Comptes Rendus
no. 3
Probability theory/Optimal control
Analysis and computation of probability density functions for a 1-D impulsively controlled diffusion process
[Analyse et calcul des fonctions de densité de probabilité pour un processus de diffusion en dimension 1 contrôlé par impulsion]

Présenté par : the Editorial Board

Yuta Yaegashi1 ; Hidekazu Yoshioka2 ; Kentaro Tsugihashi2 ; Masayuki Fujihara1
1 Graduate School of Agriculture, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto City, Kyoto Prefecture, 606-8502, Japan
2 Graduate School of Natural Science and Technology, Shimane University, Nishikawatsu-cho 1060, Matsue City, Shimane Prefecture, 690-8504, Japan
Comptes Rendus. Mathématique, Volume 357 (2019) no. 3, pp. 306-315.

This paper proposes appropriate boundary conditions to be equipped with Kolmogorov's Forward Equation that governs a stationary probability density function for a 1-D impulsively controlled diffusion process and derives an exact probability density function. The boundary conditions are verified numerically with a Monte Carlo approach. A finite-volume method for solving the equation is also presented and its accuracy is investigated through numerical experiments.

Cette Note propose des conditions aux limites appropriées pour l'équation de Kolmogorov antérograde gouvernant une fonction de densité de probabilité stationnaire d'un processus de diffusion contrôlé par impulsion, en dimension 1. Nous obtenons une fonction de densité de probabilité exacte. La condition aux limites est vérifiée numériquement pour l'approche de Monte Carlo. Nous présentons également une méthode de volumes finis pour résoudre l'équation et nous étudions sa précision au moyen de simulations numériques.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2019.02.007

Yuta Yaegashi 1 ; Hidekazu Yoshioka 2 ; Kentaro Tsugihashi 2 ; Masayuki Fujihara 1

1 Graduate School of Agriculture, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto City, Kyoto Prefecture, 606-8502, Japan
2 Graduate School of Natural Science and Technology, Shimane University, Nishikawatsu-cho 1060, Matsue City, Shimane Prefecture, 690-8504, Japan 
@article{CRMATH_2019__357_3_306_0,
     author = {Yuta Yaegashi and Hidekazu Yoshioka and Kentaro Tsugihashi and Masayuki Fujihara},
     title = {Analysis and computation of probability density functions for a {1-D} impulsively controlled diffusion process},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {306--315},
     publisher = {Elsevier},
     volume = {357},
     number = {3},
     year = {2019},
     doi = {10.1016/j.crma.2019.02.007},
     language = {en},
}
TY  - JOUR
AU  - Yuta Yaegashi
AU  - Hidekazu Yoshioka
AU  - Kentaro Tsugihashi
AU  - Masayuki Fujihara
TI  - Analysis and computation of probability density functions for a 1-D impulsively controlled diffusion process
JO  - Comptes Rendus. Mathématique
PY  - 2019
SP  - 306
EP  - 315
VL  - 357
IS  - 3
PB  - Elsevier
DO  - 10.1016/j.crma.2019.02.007
LA  - en
ID  - CRMATH_2019__357_3_306_0
ER  - 
%0 Journal Article
%A Yuta Yaegashi
%A Hidekazu Yoshioka
%A Kentaro Tsugihashi
%A Masayuki Fujihara
%T Analysis and computation of probability density functions for a 1-D impulsively controlled diffusion process
%J Comptes Rendus. Mathématique
%D 2019
%P 306-315
%V 357
%N 3
%I Elsevier
%R 10.1016/j.crma.2019.02.007
%G en
%F CRMATH_2019__357_3_306_0
Yuta Yaegashi; Hidekazu Yoshioka; Kentaro Tsugihashi; Masayuki Fujihara. Analysis and computation of probability density functions for a 1-D impulsively controlled diffusion process. Comptes Rendus. Mathématique, Volume 357 (2019) no. 3, pp. 306-315. doi : 10.1016/j.crma.2019.02.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.02.007/

[1] S. Baccarin Optimal impulse control for a multidimensional cash management system with generalized cost functions, Eur. J. Oper. Res., Volume 196 (2009) no. 1, pp. 198-206

[2] A. Bensoussan; H. Long; S. Perera; S. Sethi Impulse control with random reaction periods: a central bank intervention problem, Oper. Res. Lett., Volume 40 (2012) no. 6, pp. 425-430

[3] G.E. Box; M.E. Muller A note on the generation of random normal deviates, Ann. Math. Stat., Volume 29 (1958) no. 2, pp. 610-611

[4] A. Cadenillas Optimal central bank intervention in the foreign exchange market, J. Econ. Theory, Volume 87 (1999), pp. 218-242

[5] Z. Chen; P.A. Forsyth A numerical scheme for the impulse control formulation for pricing variable annuities with a guaranteed minimum withdrawal benefit (GMWB), Numer. Math., Volume 109 (2008) no. 4, pp. 535-569

[6] S. Federico; P. Gassiat Viscosity characterization of the value function of an investment-consumption problem in presence of an illiquid asset, J. Optim. Theory Appl., Volume 160 (2014) no. 3, pp. 966-991

