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… 
View on Springer
arxiv.org
73 Citations
Highly Influential Citations
22
Background Citations
44
Methods Citations
23
Results Citations
1

73 Citations

Polynomial Jump-Diffusion Models

A large class of novel financial asset pricing models that are based on polynomial jump-diffusions are introduced, including a generic method for option pricing based on moment expansions.

Correlators of Polynomial Processes

In the setting of polynomial jump-diffusion dynamics, we provide a formula for computing correlators, namely, cross-moments of the process at different time points along its path. The formula…

Polynomial jump-diffusions on the unit simplex

Polynomial jump-diffusions constitute a class of tractable stochastic models with wide applicability in areas such as mathematical finance and population genetics. We provide a full parameterization…

Polynomial Diffusion Models for Life Insurance Liabilities

M F ] 2 1 Ju l 2 01 9 Polynomial Jump-Diffusion Models ∗

We develop a comprehensive mathematical framework for polynomial jump-diffusions in a semimartingale context, which nest affine jump-diffusions and have broad applications in finance. We show that…

Measure-valued affine and polynomial diffusions

It is shown the so-called moment formula, i.e. a representation of the conditional marginal moments via a system of finite dimensional linear PDEs, and characterize the corresponding infinitesimal generators and obtain a representation analogous to polynomial diffusions on R+, in cases where their domain is large enough.

Polynomial models in finance

This thesis presents new flexible dynamic stochastic models for the evolution of market prices and new methods for the valuation of derivatives. These models and methods build on the recently…

Projection scheme for polynomial diffusions on the unit ball

The main idea to consider the numerical scheme is the transformation argument introduced by Swart for proving the pathwise uniqueness for some stochastic differential equation with a non-Lipschitz diffusion coefficient.

M F ] 2 1 N ov 2 01 7 Polynomial Jump-Diffusion Models ∗

We develop a comprehensive mathematical framework for polynomial jump-diffusions, which nest affine jump-diffusions and have broad applications in finance. We show that the polynomial property is…

Chaotic Hedging with Iterated Integrals and Neural Networks

In this paper, we extend the Wiener-Ito chaos decomposition to the class of diffusion processes, whose drift and diffusion coefficient are of linear growth. By omitting the orthogonality in the chaos…
...

References

SHOWING 1-10 OF 50 REFERENCES

On strong solutions for positive definite jump diffusions

Polynomial processes and their applications to mathematical finance

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

  • B. Hajek
  • Mathematics
    The 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

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…

Multivariate Jacobi process with application to smooth transitions

The Jacobi stochastic volatility model

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

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

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

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

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…