to show that for any t > 0, the law of the pair formed by the random variable Xt and the running supremum X*t of X at time t can be characterized as a weak solution of a partial differential equation concerning the distribution of the pair (X*t, Xt). Then we obtain an expression of the marginal density of X*t for all t > 0.

On the Dynamics of the Dynabee

1999 ◽  
Vol 67 (2) ◽  
pp. 321-325 ◽  
Author(s):  
D. W. Gulick ◽  
O. M. O’Reilly
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Successful Operation ◽  
Additional Work ◽  
The Moment

The Dynabee is a gyroscopic device that is marketed as a wrist exerciser. In this paper, a model for the dynamics of this device is presented. With some additional work, we find that the dynamics are governed by a single ordinary differential equation. The solution of this equation also provides the moment required to operate the device. Specifically, we find that this moment is proportional to the square of the rotor’s spin rate. We also show why it is necessary to give the device a large initial spin rate for its successful operation. [S0021-8936(00)02602-7]

Commande optimale du processus de wiener

1989 ◽  
Vol 26 (1) ◽  
pp. 50-57 ◽  
Author(s):  
Mario Lefebvre
Keyword(s):  
Optimal Control ◽  
Wiener Process ◽  
Mathematical Expectation ◽  
Optimal Value ◽  
The Moment ◽  
First Time

The problem of the optimal control of the Wiener process in Rn is considered. The optimal value of the control is obtained from the mathematical expectation of a quantity defined in terms of the moment and the place where the uncontrolled process hits the boundary of the continuation region for the first time. Explicit results are presented.

scholarly journals On the first hitting place of the integrated Wiener process

1989 ◽  
Vol 21 (4) ◽  
pp. 945-948 ◽  
Author(s):  
Mario Lefebvre ◽  
Éric Léonard
Keyword(s):  
Wiener Process ◽  
Generating Function ◽  
Moment Generating Function ◽  
One Dimensional ◽  
The Moment

Let dx(t) = y(t) dt, where y(t) is a one-dimensional Wiener process. In this note, we obtain a formula for the moment-generating function of y(T), where T is the 1/2-winding time about the origin of the integrated Wiener process x(t).

scholarly journals First Passage Problems for Asymmetric Wiener Processes

2006 ◽  
Vol 43 (1) ◽  
pp. 175-184 ◽  
Author(s):  
Mario Lefebvre
Keyword(s):  
Wiener Process ◽  
Generating Function ◽  
First Passage Time ◽  
Moment Generating Function ◽  
Passage Time ◽  
Expected Value ◽  
First Passage ◽  
One Dimensional ◽  
Wiener Processes ◽  
The Moment

The problem of computing the moment generating function of the first passage time T to a > 0 or −b < 0 for a one-dimensional Wiener process {X(t), t ≥ 0} is generalized by assuming that the infinitesimal parameters of the process may depend on the sign of X(t). The probability that the process is absorbed at a is also computed explicitly, as is the expected value of T.

scholarly journals Monte Carlo Methods and New Jump Diffusion Processes and Their Application in Gold Price

2017 ◽  
Author(s):  
Kutluk Kağan Sümer
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Standard Deviation ◽  
Diffusion Process ◽  
Wiener Process ◽  
Mean Reversion ◽  
Jump Diffusion ◽  
Gold Market ◽  
Jump Diffusion Process ◽  
Standard Deviations

This study aimed to execute Monte Carlo simulation method with Wiener Process, Generalized Wiener Process, Mean Reversion Process and Mean Reversion Jump Diffusion Process and to compare them and then expended with the idea of how to include negative and positive news shocks in the gold market to the Monte Carlo simulation. By enhancing the determination of the 3 standard deviation shocks within the process of Classic Mean Jump Diffusion Process, an enchanted model for the 1,96 and 3 standard deviation shocks were being used and additionally positive and negative shocks were added to the system in a different way. This new Mean Reversion Jump Diffusion Process that have been developed by Sümer, executes Monte Carlo simulation regarding the gold market return with five random variables that are chosen from Poisson distribution and one random variable chosen from the normal distribution. Additionally, by accepting volatilities as outlies over the 1,96 and 3 standard deviations with the effect of the new and good news and the standard deviations on the traditional approximate return and the standard deviations (volatility) and the obtained new approximate return and the new standard deviation (volatility) and compares them with the Monte Carlo simulations.

scholarly journals The distribution of refracted Lévy processes with jumps having rational Laplace transforms

2017 ◽  
Vol 54 (4) ◽  
pp. 1167-1192
Author(s):  
Jiang Zhou ◽  
Lan Wu
Keyword(s):  
Laplace Transform ◽  
Diffusion Process ◽  
Jump Diffusion ◽  
Laplace Transforms ◽  
Variable Annuities ◽  
State Dependent ◽  
Approximating Procedure ◽  
Jump Diffusion Process ◽  
The Laplace Transform ◽  
Interesting Approach

Abstract We consider a refracted jump diffusion process having two-sided jumps with rational Laplace transforms. For such a process, by applying a straightforward but interesting approach, we derive formulae for the Laplace transform of its distribution. Our formulae are presented in an attractive form and the approach is novel. In particular, the idea in the application of an approximating procedure is remarkable. In addition, the results are used to price variable annuities with state-dependent fees.

scholarly journals Uniform approximation of the Cox-Ingersoll-Ross process

2015 ◽  
Vol 47 (04) ◽  
pp. 1132-1156 ◽  
Author(s):  
Grigori N. Milstein ◽  
John Schoenmakers
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Wiener Process ◽  
Point Of View ◽  
First Passage ◽  
Cir Process ◽  
Quality Of Approximation ◽  
Time Grid ◽  
Passage Times

The Doss-Sussmann (DS) approach is used for uniform simulation of the Cox-Ingersoll-Ross (CIR) process. The DS formalism allows us to express trajectories of the CIR process through solutions of some ordinary differential equation (ODE) depending on realizations of a Wiener process involved. By simulating the first-passage times of the increments of the Wiener process to the boundary of an interval and solving the ODE, we uniformly approximate the trajectories of the CIR process. In this respect special attention is payed to simulation of trajectories near 0. From a conceptual point of view the proposed method gives a better quality of approximation (from a pathwise point of view) than standard, even exact, simulation of the stochastic differential equation at some deterministic time grid.

scholarly journals First Passage Problems for Asymmetric Wiener Processes

2006 ◽  
Vol 43 (01) ◽  
pp. 175-184
Author(s):  
Mario Lefebvre
Keyword(s):  
Wiener Process ◽  
Generating Function ◽  
First Passage Time ◽  
Moment Generating Function ◽  
Passage Time ◽  
Expected Value ◽  
First Passage ◽  
One Dimensional ◽  
Wiener Processes ◽  
The Moment

The problem of computing the moment generating function of the first passage time T to a &gt; 0 or −b &lt; 0 for a one-dimensional Wiener process {X(t), t ≥ 0} is generalized by assuming that the infinitesimal parameters of the process may depend on the sign of X(t). The probability that the process is absorbed at a is also computed explicitly, as is the expected value of T.

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