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Affine diffusions with non-canonical state space. (English) Zbl 1260.60112
The authors deal with multi-dimensional affine diffusions on state spaces which are more general than the canonical state space \(\mathbb{R}_{\geq 0}^m \times \mathbb{R}^{p-m}\). More concretely, for a given state space \(\mathcal{X} \subset \mathbb{R}^p\), the authors determine affine functions \(\mu : \mathbb{R}^p \rightarrow \mathbb{R}^p\) and \(\theta : \mathbb{R}^{p \times p} \rightarrow \mathbb{R}^p\) such that there exists a continuous square root \(\sigma\) of \(\theta\) on \(\mathcal{X}\), for which \[ \text{d}X_t = \mu(X_t)\text{d}t + \sigma(X_t)\text{d}W_t \] is an affine SDE. This problem is linked with stochastic invariance problems, which the authors study in the present paper. Then, the authors consider polyhedral state spaces \[ \mathcal{X} = \bigcap_{i=1}^q \{ u_i \geq 0 \}, \] where \(u : \mathbb{R}^p \rightarrow \mathbb{R}^q\) denotes an affine function \(u(x) = \gamma x + \delta\) with \(\gamma \in \mathbb{R}^{q \times p}\) and \(\delta \in \mathbb{R}^q\). For these spaces, they also study the problem of diagonalizing the diffusion matrix \(\theta(x)\) and revisit a classical model by Duffie and Kan. Afterwards, they consider quadratic state spaces and show that there are only two types of possible state spaces, i.e., parabolic state spaces and Lorentz cones. For both types of spaces, the authors derive conditions for the existence of affine diffusions.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J25 Continuous-time Markov processes on general state spaces
91G80 Financial applications of other theories
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References:
[1] Bhatia R., Matrix Analysis (1997) · Zbl 0863.15001 · doi:10.1007/978-1-4612-0653-8
[2] Cheridito P., Journal of Financial Economics 83 pp 123– (2007) · doi:10.1016/j.jfineco.2005.09.008
[3] Cheridito P., Mathematical Finance 20 pp 509– (2010) · Zbl 1194.91185 · doi:10.1111/j.1467-9965.2010.00408.x
[4] Cuchiero , C. 2011 . Affine and polynomial processes. PhD thesis, ETH Zürich, No. 19652.
[5] Cuchiero C., Annals of Applied Probability 21 pp 397– (2011) · Zbl 1219.60068 · doi:10.1214/10-AAP710
[6] Da Prato G., Journal of Mathematical Analysis and Applications 333 pp 151– (2007) · Zbl 1117.60060 · doi:10.1016/j.jmaa.2006.08.057
[7] Dai Q., Journal of Finance 55 pp 1943– (2000) · doi:10.1111/0022-1082.00278
[8] Duffee G.-R., Journal of Finance 57 pp 405– (2002) · doi:10.1111/1540-6261.00426
[9] Duffie D., Annals of Applied Probability 13 pp 984– (2003) · Zbl 1048.60059 · doi:10.1214/aoap/1060202833
[10] Duffie D., Mathematical Finance 6 pp 379– (1996) · Zbl 0915.90014 · doi:10.1111/j.1467-9965.1996.tb00123.x
[11] Filipović , D. , and Mayerhofer , E. 2009 . Affine diffusion processes: theory and applications. InAdvanced Financial Modelling.Vol. 8 of Radon Series on Computational and Applied Mathematics. Walter de Gruyter, Berlin, pp. 125–164. · Zbl 1205.91068
[12] Gourieroux C., Journal of Financial Econometrics 4 pp 31– (2006) · doi:10.1093/jjfinec/nbj003
[13] Gourieroux C., Journal of Economic Dynamics and Control 35 pp 815– (2011) · Zbl 1231.91455 · doi:10.1016/j.jedc.2011.01.007
[14] Grasselli M., Mathematical Finance 18 pp 135– (2008) · Zbl 1138.91547 · doi:10.1111/j.1467-9965.2007.00325.x
[15] Ikeda N., Stochastic Differential Equations and Diffusion Processes (1981) · Zbl 0495.60005
[16] Kallenberg O., Foundations of Modern Probability., 2. ed. (2002) · Zbl 0996.60001
[17] Karatzas I., Brownian Motion and Stochastic Calculus (1991) · Zbl 0734.60060 · doi:10.1007/978-1-4612-0949-2
[18] Mayerhofer E., Stochastic Processes and Their Applications 121 pp 2072– (2011) · Zbl 1225.60096 · doi:10.1016/j.spa.2011.05.006
[19] Milian A., Colloquium Mathematicum 68 pp 297– (1995)
[20] Spreij , P.J.C. , and Veerman , E. 2010 .The Affine Transform Formula for Affine Diffusions with Convex State Space.arXiv:1005.1099v2. · Zbl 1260.60112
[21] Tappe , S. 2009 .Stochastic Invariance of Closed, Convex Sets with Respect to Jump-Diffusions.Universität Wien working paper 18.
[22] Veerman E., Affine Markov Processes on a General State Space (2011)
[23] Watanabe S., Mathematics of Kyoto University 11 pp 155– (1971) · Zbl 0236.60037 · doi:10.1215/kjm/1250523691
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