Related papers: Change of measure up to a random time: Details
Invariance times are stopping times $\tau$ such that local martingales with respect to some reduced filtration and an equivalently changed probability measure, stopped before $\tau$ , are local martingales with respect to the original model…
Given a random time, we characterize the set of martingales for which the stopping theorems still hold. We also investigate how the stopping theorems are modified when we consider arbitrary random times. To this end, we introduce some…
We study the stability of several no-arbitrage conditions with respect to absolutely continuous, but not necessarily equivalent, changes of measure. We first consider models based on continuous semimartingales and show that no-arbitrage…
We consider a change of measure by a martingale $Z_t$ and clarify that in general $1/Z_t$ is only a supermartingale under the changed measure. We then give a necessary and sufficient condition for the event that the limit of the martingale…
In this note we introduce a new kind of augmentation of filtrations along a sequence of stopping times. This augmentation is suitable for the construction of new probability measures associated to a positive strict local martingale as done…
This paper completes the two studies undertaken in \cite{aksamit/choulli/deng/jeanblanc2} and \cite{aksamit/choulli/deng/jeanblanc3}, where the authors quantify the impact of a random time on the No-Unbounded-Risk-with-Bounded-Profit…
Under short sales prohibitions, no free lunch with vanishing risk (NFLVR-S) is known to be equivalent to the existence of an equivalent supermartingale measure for the price processes (Pulido [22]). For two given price processes, we…
For any discrete-time $P$--local martingale $S$ there exists a probability measure $Q \sim P$ such that $S$ is a $Q$--martingale. A new proof for this result is provided. The core idea relies on an appropriate modification of an argument by…
Recently, D. Williams \cite{williams} gave an explicit example of a random time $\rho $ associated with Brownian motion such that $\rho $ is not a stopping time but $\mathbb{E}M_{\rho}=\mathbb{E}M_{0}$ for every bounded martingale $M$. The…
The hypothesis of randomness is fundamental in statistical machine learning and in many areas of nonparametric statistics; it says that the observations are assumed to be independent and coming from the same unknown probability…
We give a new characterization for mutual absolute continuity of probability measures on a filtered space. For this, we introduce a martingale limit $M$ that measures the similarity between the tails of the probability measures restricted…
On a probability space $(\Omega,\mathcal{A},\mathbb{Q})$ we consider two filtrations $\mathbb{F}\subset \mathbb{G}$ and a $\mathbb{G}$ stopping time $\theta$ such that the $\mathbb{G}$ predictable processes coincide with $\mathbb{F}$…
An important question for a probabilistic program is whether the probability mass of all its diverging runs is zero, that is that it terminates "almost surely". Proving that can be hard, and this paper presents a new method for doing so; it…
We propose two nonparametric tests for investigating the pathwise properties of a signal modeled as the sum of a L\'{e}vy process and a Brownian semimartingale. Using a nonparametric threshold estimator for the continuous component of the…
In this paper, we provide a solution to two problems which have been open in default time modeling in credit risk. We first show that if $\tau$ is an arbitrary random (default) time such that its Az\'ema's supermartingale…
In a Monte-Carlo test, the observed dataset is fixed, and several resampled or permuted versions of the dataset are generated in order to test a null hypothesis that the original dataset is exchangeable with the resampled/permuted ones.…
No-arbitrage models of term structure have the feature that the return on zero-coupon bonds is the sum of the short rate and the product of volatility and market price of risk. Well known models restrict the behavior of the market price of…
The equivalence between multiportfolio time consistency of a dynamic multivariate risk measure and a supermartingale property is proven. Furthermore, the dual variables under which this set-valued supermartingale is a martingale are…
In this paper we estimate the rest of the approximation of a stationary process by a martingale in terms of the projections of partial sums. Then, based on this estimate, we obtain almost sure approximation of partial sums by a martingale…
The typical central limit theorems in high-frequency asymptotics for semimartingales are results on stable convergence to a mixed normal limit with an unknown conditional variance. Estimating this conditional variance usually is a hard…