zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On risk minimizing portfolios under a Markovian regime-switching Black-Scholes economy. (English) Zbl 1233.91242
Summary: We consider a risk minimization problem in a continuous-time Markovian regime-switching financial model modulated by a continuous-time, observable and finite-state Markov chain whose states represent different market regimes. We adopt a particular form of convex risk measure, which includes the entropic risk measure as a particular case, as a measure of risk. The risk-minimization problem is formulated as a Markovian regime-switching version of a two-player, zero-sum stochastic differential game. One important feature of our model is to allow the flexibility of controlling both the diffusion process representing the financial risk and the Markov chain representing macro-economic risk. This is novel and interesting from both the perspectives of stochastic differential game and stochastic control. A verification theorem for the Hamilton-Jacobi-Bellman (HJB) solution of the game is provided and some particular cases are discussed.
MSC:
91G10Portfolio theory
93E20Optimal stochastic control (systems)
91A23Differential games (game theory)
91B30Risk theory, insurance
References:
[1]Artzner, P., Delbaen, F., Eber, J., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9, 203–228. · Zbl 0980.91042 · doi:10.1111/1467-9965.00068
[2]Barrieu, P., & El Karoui, N. (2005). Inf-convolution of risk measures and optimal risk transfer. Finance and Stochastics, 9(2), 269–298. · Zbl 1088.60037 · doi:10.1007/s00780-005-0152-0
[3]Barrieu, P., & El Karoui, N. (2007). Pricing, hedging and optimally designing derivatives via minimization of risk measures. In R. Carmona (Ed.), Volume on indifference Pricing. Princeton: Princeton University Press. Forthcoming.
[4]Buffington, J., & Elliott, R. J. (2002a). Regime switching and European options. In K. S. Lawrence (Ed.), Stochastic theory and control, proceedings of a workshop (pp. 73–81). Berlin: Springer.
[5]Buffington, J., & Elliott, R. J. (2002b). American options with regime switching. International Journal of Theoretical and Applied Finance, 5, 497–514. · Zbl 1107.91325 · doi:10.1142/S0219024902001523
[6]Delbaen, F. (2002). Coherent risk measures on general probability spaces. In K. Sandmann (Ed.), Advances in finance and stochastics, essays in honor of Dieter Sondermann (pp. 1–37). Berlin-Heidelberg-New York: Springer.
[7]Duffie, D., & Pan, J. (1997). An overview of value at risk. Journal of Derivatives, Spring, 7–49. · doi:10.3905/jod.1997.407971
[8]Dufour, F., & Elliott, R. J. (1999). Filtering with discrete state observations. Applied Mathematics and Optimization, 40, 259–272. · Zbl 0955.62094 · doi:10.1007/s002459900125
[9]Elliott, R. J. (1982). Stochastic calculus and applications. Berlin-Heidelberg-New York: Springer.
[10]Elliott, R. J., & Hinz, J. (2002). Portfolio analysis, hidden Markov models and chart analysis by PF-Diagrams. International Journal of Theoretical and Applied Finance, 5, 385–399. · Zbl 1107.91331 · doi:10.1142/S0219024902001493
[11]Elliott, R. J., & Kopp, P. E. (2004). Mathematics of financial markets. Berlin-Heidelberg-New York: Springer.
[12]Elliott, R. J., & van der Hoek, J. (1997). An application of hidden Markov models to asset allocation problems. Finance and Stochastics, 3, 229–238. · Zbl 0907.90022 · doi:10.1007/s007800050022
[13]Elliott, R. J., Aggoun, L., & Moore, J. B. (1994). Hidden Markov models: estimation and control. Berlin-Heidelberg-New York: Springer.
[14]Elliott, R. J., Hunter, W. C., & Jamieson, B. M. (2001). Financial signal processing. International Journal of Theoretical and Applied Finance, 4, 567–584. · Zbl 1153.91491 · doi:10.1142/S0219024901001140
[15]Elliott, R. J., Malcolm, W. P., & Tsoi, A. H. (2003). Robust parameter estimation for asset price models with Markov modulated volatilities. Journal of Economics Dynamics and Control, 27, 1391–1409. · Zbl 1178.91222 · doi:10.1016/S0165-1889(02)00064-7
[16]Elliott, R. J., Chan, L. L., & Siu, T. K. (2005). Option pricing and Esscher transform under regime switching. Annals of Finance, 1(4), 423–432. · Zbl 1233.91270 · doi:10.1007/s10436-005-0013-z
[17]Elliott, R. J., Chan, L. L., & Siu, T. K. (2006). Risk measures for derivatives with Markov-modulated pure jump processes. Asia-Pacific Financial Markets, 13, 129–149. · Zbl 05182996 · doi:10.1007/s10690-007-9038-9
[18]Elliott, R. J., Siu, T. K., & Chan, L. L. (2008). A P.D.E. approach for risk measures for derivatives with regime switching. Annals of Finance, 4(1), 55–74. · Zbl 1233.91271 · doi:10.1007/s10436-006-0068-5
[19]Fleming, W. H., & Rishel, R. W. (1975). Deterministic and stochastic optimal control. Berlin-Heidelberg-New York: Springer.
[20]Föllmer, H., & Schied, A. (2002). Convex measures of risk and trading constraints. Finance and Stochastics, 6, 429–447. · Zbl 1041.91039 · doi:10.1007/s007800200072
[21]Frittelli, M. (2000). Introduction to a theory of value coherent to the no arbitrage principle. Finance and Stochastics, 4(3), 275–297. · Zbl 0965.60046 · doi:10.1007/s007800050074
[22]Frittelli, M., & Rosazza Gianin, E. (2002). Putting order in risk measures. Journal of Banking and Finance, 26(7), 1473–1486. · doi:10.1016/S0378-4266(02)00270-4
[23]Guo, X. (2001). Information and option pricing. Quantitative Finance, 1, 38–44. · doi:10.1080/713665550
[24]Mataramvura, S., & Øksendal, B. (2007). Risk minimizing portfolios and HJB equations for stochastic differential games. University of Oslo. Preprint. http://www.math.uio.no/eprint/pure_math/2005/40-05.html .
[25]Morgan, J. P. (1996). RiskMetrics–technical document (4th ed.). New York.
[26]Øksendal, B. (2003). Stochastic differential equations: an introduction with applications. Berlin-Heidelberg-New York: Springer.
[27]Øksendal, B., & Sulem, A. (2004). Applied stochastic control of jump diffusions. Berlin-Heidelberg-New York: Springer.
[28]Pliska, S. R. (1997). Introduction to mathematical finance. Malden: Blackwell.