Abstract
In this paper we prove a weak necessary and sufficient maximum principle for Markovian regime switching stochastic optimal control problems. Instead of insisting on the maximum condition of the Hamiltonian, we show that \( 0 \) belongs to the sum of Clarke’s generalized gradient of the Hamiltonian and Clarke’s normal cone of the control constraint set at the optimal control. Under a joint concavity condition on the Hamiltonian and a convexity condition on the terminal objective function, the necessary condition becomes sufficient. We give four examples to demonstrate the weak stochastic maximum principle.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Aubin, J.-P., Frankowska, H.: Set-valued analysis. Birkhäuser (1990)
Bensoussan, A.: Lectures on stochastic control. Lect. Notes Math. 972, 1–62 (1981)
Bismut, J.M.: Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl. 44, 384–404 (1973)
Bismut, J.M.: Linear quadratic optimal control with random coefficients. SIAM J. Control Optim. 14, 419–444 (1976)
Bismut, J.M.: An introductory approach to duality in optimal stochastic control. SIAM Rev. 20, 62–78 (1978)
Cadenillas, A., Karatzas, I.: The stochastic maximum principle for linear convex optimal control with random coefficients. SIAM J. Control Optim. 33, 590–624 (1995)
Clarke, F.H.: Generalized gradients and applications. Trans. Am. Math. Soc. 205, 247 (1975)
Clarke, F.H.: Shadow prices and duality for a class of optimal control problems. SIAM J. Control Optim. 17, 567 (1979)
Clarke, F.H.: Generalized gradients of lipschitz functionals. Adv. Math. 40, 52–67 (1981)
Clarke, F.H.: Optimization and nonsmooth analysis. SIAM (1990)
Cohen, S., Elliott, R.: Solutions of backward stochastic differential equations on markov chains. Commun. Stoch. Anal. 2, 251–262 (2008)
Cohen, S., Elliott, R.: Comparisons for backward stochastic dierential equations on markov chains and related no-arbitrage conditions. Ann. Appl. Probab. 20, 267–311 (2010)
Crepey, S.: About the pricing equations in finance. In: Ekeland, I., Jouini, E., Scheinkman, J., Touzi, N., Carmona, R., Cinlar, E. (eds.) Paris-Princeton Lectures on Mathematical Finance, pp. 63–203. Springer, Berlin (2010)
Donnelly, C: Convex duality in constrained mean-variance portfolio optimization under a regime-switching model. PhD thesis, University of Waterloo (2008)
Donnelly, C.: Sufficient stochastic maximum principle in the regime-switching diffusion model. Appl. Math. Optim. 62(2), 155–169 (2011)
Donnelly, C., Heunis, A.J.: Quadratic risk minimization in a regime-switching model with portfolio constraints. SIAM J. Control Optim. 50(4), 2431–2461 (2012)
Fleming, W.H., Rishel, R.W.: Deterministic and Stochastic Optimal Control. Springer, Berlin (1975)
Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Springer, Berlin (2006)
Framstad, N.C., Oksendal, B., Sulem, A.: Sufficient stochastic maximum principle for the optimal control of jump difussions and applications to finance. J. Optim. Theory Appl. 121(1), 77–98 (2004)
Haussmann, U.G.: A Stochastic Maximum Principle for Optimal Control of Diffusions. Longman Scientific and Technical (1986)
Kushner, H.J.: On the stochastic maximum principle: fixed time of control. J. Math. Anal. Appl. 11, 78–92 (1965)
Kushner, H.J.: Necessary conditons for continuous parameter stochastic optimization problems. SIAM J. Control Optim. 10, 550–565 (1972)
Mao, X., Yuan, C.: Stochastic Differential Equations with Markovian Switching. Imperial College Press, London (2006)
Peng, S.: A general stochastic maximum principle for optimal control problems. SIAM J. Control Optim. 28, 966–979 (1990)
Peng, S.: Nonlinear expectations, nonlinear evaluations and risk measures. In: Hipp, C., Peng, S., Schachermayer, W., Back, K., Bielecki, T.R. (eds.) Stochastic Methods in Finance, pp. 165–253. Springer, Berlin (2004)
Pham, H.: Continuous-Time Stochastic Control and Optimization with Financial Applications. Springer, Berlin (2009)
Rockafeller, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Rogers, L.C.G.: Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus. Cambridge University Press, Cambridge (2000)
Tao, R., Wu, Z.: Maximum principle for optimal control problems of forwardcbackward regime-switching system and applications. Syst. Control Lett. 61, 911–917 (2012)
Tao, R., Wu, Z., Zhang, Q.: Bsdes with regime switching: weak convergence and applications. J. Math. Anal. Appl. 407, 97–111 (2013)
Tang, S., Li, X.: Necessary conditions for optimality control of stochastic systems with random jumps. SIAM J. Control Optim. 32, 1447–1475 (1994)
Wu, Z.: A general maximum principle for optimal control of forward–backward stochastic systems. Automatica 49, 1473–1480 (2013)
Yong, J., Zhou, X.Y.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York (1999)
Zhang, X., Elliott, R.J., Siu, T.K.: A stochastic maximum principle for a markov regime-switching jump-diffusion model and its application to finance. SIAM J. Control Optim. 50, 964–990 (2012)
Zhou, X.Y.: A unified treatment of maximum principle and dynamic programming in stochastic controls. Stoch. Stoch. Rep. 36, 137–161 (1991)
Zhou, X.Y.: Sufficient conditions of optimality for stochastic systems with controllable diffusions. IEEE Trans. Autom. Control 41, 1176–1179 (1996)
Acknowledgments
The authors are grateful to Professor Nicole El Karoui for the useful discussions on the paper, especially on the contents of the measurability of stochastic processes. The authors also thank two anonymous referees for their comments that have helped to improve the previous version.
