The minimal entropy martingale measure for general Barndorff-Nielsen/Shephard models

Generalization of the theorems of Barndorff-Nielsen and Balakrishnan-Stepanov to Riesz spaces

In a Dedekind complete Riesz space, $E$, we show that if $(P_n)$ is a sequence of band projections in $E$ then $$\limsup\limits_{n\to \infty} P_n - \liminf\limits_{n\to \infty} P_n = \limsup\limits_{n\to \infty} P_n(I-P_{n+1}).$$ This identity is used to obtain conditional extensions in a Dedekind complete Riesz spaces with weak order unit and conditional expectation operator of the Barndorff-Nielsen and Balakrishnan-Stepanov generalizations of the First Borel-Cantelli Theorem.

ambit stochastics（范围随机过程）研究前沿：纪念Ole E. Barndorff-Nielsen

Research frontiers in ambit stochastics: In memory of Ole E. Barndorff-Nielsen

本文为纪念Ole E. Barndorff-Nielsen，综述了ambit stochastics领域的关键内容、他对该领域的重要贡献，以及该领域的新兴研究趋势。

How singular are moment generating functions?

This short note concerns the possible singular behaviour of moment generating functions of finite measures at the boundary of their domain of existence. We look closer at Example 7.3 in O. Barndorff-Nielsen's book "Information and Exponential Families in Statistical Theory (1978)" and elaborate on the type of exhibited singularity. Finally, another regularity problem is discussed and it is solved through tensorizing two Barndorff- Nielsen's distributions.

A Note on the Borel-Cantelli Lemma

In this short note, we discuss the Barndorff-Nielsen lemma, which is a generalization of well-known Borel-Cantelli lemma. Although the result stated in the Barndorff-Nielsen lemma is correct, it does not follow from the argument proposed in the corresponding proof. In this note, we show this and offer an alternative proof of this lemma. We also propose a new generalization of Borel-Cantelli lemma.

Analysis of a Class of Likelihood Based Continuous Time Stochastic Volatility Models including Ornstein-Uhlenbeck Models in Financial Economics

In a series of recent papers Barndorff-Nielsen and Shephard introduce an attractive class of continuous time stochastic volatility models for financial assets where the volatility processes are functions of positive Ornstein-Uhlenbeck(OU) processes. This models are known to be substantially more flexible than Gaussian based models. One current problem of this approach is the unavailability of a tractable exact analysis of likelihood based stochastic volatility models for the returns of log prices of stocks. With this point in mind, the likelihood models of Barndorff-Nielsen and Shephard are vi…

Moment Explosions and Long-Term Behavior of Affine Stochastic Volatility Models

We consider a class of asset pricing models, where the risk-neutral joint process of log-price and its stochastic variance is an affine process in the sense of Duffie, Filipovic and Schachermayer [2003]. First we obtain conditions for the price process to be conservative and a martingale. Then we present some results on the long-term behavior of the model, including an expression for the invariant distribution of the stochastic variance process. We study moment explosions of the price process, and provide explicit expressions for the time at which a moment of given order becomes infinite. We d…

Robustness of Hilbert space-valued stochastic volatility models

In this paper we show that Hilbert space-valued stochastic models are robust with respect to perturbation, due to measurement or approximation errors, in the underlying volatility process. Within the class of stochastic volatility modulated Ornstein-Uhlenbeck processes, we quantify the error induced by the volatility in terms of perturbations in the parameters of the volatility process. We moreover study the robustness of the volatility process itself with respect to finite dimensional approximations of the driving compound Poisson process and semigroup generator respectively, when considering…

An infinite-dimensional affine stochastic volatility model

We introduce a flexible and tractable infinite-dimensional stochastic volatility model. More specifically, we consider a Hilbert space valued Ornstein-Uhlenbeck-type process, whose instantaneous covariance is given by a pure-jump stochastic process taking values in the cone of positive self-adjoint Hilbert-Schmidt operators. The tractability of our model lies in the fact that the two processes involved are jointly affine, i.e., we show that their characteristic function can be given explicitly in terms of the solutions to a set of generalised Riccati equations. The flexibility lies in the fact…

A note on exact likelihoods of the Carr-Wu models for leverage effects and volatility in financial economics

Recently Carr and Wu (2004, 2005) and also Huang and Wu (2004) show that most stochastic processes used in traditional option pricing models can be cast as special cases of time-changed Lévy processes. In particular these are models which can be tailored to exhibit correlated jumps in both the log price of assets and the instantaneous volatility. Naturally similar to a recent work of Barndorff-Nielsen and Shephard (2001a, b), such models may be used in a likelihood based framework. These likelihoods are based on the unobserved integrated volatility, rather than the instantaneous volatility. Ja…

Ornstein-Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility

We propose a non-Gaussian operator-valued extension of the Barndorff-Nielsen and Shephard stochastic volatility dynamics, defined as the square-root of an operator-valued Ornstein-Uhlenbeck process with Levy noise and bounded drift. We derive conditions for the positive definiteness of the Ornstein-Uhlenbeck process, where in particular we must restrict to operator-valued Levy processes with "non-decreasing paths". It turns out that the volatility model allows for an explicit calculation of its characteristic function, showing an affine structure. We introduce another Hilbert space-valued Orns…

马尔可夫环境下的波动率建模：两种与 Ornstein-Uhlenbeck 相关的方法

Volatility modeling in a Markovian environment: Two Ornstein-Uhlenbeck-related approaches

本文将 COGARCH 模型和 Barndorff-Nielsen-Shephard 模型推广到马尔可夫切换环境，在马尔可夫调制的广义 Ornstein-Uhlenbeck 过程框架下推导平稳性、矩和自协方差结构，并验证其能刻画金融时间序列的典型事实。

A Class of Generalized Hyperbolic Continuous Time Integrated Stochastic Volatility Likelihood Models

This paper discusses and analyzes a class of likelihood models which are based on two distributional innovations in financial models for stock returns. That is, the notion that the marginal distribution of aggregate returns of log-stock prices are well approximated by generalized hyperbolic distributions, and that volatility clustering can be handled by specifying the integrated volatility as a random process such as that proposed in a recent series of papers by Barndorff-Nielsen and Shephard (BNS). The BNS models produce likelihoods for aggregate returns which can be viewed as a subclass of l…