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Polynomial diffusions and applications in finance. (English) Zbl 1386.60237
The article analyzes diffusions with polynomial drift and diffusion matrix of degrees at most 1 and 2 resp., evolving in semi-algebraic sets defined by polynomial inequalities. The main result is well-posedness of the solution and conditions for a non-sticky boundary. Specific instances of the state space given by certain quadric sets, product of the unit cube and non-negative orthant, or the unit simplex, are further analyzed. Applications to finance are also described.

MSC:
60H30 Applications of stochastic analysis (to PDEs, etc.)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
91G99 Actuarial science and mathematical finance
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[1] Ackerer, D., Filipović, D.: Linear credit risk models. Swiss Finance Institute Research Paper No. 16-34 (2016). Available online at http://ssrn.com/abstract=2782455
[2] Ackerer, D., Filipović, D., Pulido, S.: The Jacobi stochastic volatility model. Swiss Finance Institute Research Paper No. 16-35 (2016). Available online at http://ssrn.com/abstract=2782486 · Zbl 0514.44007
[3] Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Oliver & Boyd, Edinburgh (1965) · Zbl 0135.33803
[4] Bakry, D.; Émery, M.; Yor, M. (ed.); Azéma, J. (ed.), Diffusions hypercontractives, No. 1123, 177-206, (1985), Berlin · Zbl 0561.60080
[5] Berg, C.; Christensen, J.P.R.; Jensen, C.U., A remark on the multidimensional moment problem, Math. Ann., 243, 163-169, (1979) · Zbl 0416.46003
[6] Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer, Berlin (1998) · Zbl 0912.14023
[7] Carr, P.; Fisher, T.; Ruf, J., On the hedging of options on exploding exchange rates, Finance Stoch., 18, 115-144, (2014) · Zbl 1314.91205
[8] Cherny, A., On the uniqueness in law and the pathwise uniqueness for stochastic differential equations, Theory Probab. Appl., 46, 406-419, (2002) · Zbl 1036.60051
[9] Cuchiero, C.: Affine and polynomial processes. Ph.D. thesis, ETH Zurich (2011). Available online at http://e-collection.library.ethz.ch/eserv/eth:4629/eth-4629-02.pdf · Zbl 0537.60050
[10] Cuchiero, C.; Keller-Ressel, M.; Teichmann, J., Polynomial processes and their applications to mathematical finance, Finance Stoch., 16, 711-740, (2012) · Zbl 1270.60079
[11] Curtiss, J.H., A note on the theory of moment generating functions, Ann. Math. Stat., 13, 430-433, (1942) · Zbl 0063.01024
[12] Prato, G.; Frankowska, H., Invariance of stochastic control systems with deterministic arguments, J. Differ. Equ., 200, 18-52, (2004) · Zbl 1066.93049
[13] Prato, G.; Frankowska, H., Stochastic viability of convex sets, J. Math. Anal. Appl., 333, 151-163, (2007) · Zbl 1117.60060
[14] Delbaen, F.; Schachermayer, W., A general version of the fundamental theorem of asset pricing, Math. Ann., 300, 463-520, (1994) · Zbl 0865.90014
[15] Delbaen, F.; Shirakawa, H., An interest rate model with upper and lower bounds, Asia-Pac. Financ. Mark., 9, 191-209, (2002) · Zbl 1071.91020
[16] Dummit, D.S., Foote, R.M.: Abstract Algebra, 3rd edn. Wiley, Hoboken (2004) · Zbl 1037.00003
[17] Dunkl, C.F., Hankel transforms associated to finite reflection groups, Contemp. Math., 138, 123-138, (1992) · Zbl 0789.33008
[18] Ethier, S.N., A class of degenerate diffusion processes occurring in population genetics, Commun. Pure Appl. Math., 29, 483-493, (1976) · Zbl 0333.60075
[19] Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. Wiley, Hoboken (2005) · Zbl 1089.60005
[20] Filipović, D.; Mayerhofer, E.; Schneider, P., Density approximations for multivariate affine jump-diffusion processes, J. Econom., 176, 93-111, (2013) · Zbl 1284.62110
[21] Filipović, D., Larsson, M., Trolle, A.: Linear-rational term structure models. J. Finance. Forthcoming. Available at SSRN http://ssrn.com/abstract=2397898
