scholarly journals Weak convergence of a numerical scheme for stochastic differential equations

2018 ◽  
Vol 37 (1) ◽  
pp. 201-215
Author(s):  
Esteban Aguilera ◽  
Raúl Fierro
Keyword(s):  
Differential Equation ◽  
Weak Convergence ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
State Space ◽  
Numerical Scheme ◽  
Step Size ◽  
Finite State ◽  
Convergence Of The Scheme ◽  
Finite State Space

WEAK CONVERGENCE OF A NUMERICAL SCHEME FOR STOCHASTIC DIFFERENTIAL EQUATIONSIn this paper a numerical scheme approximating the solution to a stochastic differential equation is presented. On bounded subsets of time, this scheme has a finite state space, which allows us to decrease the round-off error when the algorithm is implemented. At the same time, the scheme introduced turns out locally consistent for any step size of time. Weak convergence of the scheme to the solution of the stochastic differential equation is shown.

scholarly journals Conditioning an additive functional of a Markov chain to stay nonnegative. I. Survival for a long time

2005 ◽  
Vol 37 (4) ◽  
pp. 1015-1034 ◽  
Author(s):  
Saul D. Jacka ◽  
Zorana Lazic ◽  
Jon Warren
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
State Space ◽  
Continuous Time ◽  
Additive Functional ◽  
Finite State ◽  
Long Time ◽  
Negative Drift ◽  
Irreducible Markov Chain ◽  
Finite State Space

Let (Xt)t≥0 be a continuous-time irreducible Markov chain on a finite state space E, let v be a map v: E→ℝ\{0}, and let (φt)t≥0 be an additive functional defined by φt=∫0tv(Xs)d s. We consider the case in which the process (φt)t≥0 is oscillating and that in which (φt)t≥0 has a negative drift. In each of these cases, we condition the process (Xt,φt)t≥0 on the event that (φt)t≥0 is nonnegative until time T and prove weak convergence of the conditioned process as T→∞.

scholarly journals Mean reflected stochastic differential equations with jumps

2020 ◽  
Vol 52 (2) ◽  
pp. 523-562
Author(s):  
Phillippe Briand ◽  
Abir Ghannoum ◽  
Céline Labart
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Numerical Scheme ◽  
The Law

AbstractIn this paper, a reflected stochastic differential equation (SDE) with jumps is studied for the case where the constraint acts on the law of the solution rather than on its paths. These reflected SDEs have been approximated by Briand et al. (2016) using a numerical scheme based on particles systems, when no jumps occur. The main contribution of this paper is to prove the existence and the uniqueness of the solutions to this kind of reflected SDE with jumps and to generalize the results obtained by Briand et al. (2016) to this context.

Extreme values, range and weak convergence of integrals of Markov chains

1982 ◽  
Vol 19 (02) ◽  
pp. 272-288 ◽  
Author(s):  
P. J. Brockwell ◽  
S. I. Resnick ◽  
N. Pacheco-Santiago
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Markov Chains ◽  
Asymptotic Behaviour ◽  
State Space ◽  
Extreme Values ◽  
Finite State ◽  
Integral Process ◽  
Maximum Minimum ◽  
Finite State Space

A study is made of the maximum, minimum and range on [0,t] of the integral processwhereSis a finite state-space Markov chain. Approximate results are derived by establishing weak convergence of a sequence of such processes to a Wiener process. For a particular family of two-state stationary Markov chains we show that the corresponding centered integral processes exhibit the Hurst phenomenon to a remarkable degree in their pre-asymptotic behaviour.

Doubly stochastic Poisson processes and process control

1972 ◽  
Vol 4 (02) ◽  
pp. 318-338 ◽  
Author(s):  
Mats Rudemo
Keyword(s):  
Markov Chain ◽  
Differential Equations ◽  
State Space ◽  
Point Process ◽  
Doubly Stochastic ◽  
Long Run ◽  
Finite State ◽  
Control State ◽  
Special Case ◽  
Finite State Space

Consider a Poisson point process with an intensity parameter forming a Markov chain with continuous time and finite state space. A system of ordinary differential equations is derived for the conditional distribution of the Markov chain given observations of the point process. An estimate of the current intensity, optimal in the least-squares sense, is computed from this distribution. Applications to reliability and replacement theory are given. A special case with two states, corresponding to a process in control and out of control, is discussed at length. Adjustment rules, based on the conditional probability of the out of control state, are studied. Regarded as a function of time, this probability forms a Markov process with the unit interval as state space. For the distribution of this process, integro-differential equations are derived. They are used to compute the average long run cost of adjustment rules.

