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Ashkan Nikeghbali - One of the best experts on this subject based on the ideXlab platform.
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A NEW CONSTRUCTION OF THE σ-FINITE MEASURES ASSOCIATED WITH Submartingales OF CLASS (Σ)
2015Co-Authors: Joseph Najnudel, Ashkan NikeghbaliAbstract:Abstract. In [7], we prove that for any submartingale (Xt)t≥0 of class (Σ), defined on a filtered probability space (Ω,F,P, (Ft)t≥0), which satisfies some technical conditions, one can construct a σ-finite measure Q on (Ω,F), such that for all t ≥ 0, and for all events Λt ∈ Ft: Q[Λt, g ≤ t] = EP[1ΛtXt] where g is the last hitting time of zero of the process X. Some particular cases of this construction are related with Brownian penalisation or mathematical finance. In this note, we give a simpler construction of Q, and we show that an analog of this measure can also be defined for discrete-time Submartingales. Dans [7], nous prouvons que pour toute sous-martingale (Xt)t≥0 de classe (Σ), définie sur un espace de probabilite ́ filtre ́ (Ω,F,P, (Ft)t≥0), satisfaisant certaines conditions techniques, on peut construire une mesure σ-finie Q sur (Ω,F), telle que pour tout t ≥ 0, et pour tout événement Λt ∈ Ft: Q[Λt, g ≤ t] = EP[1ΛtXt] ou ̀ g est le dernier zéro de X. Certains cas particuliers de cette construction sont liés au
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On some universal σ-finite measures related to a remarkable class of Submartingales
Stochastic Processes and their Applications, 2012Co-Authors: Joseph Najnudel, Ashkan NikeghbaliAbstract:Abstract In this paper, for any submartingale of class ( Σ ) defined on a filtered probability space ( Ω , F , P , ( F t ) t ≥ 0 ) satisfying some technical conditions, we associate a σ -finite measure Q on ( Ω , F ) , such that for all t ≥ 0 , and for all events Λ t ∈ F t : Q [ Λ t , g ≤ t ] = E P [ 1 Λ t X t ] , where g is the last time for which the process X hits zero. The existence of Q has already been proven in several particular cases, some of them are related with Brownian penalization, and others are involved with problems in mathematical finance. More precisely, the existence of Q in the general case gives an answer to a problem stated by Madan, Roynette and Yor, in a paper about the link between the Black–Scholes formula and the last passage times of some particular Submartingales. Moreover, the equality defining Q still holds if the fixed time t is replaced by any bounded stopping time. This generalization can be considered as an extension of Doob’s optional stopping theorem.
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A new construction of the σ-finite measures associated with Submartingales of class (Σ)
Comptes Rendus Mathematique, 2010Co-Authors: Joseph Najnudel, Ashkan NikeghbaliAbstract:In a previous paper, we proved that for any submartingale $(X_t)_{t \geq 0}$ of class $(\Sigma)$, defined on a filtered probability space $(\Omega, \mathcal{F}, \mathbb{P}, (\mathcal{F}_t)_{t \geq 0})$, which satisfies some technical conditions, one can construct a $\sigma$-finite measure $\mathcal{Q}$ on $(\Omega, \mathcal{F})$, such that for all $t \geq 0$, and for all events $\Lambda_t \in \mathcal{F}_t$: $$ \mathcal{Q} [\Lambda_t, g\leq t] = \mathbb{E}_{\mathbb{P}} [\mathds{1}_{\Lambda_t} X_t]$$ where $g$ is the last hitting time of zero of the process $X$. Some particular cases of this construction are related with Brownian penalisation or mathematical finance. In this note, we give a simpler construction of $\mathcal{Q}$, and we show that an analog of this measure can also be defined for discrete-time Submartingales.