[7] M. Grigoriu Stochastic Calculus: Applications in Science and Engineering, Springer Science & Business Media, 2013

[8] P.E. Kloeden; E. Platen; H. Schurz Numerical Solution of SDE Through Computer Experiments, Springer Science & Business Media, 2012

[9] R. Korn Portfolio optimisation with strictly positive transaction costs and impulse control, Finance Stoch., Volume 2 (1998) no. 2, pp. 85-114

[10] P.G. LeFloch; J.-M. Mercier A new method for solving Kolmogorov equations in mathematical finance, C. R. Acad. Sci. Paris, Ser. I, Volume 355 (2017) no. 6, pp. 680-686

[11] R.J. LeVeque Finite Volume Methods for Hyperbolic Problems, vol. 31, Cambridge University Press, 2002

[12] F. Marpeau; A. Barua; K. Josić A finite volume method for stochastic integrate-and-fire models, J. Comput. Neurosci., Volume 26 (2009) no. 3, pp. 445-457

[13] M. Matsumoto; T. Nishimura Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator, ACM Trans. Model. Comput. Simul., Volume 8 (1998) no. 1, pp. 3-30

[14] J.J. Miller; S. Wang An exponentially fitted finite volume method for the numerical solution of 2D unsteady incompressible flow problems, J. Comput. Phys., Volume 115 (1994) no. 1, pp. 56-64

[15] M. Ohnishi; M. Tsujimura An impulse control of a geometric Brownian motion with quadratic costs, Eur. J. Oper. Res., Volume 168 (2006) no. 2, pp. 311-321

[16] B. Øksendal; A. Sulem Optimal consumption and portfolio with both fixed and proportional transaction costs, SIAM J. Control Optim., Volume 40 (2002) no. 6, pp. 1765-1790

[17] B. Øksendal Stochastic Differential Equations, Springer, 2003

[18] B. Øksendal; A. Sulem Applied Stochastic Control of Jump Diffusions, vol. 498, Springer, 2005

[19] H. Risken The Fokker–Planck Equation, Springer, 1996

[20] G.D. Smith Numerical Solution of Partial Differential Equations: Finite Difference Methods, Oxford University Press, 1985

[21] E.F. Toro Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, Springer Science & Business Media, 2013

[22] M. Tsujimura; A. Maeda Stochastic Control: Theory and Applications, Asakura Shoten, 2016 (in Japanese)

[23] S.P. Van der Pijl; C.W. Oosterlee An ENO-based method for second order equations and application to the control of dike levels, J. Sci. Comput., Volume 50 (2012) no. 2, pp. 462-492

[24] Y. Yaegashi; H. Yoshioka; K. Unami; M. Fujihara Impulse and singular stochastic control approaches for population management of a fish-eating bird (P. Daniele; L. Scrimali, eds.), New Trends in Emerging Complex Real Life Problems, AIRO Springer Series, vol. 1, Springer Nature Switzerland AG, 2018, pp. 493-500 | DOI

[25] H. Yoshioka; K. Unami A cell-vertex finite volume scheme for solute transport equations in open channel networks, Probab. Eng. Mech., Volume 31 (2013), pp. 30-38

  • Hidekazu Yoshioka; Motoh Tsujimura; Kunihiko Hamagami; Yuta Yaegashi; Yumi Yoshioka HJB and Fokker-Planck equations for river environmental management based on stochastic impulse control with discrete and random observation, Computers Mathematics with Applications, Volume 96 (2021), p. 131 | DOI:10.1016/j.camwa.2021.05.015
  • Hidekazu Yoshioka; Yuta Yaegashi Mathematical and numerical analyses of a stochastic impulse control model with imperfect interventions, Journal of Mathematics in Industry, Volume 11 (2021) no. 1 | DOI:10.1186/s13362-021-00112-9
  • Hidekazu Yoshioka; Yuta Yaegashi Stochastic impulse control of nonsmooth dynamics with partial observation and execution delay: Application to an environmental restoration problem, Optimal Control Applications and Methods, Volume 42 (2021) no. 5, p. 1226 | DOI:10.1002/oca.2723
  • Charles Bertucci Fokker-Planck equations of jumping particles and mean field games of impulse control, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, Volume 37 (2020) no. 5, p. 1211 | DOI:10.1016/j.anihpc.2020.04.006
  • Yuta Yaegashi; Hidekazu Yoshioka; Motoh Tsujimura; Masayuki Fujihara Finite volume computation for the non-stationary probability density function of an impulsively controlled 1-D diffusion process, Journal of Advanced Simulation in Science and Engineering, Volume 7 (2020) no. 2, p. 262 | DOI:10.15748/jasse.7.262
  • Hidekazu Yoshioka; Yuta Yaegashi; Motoh Tsujimura; Masayuki Fujihara Non-Local Fokker-Planck Equation of Imperfect Impulsive Interventions and its Effectively Super-Convergent Numerical Discretization, Methods and Applications for Modeling and Simulation of Complex Systems, Volume 1094 (2019), p. 79 | DOI:10.1007/978-981-15-1078-6_7

Cité par 6 documents. Sources : Crossref

Commentaires - Politique