Fast approximation of solutions of sde’s with oblique reflection on an orthant

• Autorzy: Czarkowski K. T.
• Języki publikacji: EN

Abstrakty

EN

We consider the discrete “fast” penalization scheme for SDE’s driven by general semimartingale on orthant R^d₊ with oblique reflection.

Słowa kluczowe

EN

Skorokhod problem; reflected stochastic differential equations; discrete approximation methods

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2010
• Tom: Vol. 30, Fasc. 1
• Strony: 167--182
• Opis fizyczny: Bibliogr. 31 poz.

Twórcy

autor

Czarkowski K. T.

• Faculty of Mathematics and Computer Science, Nicholas Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland

Bibliografia

• [1] H. Chen and A. Mandelbaum, Stochastic discrete flow newtworks: diffusion approximations and bottlenecks, Ann. Probab. 19 (1991), pp. 1463-1519.
• [2] R. J. Chitashvili and N. L. Lazrieva, Strong solutions of stochastic differential equations with boundary conditions, Stochastics 5 (1981), pp. 225-309.
• [3] K. Czarkowski and L. Słomiński, Approximation of solutions of SDE’s with oblique reflection on an orthant, Probab. Math. Statist. 22 (2002), pp. 29-49.
• [4] C. Dellacherie and P. A. Meyer, Probabilités et potentiel, Hermann, Paris 1980.
• [5] P. Dupuis and H. Ishi, On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications, Stochastics Stochastics Rep. 35 (1991), pp. 31-62.
• [6] P. Dupuis and H. Ishi, SDEs with oblique reflection on nonsmooth domains, Ann. Probab. 21 (1993), pp. 554-580.
• [7] P. Dupuis and K. Ramanan, Convex duality and the Skorokhod problem. I, Probab. Theory Related Fields 115 (1999), pp. 153-197.
• [8] P. Dupuis and K. Ramanan, Convex duality and the Skorokhod problem. II, Probab. Theory Related Fields 115 (1999), pp. 197-237.
• [9] J. M. Harrison and M. I. Reiman, Reflected Brownian motion on an orthant, Ann. Probab. 9 (1981), pp. 302-308.
• [10] J. M. Harrison and R. J. Williams, A multiclass closed queueing network with unconventional heavy traffic behavior, Ann. Appl. Probab. 6 (1996), pp. 1-47.
• [11] A. Jakubowski, A non-Skorohod topology on the Skorohod space, EJP 2 (1997), pp. 1-21.
• [12] A. Jakubowski, J. Mémin and G. Pages, Convergence en loi des suites d’intégrales stochastiques sur l’espace D1 de Skorokhod, Probab. Theory Related Fields 81 (1989), pp. 111-137.
• [13] O. Kella, Stability and nonproduct form of stochastic fluid networks with Lévy inputs, Ann. Appl. Probab. 6 (1996), pp. 186-199.
• [14] T. G. Kurtz and P. Protter, Weak limit theorems for stochastic integrals and stochastic differential equations, Ann. Probab. 19 (1991), pp. 1035-1070.
• [15] T. G. Kurtz and P. Protter, Wong-Zakai Corrections, Random Evolutions, and Simulation Schemes for SDE’s, Proc. Conference in Honor Moshe Zakai 65th Birthday, Haifa, Stochastic Analysis (1991), pp. 331-346.
• [16] D. Lépingle, Un schéma d’Euler pour équations différentielles stochastiques réfléchies, C. R. Acad. Sci. Paris 316 (1993), pp. 601-605.
• [17] Y. Liu, Numerical approches to stochastic differential equations with boundary conditions, Thesis, Purdue University, 1993.
• [18] J. Mémin and L. Słomiński, Condition UT et stabilité en loi des solutions d’équations différentielles stochastiques, Sém. de Probab. XXV, Lecture Notes in Math. No 1485, Springer, Berlin 1991, pp. 162-177.
• [19] M. Métivier and J. Pellaumail, Une formule de majoration pour martingales, C. R. Acad. Sci. Paris, Sér. A 285 (1977), pp. 685-688.
• [20] R. Pettersson, Approximations for stochastic differential equation with reflecting convex boundaries, Stochastic Process. Appl. 59 (1995), pp. 295-308.
• [21] R. Pettersson, Penalization schemes for reflecting stochastic differential equations, Bernoulli 3 (4) (1997), pp. 403-414.
• [22] P. Protter, Stochastic Integration and Differential Equations, Springer, Berlin 1990.
• [23] M. A. Shashiashvili, On the variation of the difference of singular components in the Skorokhod problem and on stochastic differential systems in a half-space, Stochastics 24 (1988), pp. 151-169.
• [24] L. Słomiński, Stability of strong solutions of stochastic differential equations, Stochastic Process. Appl. 31 (1989), pp. 173-202.
• [25] L. Słomiński, On approximation of solutions of multidimensional SDE’s with reflecting boundary conditions, Stochastic Process. Appl. 50 (1994), pp. 197-219.
• [26] L. Słomiński, Stability of stochastic differential equations driven by general semimartingales, Dissertationes Math. 349 (1996), pp. 1-113.
• [27] L. Słomiński, Euler’s approximations of solutions of SDE’s with reflecting boundary, Stochastic Process. Appl. 94 (2001), pp. 317-337.
• [28] C. Stricker, Loi de semimartingales et critères de compacité, Sém. de Probab. XIX, Lecture Notes in Math. No 1123, Springer, Berlin 1985.
• [29] R. J. Williams, Semimartingale reflecting Brownian motions in the orthant, in: Stochastic Networks, F. P. Kelly and R. J. Williams (Eds.), Springer, 1995.
• [30] R. J. Williams and S. R. S. Varadhan, Brownian motion in a wedge with oblique reflection, Comm. Pure Appl. Math. 12 (1985), pp. 147-225.
• [31] K. Yamada, Diffusion approximation for open state-dependent queueing networks in the heavy traffic situation, Ann. Appl. Probab. 5 (1995), pp. 958-982.