Extreme values, range and weak convergence of integrals of Markov chains

1982 ◽  
Vol 19 (2) ◽  
pp. 272-288 ◽  
Author(s):  
P. J. Brockwell ◽  
S. I. Resnick ◽  
N. Pacheco-Santiago
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Markov Chains ◽  
Asymptotic Behaviour ◽  
State Space ◽  
Extreme Values ◽  
Finite State ◽  
Integral Process ◽  
Maximum Minimum ◽  
Finite State Space

A study is made of the maximum, minimum and range on [0, t] of the integral process where S is a finite state-space Markov chain. Approximate results are derived by establishing weak convergence of a sequence of such processes to a Wiener process. For a particular family of two-state stationary Markov chains we show that the corresponding centered integral processes exhibit the Hurst phenomenon to a remarkable degree in their pre-asymptotic behaviour.

scholarly journals Conditioning an additive functional of a Markov chain to stay nonnegative. I. Survival for a long time

2005 ◽  
Vol 37 (04) ◽  
pp. 1015-1034 ◽  
Author(s):  
Saul D. Jacka ◽  
Zorana Lazic ◽  
Jon Warren
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
State Space ◽  
Continuous Time ◽  
Additive Functional ◽  
Finite State ◽  
Long Time ◽  
Negative Drift ◽  
Irreducible Markov Chain ◽  
Finite State Space

Let (X t ) t≥0 be a continuous-time irreducible Markov chain on a finite state space E, let v be a map v: E→ℝ\{0}, and let (φ t ) t≥0 be an additive functional defined by φ t =∫0 t v(X s )d s. We consider the case in which the process (φ t ) t≥0 is oscillating and that in which (φ t ) t≥0 has a negative drift. In each of these cases, we condition the process (X t ,φ t ) t≥0 on the event that (φ t ) t≥0 is nonnegative until time T and prove weak convergence of the conditioned process as T→∞.

scholarly journals Weak Convergence Analysis and Improved Error Estimates for Decoupled Forward-Backward Stochastic Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 848
Author(s):  
Wei Zhang ◽  
Hui Min
Keyword(s):  
Differential Equation ◽  
Weak Convergence ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Convergence Analysis ◽  
Error Estimates ◽  
Taylor Expansion ◽  
Backward Stochastic Differential Equations ◽  
Error Terms

In this paper, we mainly investigate the weak convergence analysis about the error terms which are determined by the discretization for solving the stochastic differential equation (SDE, for short) in forward-backward stochastic differential equations (FBSDEs, for short), which is on the basis of Itô Taylor expansion, the numerical SDE theory, and numerical FBSDEs theory. Under the weak convergence analysis of FBSDEs, we further establish better error estimates of recent numerical schemes for solving FBSDEs.

Doubly stochastic Poisson processes and process control

1972 ◽  
Vol 4 (2) ◽  
pp. 318-338 ◽  
Author(s):  
Mats Rudemo
Keyword(s):  
Markov Chain ◽  
Differential Equations ◽  
State Space ◽  
Point Process ◽  
Doubly Stochastic ◽  
Long Run ◽  
Finite State ◽  
Control State ◽  
Special Case ◽  
Finite State Space

Consider a Poisson point process with an intensity parameter forming a Markov chain with continuous time and finite state space. A system of ordinary differential equations is derived for the conditional distribution of the Markov chain given observations of the point process. An estimate of the current intensity, optimal in the least-squares sense, is computed from this distribution. Applications to reliability and replacement theory are given. A special case with two states, corresponding to a process in control and out of control, is discussed at length. Adjustment rules, based on the conditional probability of the out of control state, are studied. Regarded as a function of time, this probability forms a Markov process with the unit interval as state space. For the distribution of this process, integro-differential equations are derived. They are used to compute the average long run cost of adjustment rules.

Sign in / Sign up

Export Citation Format

Share Document