STRICT LOCAL MARTINGALES VIA FILTRATION ENLARGEMENT

A strict local martingale is a local martingale that is not a martingale. We investigate how such a process might arise from a true martingale as a result of an enlargement of the filtration and a change of measure. We study and implement a particular type of enlargement, initial expansion of filtration, for stochastic volatility models with and without jumps and provide sufficient conditions in each of these cases such that initial expansion can create a strict local martingale. We provide examples of initial enlargement that effect this change.

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Complications with stochastic volatility models

Advances in Applied Probability ◽

10.1017/s0001867800008193 ◽

1998 ◽

Vol 30 (01) ◽

pp. 256-268 ◽

Cited By ~ 12

Author(s):

Carlos A. Sin

Keyword(s):

Stochastic Volatility ◽

Stock Price ◽

Local Martingale ◽

Stochastic Volatility Models ◽

Martingale Measures ◽

Equivalent Martingale Measures ◽

Volatility Models ◽

Equivalent Martingale

We show a class of stock price models with stochastic volatility for which the most natural candidates for martingale measures are only strictly local martingale measures, contrary to what is usually assumed in the finance literature. We also show the existence of equivalent martingale measures, and provide one explicit example.

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Noncausality in Continuous Time Models

Econometric Theory ◽

10.1017/s0266466600006575 ◽

1996 ◽

Vol 12 (2) ◽

pp. 215-256 ◽

Cited By ~ 22

Author(s):

F. Comte ◽

E. Renault

Keyword(s):

Stochastic Volatility ◽

Continuous Time ◽

Moving Average ◽

Sufficient Conditions ◽

Stochastic Volatility Models ◽

Discrete Sample ◽

Volatility Models ◽

Moving Average Representation ◽

Canonical Representations ◽

Average Representation

In this paper, we study new definitions of noncausality, set in a continuous time framework, illustrated by the intuitive example of stochastic volatility models. Then, we define CIMA processes (i.e., processes admitting a continuous time invertible moving average representation), for which canonical representations and sufficient conditions of invertibility are given. We can provide for those CIMA processes parametric characterizations of noncausality relations as well as properties of interest for structural interpretations. In particular, we examine the example of processes solutions of stochastic differential equations, for which we study the links between continuous and discrete time definitions, find conditions to solve the possible problem of aliasing, and set the question of testing continuous time noncausality on a discrete sample of observations. Finally, we illustrate a possible generalization of definitions and characterizations that can be applied to continuous time fractional ARMA processes.

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MIXING AND MOMENT PROPERTIES OF VARIOUS GARCH AND STOCHASTIC VOLATILITY MODELS

Econometric Theory ◽

10.1017/s0266466602181023 ◽

2002 ◽

Vol 18 (1) ◽

pp. 17-39 ◽

Cited By ~ 312

Author(s):

Marine Carrasco ◽

Xiaohong Chen

Keyword(s):

Stochastic Volatility ◽

Sufficient Conditions ◽

Higher Order ◽

Decay Rates ◽

Stochastic Volatility Models ◽

Higher Order Moments ◽

Second Moments ◽

Volatility Models ◽

Strict Stationarity ◽

Autoregressive Conditional Duration

This paper first provides some useful results on a generalized random coefficient autoregressive model and a generalized hidden Markov model. These results simultaneously imply strict stationarity, existence of higher order moments, geometric ergodicity, and β-mixing with exponential decay rates, which are important properties for statistical inference. As applications, we then provide easy-to-verify sufficient conditions to ensure β-mixing and finite higher order moments for various linear and nonlinear GARCH(1,1), linear and power GARCH(p,q), stochastic volatility, and autoregressive conditional duration models. For many of these models, our sufficient conditions for existence of second moments and exponential β-mixing are also necessary. For several GARCH(1,1) models, our sufficient conditions for existence of higher order moments again coincide with the necessary ones in He and Terasvirta (1999, Journal of Econometrics 92, 173–192).

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