[22] Filipović, D.; Tappe, S.; Teichmann, J., Invariant manifolds with boundary for jump-diffusions, Electron. J. Probab., 19, 1-28, (2014) · Zbl 1301.60072
[23] Filipović, D.; Gourier, E.; Mancini, L., Quadratic variance swap models, J. Financ. Econ., 119, 44-68, (2016)
[24] Forman, J.L.; Sørensen, M., The Pearson diffusions: a class of statistically tractable diffusion processes, Scand. J. Stat., 35, 438-465, (2008) · Zbl 1198.62078
[25] Gallardo, L.; Yor, M., A chaotic representation property of the multidimensional Dunkl processes, Ann. Probab., 34, 1530-1549, (2006) · Zbl 1107.60015
[26] Göing-Jaeschke, A.; Yor, M., A survey and some generalizations of Bessel processes, Bernoulli, 9, 313-349, (2003) · Zbl 1038.60079
[27] Gouriéroux, C.; Jasiak, J., Multivariate Jacobi process with application to smooth transitions, J. Econom., 131, 475-505, (2006) · Zbl 1337.62365
[28] Hajek, B., Mean stochastic comparison of diffusions, Z. Wahrscheinlichkeitstheor. Verw. Geb., 68, 315-329, (1985) · Zbl 0537.60050
[29] Heyde, C.C., On a property of the lognormal distribution, J. R. Stat. Soc., Ser. B, Stat. Methodol., 25, 392-393, (1963) · Zbl 0114.33802
[30] Horn, R.A., Johnson, C.A.: Matrix Analysis. Cambridge University Press, Cambridge (1985) · Zbl 0576.15001
[31] Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam (1981) · Zbl 0495.60005
[32] Kleiber, C.; Stoyanov, J., Multivariate distributions and the moment problem, J. Multivar. Anal., 113, 7-18, (2013) · Zbl 1253.60015
[33] Larsen, K.S.; Sørensen, M., Diffusion models for exchange rates in a target zone, Math. Finance, 17, 285-306, (2007) · Zbl 1186.91232
[34] Larsson, M., Ruf, J.: Convergence of local supermartingales and Novikov-Kazamaki type conditions for processes with jumps (2014). arXiv:1411.6229
[35] Lord, R.; Koekkoek, R.; Dijk, D., A comparison of biased simulation schemes for stochastic volatility models, Quant. Finance, 10, 177-194, (2012) · Zbl 1198.91240
[36] Maisonneuve, B.; Dellacherie, C. (ed.); etal., Une mise au point sur LES martingales locales continues définies sur un intervalle stochastique, No. 581, 435-445, (1977), Berlin
[37] Mayerhofer, E.; Pfaffel, O.; Stelzer, R., On strong solutions for positive definite jump diffusions, Stoch. Process. Appl., 121, 2072-2086, (2011) · Zbl 1225.60096
[38] Mazet, O.; Azéma, J. (ed.); etal., Classification des semi-groupes de diffusion sur ℝ associés à une famille de polynômes orthogonaux, No. 1655, 40-53, (1997), Berlin · Zbl 0883.60072
[39] Penrose, R., A generalized inverse for matrices, Math. Proc. Camb. Philos. Soc., 51, 406-413, (1955) · Zbl 0065.24603
[40] Petersen, L.C., On the relation between the multidimensional moment problem and the one-dimensional moment problem, Math. Scand., 51, 361-366, (1982) · Zbl 0514.44007
[41] Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (1999) · Zbl 0917.60006
[42] Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes and Martingales. Cambridge University Press, Cambridge (1994) · Zbl 0826.60002
[43] Schmüdgen, K., The \(K\)-moment problem for compact semi-algebraic sets, Math. Ann., 289, 203-206, (1991) · Zbl 0744.44008
[44] Spreij, P.; Veerman, E., Affine diffusions with non-canonical state space, Stoch. Anal. Appl., 30, 605-641, (2012) · Zbl 1260.60112
[45] Stieltjes, T.J.: Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse 8(4), 1-122 (1894) · JFM 25.0326.01
[46] Stoyanov, J., Krein condition in probabilistic moment problems, Bernoulli, 6, 939-949, (2000) · Zbl 0971.60017
[47] Willard, S.: General Topology. Courier Corporation, North Chelmsford (2004) · Zbl 1052.54001
[48] Wong, E.; Bellman, R. (ed.), The construction of a class of stationary markoff processes, 264-276, (1964), Providence
[49] Zhou, H., Itô conditional moment generator and the estimation of short-rate processes, J. Financ. Econom., 1, 250-271, (2003)
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