scholarly journals A new kind of augmentation of filtrations suitable for a change of probability measure by a strict local martingale

2015 ◽  
Vol 104 ◽  
pp. 94-101 ◽  
Author(s):  
Dörte Kreher ◽  
Ashkan Nikeghbali
Keyword(s):  
Probability Measure ◽  
Local Martingale ◽  
Strict Local Martingale ◽  
Change Of Probability Measure

scholarly journals Bayesian credibility premium with GB2 copulas

2020 ◽  
Vol 8 (1) ◽  
pp. 157-171 ◽  
Author(s):  
Himchan Jeong ◽  
Emiliano A. Valdez
Keyword(s):  
Probability Measure ◽  
Explicit Form ◽  
Insurance Premium ◽  
Response Variable ◽  
Generalized Pareto ◽  
Special Case ◽  
Change Of Probability Measure

AbstractFor observations over a period of time, Bayesian credibility premium may be used to predict the value of a response variable for a subject, given previously observed values. In this article, we formulate Bayesian credibility premium under a change of probability measure within the copula framework. Such reformulation is demonstrated using the multivariate generalized beta of the second kind (GB2) distribution. Within this family of GB2 copulas, we are able to derive explicit form of Bayesian credibility premium. Numerical illustrations show the application of these estimators in determining experience-rated insurance premium. We consider generalized Pareto as a special case.

scholarly journals Uniform convergence of conditional distributions for absorbed one-dimensional diffusions

2018 ◽  
Vol 50 (01) ◽  
pp. 178-203 ◽  
Author(s):  
Nicolas Champagnat ◽  
Denis Villemonais
Keyword(s):  
Sufficient Conditions ◽  
Initial Distribution ◽  
Initial Position ◽  
Local Martingale ◽  
One Dimensional ◽  
Positive Time ◽  
Strict Local Martingale ◽  
Necessary And Sufficient ◽  
Stationary Behavior ◽  
Uniformly Bounded

Abstract In this paper we study the quasi-stationary behavior of absorbed one-dimensional diffusions. We obtain necessary and sufficient conditions for the exponential convergence to a unique quasi-stationary distribution in total variation, uniformly with respect to the initial distribution. An important tool is provided by one-dimensional strict local martingale diffusions coming down from infinity. We prove, under mild assumptions, that their expectation at any positive time is uniformly bounded with respect to the initial position. We provide several examples and extensions, including the sticky Brownian motion and some one-dimensional processes with jumps.

scholarly journals STRICT LOCAL MARTINGALES VIA FILTRATION ENLARGEMENT

2019 ◽  
Vol 23 (01) ◽  
pp. 2050001
Author(s):  
ADITI DANDAPANI ◽  
PHILIP PROTTER
Keyword(s):  
Stochastic Volatility ◽  
Sufficient Conditions ◽  
Local Martingale ◽  
Change Of Measure ◽  
Stochastic Volatility Models ◽  
Strict Local Martingales ◽  
Volatility Models ◽  
Strict Local Martingale

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.

scholarly journals DUPIRE'S EQUATION FOR BUBBLES

2012 ◽  
Vol 15 (06) ◽  
pp. 1250041 ◽  
Author(s):  
ERIK EKSTRÖM ◽  
JOHAN TYSK
Keyword(s):  
Option Price ◽  
Local Martingale ◽  
Option Prices ◽  
Uniqueness Of Solutions ◽  
Underlying Asset ◽  
Put Options ◽  
Price Process ◽  
Usual Form ◽  
Volatility Models ◽  
Strict Local Martingale

We study Dupire's equation for local volatility models with bubbles, i.e. for models in which the discounted underlying asset follows a strict local martingale. If option prices are given by risk-neutral valuation, then the discounted option price process is a true martingale, and we show that the Dupire equation for call options contains extra terms compared to the usual equation. However, the Dupire equation for put options takes the usual form. Moreover, uniqueness of solutions to the Dupire equation is lost in general, and we show how to single out the option price among all possible solutions. The Dupire equation for models in which the discounted derivative price process is merely a local martingale is also studied.

scholarly journals Implied Volatility in Strict Local Martingale Models

2018 ◽  
Vol 9 (1) ◽  
pp. 171-189 ◽  
Author(s):  
Antoine Jacquier ◽  
Martin Keller-Ressel
Keyword(s):  
Implied Volatility ◽  
Local Martingale ◽  
Strict Local Martingale
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