Polynomial diffusions and applications in finance

  title={Polynomial diffusions and applications in finance},
  author={Damir Filipovi{\'c} and Martin Larsson},
  journal={Finance and Stochastics},
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… 
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
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
Probability measure-valued polynomial diffusions
We introduce a class of probability measure-valued diffusions, coined polynomial, of which the well-known Fleming--Viot process is a particular example. The defining property of finite dimensional
Asian option pricing with orthogonal polynomials
In this paper we derive a series expansion for the price of a continuously sampled arithmetic Asian option in the Black–Scholes setting. The expansion is based on polynomials that are orthogonal with


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
Density Approximations for Multivariate Affine Jump-Diffusion Processes
We introduce closed-form transition density expansions for multivariate affine jump-diffusion processes. The expansions rely on a general approximation theory which we develop in weighted Hilbert
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
Linear credit risk models
We introduce a novel class of credit risk models in which the drift of the survival process of a firm is a linear function of the factors. The prices of defaultable bonds and credit default swaps
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