# Polynomial diffusions and applications in finance

@article{Filipovi2016PolynomialDA, title={Polynomial diffusions and applications in finance}, author={Damir Filipovi{\'c} and Martin Larsson}, journal={Finance and Stochastics}, year={2016}, volume={20}, pages={931-972} }

This paper provides the mathematical foundation for polynomial diffusions. They play an important role in a growing range of applications in finance, including financial market models for interest rates, credit risk, stochastic volatility, commodities and electricity. Uniqueness of polynomial diffusions is established via moment determinacy in combination with pathwise uniqueness. Existence boils down to a stochastic invariance problem that we solve for semialgebraic state spaces. Examples…

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## References

SHOWING 1-10 OF 50 REFERENCES

### Polynomial processes and their applications to mathematical finance

- MathematicsFinance Stochastics
- 2012

We introduce a class of Markov processes, called m-polynomial, for which the calculation of (mixed) moments up to order m only requires the computation of matrix exponentials. This class contains…

### Mean stochastic comparison of diffusions

- MathematicsThe 23rd IEEE Conference on Decision and Control
- 1984

SummaryStochastic bounds are derived for one dimensional diffusions (and somewhat more general random processes) by dominating one process pathwise by a convex combination of other processes. The…

### A survey and some generalizations of Bessel processes

- Mathematics
- 2003

Bessel processes play an important role in financial mathematics because of their strong relation to financial processes like geometric Brownian motion or CIR processes. We are interested in the…

### The Jacobi stochastic volatility model

- Mathematics, EconomicsFinance Stochastics
- 2018

A novel stochastic volatility model where the squared volatility of the asset return follows a Jacobi process is introduced and the Heston model is included as a limit case.

### Affine Diffusions with Non-Canonical State Space

- Mathematics
- 2010

Multidimensional affine diffusions have been studied in detail for the case of a canonical state space. We present results for general state spaces and provide a complete characterization of all…

### Diffusions, Markov processes, and martingales

- Mathematics
- 1979

This celebrated book has been prepared with readers' needs in mind, remaining a systematic treatment of the subject whilst retaining its vitality. The second volume follows on from the first,…

### Convergence of local supermartingales and Novikov-Kazamaki type conditions for processes with jumps

- Mathematics
- 2014

We characterize the event of convergence of a local supermartingale. Conditions are given in terms of its predictable characteristics and quadratic variation. The notion of extended local…

### A chaotic representation property of the multidimensional Dunkl processes

- Mathematics
- 2006

Dunkl processes are martingales as well as cadlag homogeneous Markov processes taking values in R d and they are naturally associated with a root system. In this paper we study the jumps of these…