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
1.1 Proof of Theorem 5.15
Proof
Consider the function \( \Phi \) on \( \mathbb {S}^2([0,T])\times L^2(W,[0,T])\times L^2(Q,[0,T]) \) mapping \( (Y,Z,S)\in \mathbb {S}^2([0,T])\times L^2(W,[0,T])\times L^2(Q,[0,T]) \) to \( \left( \hat{Y},\hat{Z},\hat{S}\right) =\Phi (Y,Z,S) \) defined by
Consider the square-integrable martingale
According to Theorem 5.13, there exists unique \( \left( \hat{Z},\hat{S} \right) \in L^2(W,[0,T])\times L^2(Q,[0,T]) \) such that
We then define the process \( \hat{Y}(t) \) by
By Doob’s \( L^2 \) inequality, we have
Under the assumptions on \( (\xi ,f) \), we conclude that \( \hat{Y}\in S^2([0,T]) \). Hence \( \Phi \) is a well defined function from \( S^2([0,T])\times L^2(W,[0,T])\times L^2(Q,[0,T]) \) into itself. Next, we show that \( (\hat{Y},\hat{Z},\hat{S}) \) is a solut ion to the regime switching BSDE (5.3) if and only if it is a fixed point of \( \Phi \).
Let \( (U,V,\Gamma ) \), \( (U',V',\Gamma ') \in S^2([0,T])\times L^2(W,[0,T])\times L^2(Q,[0,T])\). Apply function \( \Phi \) and obtain \( (Y,Z,S)=\Phi (U,V,\Gamma ),\ (Y',Z',S')=\Phi (U',V',\Gamma ') \). Set \( (\bar{U},\bar{V},\bar{\Gamma }) =(U-U',V-V',\Gamma -\Gamma ')\), \( (\bar{Y},\bar{Z},\bar{S})=(Y-Y',Z-Z',S-S') \) and \( \bar{f}(t)=f(t,U(t),V(t))-f(t,U'(t),V'(t)) \). Take \( \beta >0 \) to be chosen later and apply Ito’s formula to \( e^{\beta s}\vert \bar{Y} \vert ^2 \) on \( [0,T] \),
Observe that, according to Young’s inequality
Hence \( \int _0^t e^{\beta s}\bar{Y}(s)^\intercal \bar{Z}(s)dW(s) \) and \( \int _0^te^{\beta s}\sum _{l=1}^n\sum _{i,j=1}^d\bar{Y}^{(l)}(s)\bar{S}_{ij}^{(l)}(s)dQ_{ij}(s) \) are true martingales by the Burkholder-Davis-Gundy inequality. Taking expectation in (8.1), we get
Take \( \beta =1+4C_f^2 \) and substitute into (8.2), we have
Notice that \( L^2(W,[0,T]) \) and \( L^2(Q,[0,T]) \) are Hilbert spaces and therefore the space \( \mathbb {S}^2([0,T])\times L^2(W,[0,T])\times L^2(Q,[0,T]) \) endowed with the norm
is a Banach space. We conclude that \( \Phi \) admits a unique fixed point which is the solution to the BSDE (5.3).
Rights and permissions
About this article
Cite this article
Li, Y., Zheng, H. Weak Necessary and Sufficient Stochastic Maximum Principle for Markovian Regime-Switching Diffusion Models. Appl Math Optim 71, 39–77 (2015). https://doi.org/10.1007/s00245-014-9252-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00245-014-9252-6
Keywords
- Regime switching stochastic optimal control
- Weak stochastic maximum principle
- Necessary and sufficient conditions
- Clarke’s generalized gradient
- Clarke’s normal cone
- Measurable selection