Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe Laplace transform of the integrated Volterra Wishart process
88
0
0.0
(
0
)
Added by
Eduardo Abi Jaber
Publication date
2019
fields
Financial
and research's language is
English
Authors
Eduardo Abi Jaber
Ask ChatGPT about the research
No Arabic abstract
We establish an explicit expression for the conditional Laplace transform of the integrated Volterra Wishart process in terms of a certain resolvent of the covariance function. The core ingredient is the derivation of the conditional Laplace transform of general Gaussian processes in terms of Fredholms determinant and resolvent. Furthermore , we link the characteristic exponents to a system of non-standard infinite dimensional matrix Riccati equations. This leads to a second representation of the Laplace transform for a special case of convolution kernel. In practice, we show that both representations can be approximated by either closed form solutions of conventional Wishart distributions or finite dimensional matrix Riccati equations stemming from conventional linear-quadratic models. This allows fast pricing in a variety of highly flexible models, ranging from bond pricing in quadratic short rate models with rich autocorrelation structures, long range dependence and possible default risk, to pricing basket options with covariance risk in multivariate rough volatility models.
rate research
Read More
We provide existence, uniqueness and stability results for affine stochastic Volterra equations with $L^1$-kernels and jumps. Such equations arise as scaling limits of branching processes in population genetics and self-exciting Hawkes processes in mathematical finance. The strategy we adopt for the existence part is based on approximations using stochastic Volterra equations with $L^2$-kernels combined with a general stability result. Most importantly, we establish weak uniqueness using a duality argument on the Fourier--Laplace transform via a deterministic Riccati--Volterra integral equation. We illustrate the applicability of our results on Hawkes processes and a class of hyper-rough Volterra Heston models with a Hurst index $H in (-1/2,1/2]$.
In an earlier paper (A. N. Kochubei, {it Pacif. J. Math.} 269 (2014), 355--369), the author considered a restriction of Vladimirovs fractional differentiation operator $D^alpha$, $alpha >0$, to radial functions on a non-Archimedean field. In particular, it was found to possess such a right inverse $I^alpha$ that the appropriate change of variables reduces equations with $D^alpha$ (for radial functions) to integral equations whose properties resemble those of classical Volterra equations. In other words, we found, in the framework of non-Archimedean pseudo-differential operators, a counterpart of ordinary differential equations. In the present paper, we begin an operator-theoretic investigation of the operator $I^alpha$, and study a related analog of the Laplace transform.
We build a sequence of empirical measures on the space D(R_+,R^d) of R^d-valued c`adl`ag functions on R_+ in order to approximate the law of a stationary R^d-valued Markov and Feller process (X_t). We obtain some general results of convergence of this sequence. Then, we apply them to Brownian diffusions and solutions to Levy driven SDEs under some Lyapunov-type stability assumptions. As a numerical application of this work, we show that this procedure gives an efficient way of option pricing in stochastic volatility models.
We calculate exactly the Laplace transform of the Fr{e}chet distribution in the form $gamma x^{-(1+gamma)} exp(-x^{-gamma})$, $gamma > 0$, $0 leq x < infty$, for arbitrary rational values of the shape parameter $gamma$, i.e. for $gamma = l/k$ with $l, k = 1,2, ldots$. The method employs the inverse Mellin transform. The closed form expressions are obtained in terms of Meijer G functions and their graphical illustrations are provided. A rescaled Fr{e}chet distribution serves as a kernel of Fr{e}chet integral transform. It turns out that the Fr{e}chet transform of one-sided L{e}vy law reproduces the Fr{e}chet distribution.
This article is concerned with the joint law of an integrated Wishart bridge process and the trace of an integrated inverse Wishart bridge process over the interval $ left[0,tright] $. Its Laplace transform is obtained by studying the Wishart bridge processes and the absolute continuity property of Wishart laws.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
