A Monte Carlo Method for Estimating Sensitivities of Reflected Diffusions in Convex Polyhedral Domains

Published Online:https://doi.org/10.1287/stsy.2019.0031

References

  • Asmussen S, Glynn PW (2007) Stochastic Simulation: Algorithms and Analysis (Springer, New York).Google Scholar
  • Banner AD, Fernholz R, Karatzas I (2005) Atlas models of equity markets. Ann. Appl. Probab. 15(4):2296–2330.Google Scholar
  • Blanchet J, Murthy KRA (2018) Exact simulation of multidimensional reflected Brownian motion. J. Appl. Probab. 55(1):137–156.Google Scholar
  • Bossy M, Gobet E, Talay D (2004) Symmetrized Euler scheme for an efficient approximation of reflected diffusions. J. Appl. Probab. 4(3):877–889.Google Scholar
  • Chen H, Mandelbaum A (1991) Leontief systems, RBVs and RBMs. Davis MHA, Elliott RJ, eds. Applied Stochastic Analysis, Stochastics Monographs, vol. 5 (Gordon and Breach, New York), 1–43.Google Scholar
  • Costantini C, Pacciarotti B, Sartoretto F (1998) Numerical approximation for functionals of reflecting diffusion processes. SIAM J. Appl. Math. 58(1):73–102.Google Scholar
  • Dupuis P, Ramanan K (1999) Convex duality and the Skorokhod problem. I. Probab. Theory Related Fields 115(2):153–195.Google Scholar
  • Fernholz ER (2002) Stochastic Portfolio Theory (Springer, New York).Google Scholar
  • Glasserman P (2003) Monte Carlo Methods in Financial Engineering (Springer, New York).Google Scholar
  • Gobet E (2001) Efficient schemes for the weak approximation of reflected diffusions. Monte Carlo Methods Appl. 7(1–2):193–202.Google Scholar
  • Harrison JM (2013) Brownian Models of Performance and Control (Cambridge University Press, Cambridge, UK).Google Scholar
  • Ichiba T, Karatzas I (2010) On collisions of Brownian particles. Ann. Appl. Probab. 20(3):951–977.Google Scholar
  • Ichiba T, Karatzas I, Shkolnikov M (2013) Strong solutions of stochastic equations with rank-based coefficients. Probab. Theory Related Fields 156(1–2):229–248.Google Scholar
  • Ichiba T, Papathanakos V, Banner A, Karatzas I, Fernholz R (2011) Hybrid atlas models. Ann. Appl. Probab. 21(2):609–644.Google Scholar
  • Kang W, Williams RJ (2007) An invariance principle for semimartingale reflecting Brownian motions in domains with piecewise smooth boundaries. Ann. Appl. Probab. 17(2):741–779.Google Scholar
  • Lépingle D (1995) Euler scheme for reflected stochastic differential equations. Math. Comput. Simulation 38(1–3):119–126.Google Scholar
  • Lipshutz D, Ramanan K (2018) On directional derivatives of Skorokhod maps in convex polyhedral domains. Ann. Appl. Probab. 28(2):688–750.Google Scholar
  • Lipshutz D, Ramanan K (2019a) Pathwise differentiability of reflected diffusions in convex polyhedral domains. Annales de l'Institut Henri Poincare Probab. Statist. Forthcoming.Google Scholar
  • Lipshutz D, Ramanan K (2019b) Sensitivity analysis of the stationary distribution of reflected Brownian motion in a convex polyhedral cone. Working paper, Technion, Haifa, Israel.Google Scholar
  • Liu Y (1993) Numerical approches to stochastic differential equations with boundary conditions. Unpublished doctoral dissertation, Purdue University, West Lafayette, IN.Google Scholar
  • Mandelbaum A, Pats G (1998) State-dependent stochastic networks. Part I. Approximations and applications with continuous diffusion limits. Ann. Appl. Probab. 8(2):569–646.Google Scholar
  • Milshtein GN (1995) The solving of boundary value problems by numerical integration of stochastic equations. Math. Comput. Simulation 38(1–3):77–85.Google Scholar
  • Peterson WP (1991) A heavy traffic limit theorem for networks of queues with multiple customer types. Math. Oper. Res. 16(1):90–118.LinkGoogle Scholar
  • Pettersson R (1995) Approximations for stochastic differential equations with reflecting convex boundaries. Stochastic Processes Appl. 59(2):295–308.Google Scholar
  • Pettersson R (1997) Penalization schemes for reflecting stochastic differential equations. Bernoulli 3(4):403–414.Google Scholar
  • Ramanan K (2006) Reflected diffusions defined via the extended Skorokhod map. Electronic J. Probab. 11(36):934–992.Google Scholar
  • Ramanan K, Reiman MI (2003) Fluid and heavy traffic diffusion limits for a generalized processor sharing model. Ann. Appl. Probab. 13(1):100–139.Google Scholar
  • Ramanan K, Reiman MI (2008) The heavy traffic limit of an unbalanced generalized processor sharing model. Ann. Appl. Probab. 18(1):22–58.Google Scholar
  • Reiman MI (1984) Open queueing networks in heavy traffic. Math. Oper. Res. 9(3):441–458.LinkGoogle Scholar
  • Reiman MI, Williams RJ (1988) A boundary property of semimartingale reflecting Brownian motions. Probab. Theory Related Fields 77(1):87–97.Google Scholar
  • Rogers LCG, Williams D (2000) Diffusions, Markov Processes and Martingales. Volume 2: Itô Calculus, 2nd ed. (Cambridge University Press, Cambridge, UK).Google Scholar
  • Słomínski L (1994) On approximation of solutions of multidimensional SDE’s with reflecting boundary conditions. Stochastic Processes Appl. 50(2):197–219.Google Scholar
  • Słomínski L (2001) Euler’s approximations of solutions of SDEs with reflecting boundary. Stochastic Processes Appl. 94(2):317–337.Google Scholar
  • Ward A, Glynn PW (2003) A diffusion approximation for a Markovian queue with reneging. Queueing Systems 43(1–2):103–128.Google Scholar
  • Whitt W (2002) An Introduction to Stochastic-Process Limits and Their Applications to Queues. Internet Supplement. Accessed April 30, 2019, http://www.columbia.edu/∼ww2040/supplement.html.Google Scholar
  • Yan L (2002) The Euler scheme with irregular coefficients. Ann. Probab. 30(3):1172–1194.Google Scholar
  • Yang J, Kushner HJ (1991) A Monte Carlo method for sensitivity analysis and parametric optimization of nonlinear stochastic systems. SIAM J. Control Optim. 29(5):1216–1249.Google Scholar
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