Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-17T04:18:17.646Z Has data issue: false hasContentIssue false
  • English
  • Français

A Method for Computing Double Band Policies for Switching between Two Diffusions

Published online by Cambridge University Press:  27 July 2009

Florin Avram  and
Fikri Karaesmen
Florin Avram
Affiliation:
Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
Fikri Karaesmen
Affiliation:
Department of Mechanical, Industrial and Manufacturing Engineering, Northeastern University, Boston, Massachusetts 02115
Get access Rights & Permissions [Opens in a new window]

Extract

We develop a method for computing the optimal double band [b, B] policy for switching between two diffusions with continuous rewards and switching costs. The two switch levels [b, B] are obtained as perturbations of the single optimal switching point a of the control problem with no switching costs. More precisely, we find that in the case of average reward problems the optimal switch levels can be obtained by intersecting two curves: (a) the function, γ(a), which represents the long-run average reward if we were to switch between the two diffusions at a and switches were free, and (b) a horizontal line whose height depends on the size of the transaction costs. Our semianalytical approach reduces, for example, the solution of a problem recently posed by Perry and Bar-Lev (1989, in Stochastic Analysis and Applications 7: 103–115) to the solution of one nonlinear equation.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 10 , Issue 4 , October 1996 , pp. 569 - 590
Copyright
Copyright © Cambridge University Press 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bather, J. (1966). A continuous time inventory model. Journal of Applied Probability 3: 538–549.CrossRefGoogle Scholar
2.Bather, J. (1968). A diffusion model for the control of a dam. Journal of Applied Probability 5: 55–71.CrossRefGoogle Scholar
3.Benes, V.E., Shepp, L.A., & Witsenhausen, H.S. (1980). Some solvable stochastic control problems. Slochastics 4: 39–83.CrossRefGoogle Scholar
4.Chernoff, H. & Petkau, A. (1978). Optimal control of Brownian motion. SIAM Journal on Applied Mathematics 34: 717–731.CrossRefGoogle Scholar
5.Chung, K.J. & Williams, R.J. (1990). Introduction to stochastic integration. Boston: Birkhäuser.CrossRefGoogle Scholar
6.Davis, M.H.A. & Norman, A.R. (1990). Portfolio selection with transaction costs. Mathematics of Operations Research 19: 676–713.CrossRefGoogle Scholar
7.Foschini, G.J. (1977). On heavy traffic diffusion analysis and dynamic routing in packet switched networks. In Chandy, M.K. & Reiser, M. (eds.), Computer performance. Amsterdam: North-Holland, pp. 419–514.Google Scholar
8.Harrison, J.M. (1978). The diffusion approximation for tandem queues in heavy traffic. Advances in Applied Probability 10: 886–905.CrossRefGoogle Scholar
9.Harrison, J.M. (1985). Brownian motion and stochastic flow systems. New York: John Wiley.Google Scholar
10.Harrison, J.M. & Reiman, M.I. (1981). Reflected Brownian motion in an orthant. Annals of Probability 9: 302–308.CrossRefGoogle Scholar
11.Harrison, J.M., Sellke, T.M., & Taylor, A.J. (1983). Impulse control of Brownian motion. Mathematics of Operations Research 8: 454–466.CrossRefGoogle Scholar
12.Harrison, J.M. & Taksar, M.I. (1983). Instantaneous control of Brownian motion. Mathematics of Operations Research 8: 439–453.CrossRefGoogle Scholar
13.Harrison, J.M. & Taylor, A.J. (1978). Optimal control of a Brownian storage system. Stochastic Processes with Applications 6: 179–194.CrossRefGoogle Scholar
14.Hopkins, W.E. & Blankenship, G.L. (1981). Perturbation analysis of a system of quasivariational inequalities for optimal stochastic scheduling. IEEE Transactions on Automatic Control AC-26: 1054–1070.CrossRefGoogle Scholar
15.Karlin, S. & Taylor, H. (1981). A second course in stochastic processes. San Diego: Academic Press.Google Scholar
16.Krichagina, E.V., Lou, S.X.C., Sethi, S.P., & Taksar, M.I. (1993). Production control in a failure-prone manufacturing system. Annals of Applied Probability 3: 421–453.CrossRefGoogle Scholar
17.Krichagina, E.V., Lou, S.X.C., & Taksar, M.I. (1994). Double band policy for stochastic manufacturing systems in heavy traffic. Mathematics of Operations Research 19: 560–596.CrossRefGoogle Scholar
18.Mandelbaum, A. (1987). Continuous multiarmed bandits. Annals of Probability 15: 1527–1556.CrossRefGoogle Scholar
19.Newell, G.F. (1979). Approximate behavior of tandem queues: Lecture Notes in Economics and Mathematical Systems, 171. New York: Springer-Verlag.CrossRefGoogle Scholar
20.Perry, D. & Bar-Lev, S.K. (1989). A control of a Brownian motion storage system with two switchover drifts. Stochastic Analysis and Applications 7: 103–115.CrossRefGoogle Scholar
21.Rath, J. (1975). Controlled queues in heavy traffic. Advances in Applied Probability 7: 656–671.CrossRefGoogle Scholar
22.Rath, J. (1977). The optimal policy for a controlled Brownian motion process. SIAM Journal on Applied Mathematics 32: 115–125.CrossRefGoogle Scholar
23.Reiman, M.I. (1984). Open queueing networks in heavy traffic. Mathematics of Operations Research 9: 451–458.CrossRefGoogle Scholar
24.Samuelson, P.A. & McKean, H.P. Jr, (1965). Rational theory of warrant pricing. Industrial Management Review 6: 13–39.Google Scholar
25.Shepp, L. & Shiryaev, A.N. (1993). The Russian option: Reduced regret. Advances in Applied Probability 3: 631–640.Google Scholar
26.Shreve, S.E., Soner, H.M., & Xu, G.L. (1991). Optimal investment and consumption with transaction costs. Mathematical Finance 1: 53–84.CrossRefGoogle Scholar
27.Taksar, M. (1985). Average optimal singular control and a related stopping problem. Mathematics of Operations Research 10: 63–81.CrossRefGoogle Scholar
28.Taksar, M., Klass, M.J., & Assaf, D. (1988). A diffusion model for optimal portfolio selection in the presence of brokerage fees. Mathematics of Operations Research 13: 277–294.CrossRefGoogle Scholar
29.Varadhan, S. & Williams, R.J. (1985). Brownian motion in a wedge with oblique reflection. Communications on Pure and Applied Mathematics 38: 405–443.CrossRefGoogle Scholar