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a new construction of the sigma finite measures associated with Submartingales of class sigma
Comptes rendus de l'Académie des sciences, 2010Co-Authors: Joseph Najnudel, Ashkan NikeghbaliAbstract:In a previous paper, we proved that for any submartingale $(X_t)_{t \geq 0}$ of class $(\Sigma)$, defined on a filtered probability space $(\Omega, \mathcal{F}, \mathbb{P}, (\mathcal{F}_t)_{t \geq 0})$, which satisfies some technical conditions, one can construct a $\sigma$-finite measure $\mathcal{Q}$ on $(\Omega, \mathcal{F})$, such that for all $t \geq 0$, and for all events $\Lambda_t \in \mathcal{F}_t$: $$ \mathcal{Q} [\Lambda_t, g\leq t] = \mathbb{E}_{\mathbb{P}} [\mathds{1}_{\Lambda_t} X_t]$$ where $g$ is the last hitting time of zero of the process $X$. Some particular cases of this construction are related with Brownian penalisation or mathematical finance. In this note, we give a simpler construction of $\mathcal{Q}$, and we show that an analog of this measure can also be defined for discrete-time Submartingales.
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on penalisation results related with a remarkable class of Submartingales
arXiv: Probability, 2009Co-Authors: Joseph Najnudel, Ashkan NikeghbaliAbstract:Is this paper we study penalisations of diffusions satisfying some technical conditions, generalizing a result obtained by Najnudel, Roynette and Yor. If one of these diffusions has probability distribution $\mathbb{P}$, then our result can be described as follows: for a large class of families of probability measures $(\mathbb{Q}_t)_{t \geq 0}$, each of them being absolutely continuous with respect to $\mathbb{P}$, there exists a probability $\mathbb{Q}_{\infty}$ such that for all events $\Lambda$ depending only on the canonical trajectory up to a fixed time, $\mathbb{Q}_t (\Lambda)$ tends to $\mathbb{Q}_{\infty} (\Lambda)$ when $t$ goes to infinity. In the cases we study here, the limit measure $\mathbb{Q}_{\infty}$ is absolutely continous with respect to a sigma-finite measure $\mathcal{Q}$, which does not depend on the choice of the family of probabilities $(\mathbb{Q}_t)_{t \geq 0}$, but only on $\mathbb{P}$. The relation between $\mathbb{P}$ and $\mathcal{Q}$ is obtained in a very general framework by the authors of this paper.
Wang Lasheng - One of the best experts on this subject based on the ideXlab platform.
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riesz decomposition and convergence for set valued order submartingale with discrete parameter
Journal of Mathematics, 2007Co-Authors: Wang LashengAbstract:In this paper,we study the Riesz decomposition and convergence of set-valued order submartingale with discreate parameter.By set-valued order relations and martingale methods,we give the existence and uniquess theorem of set-valued order submartingale Riesz decomposition.Based on the Riesz decomposition theorem,we obtain the convergence theorems of set-valued order submartingale.
Adam Osekowski - One of the best experts on this subject based on the ideXlab platform.
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strong differential subordinates for noncommutative Submartingales
Annals of Probability, 2019Co-Authors: Yong Jiao, Adam OsekowskiAbstract:We introduce a notion of strong differential subordination of noncommutative semimartingales, extending Burkholder’s definition from the classical case (Ann. Probab. 22 (1994) 995–1025). Then we establish the maximal weak-type $(1,1)$ inequality under the additional assumption that the dominating process is a submartingale. The proof rests on a significant extension of the maximal weak-type estimate of Cuculescu and a Gundy-type decomposition of an arbitrary noncommutative submartingale. We also show the corresponding strong-type $(p,p)$ estimate for $1
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burkholder inequalities for Submartingales bessel processes and conformal martingales
American Journal of Mathematics, 2013Co-Authors: Rodrigo Banuelos, Adam OsekowskiAbstract:The motivation for this paper comes from the following question on comparison of norms of conformal martingales $X$, $Y$ in $\Bbb{R}^d$, $d\geq 2$. Suppose that $Y$ is differentially subordinate to $X$. For $0
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logarithmic estimates for Submartingales and their differential subordinates
Journal of Theoretical Probability, 2011Co-Authors: Adam OsekowskiAbstract:In the paper we determine, for any K>0 and α∈[0,1], the optimal constant L(K,α)∈(0,∞] for which the following holds: If X is a nonnegative submartingale and Y is α-strongly differentially subordinate to X, then $$\sup_t\mathbb{E}|Y_t|\leq K\sup_t\mathbb{E}X_t\log^+X_t+L(K,\alpha).$$ Related sharp inequalities for martingales are also established. As an application, we obtain logarithmic estimates for smooth functions on Euclidean domains.
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a maximal inequality for nonnegative sub and supermartingales
Mathematical Inequalities & Applications, 2011Co-Authors: Adam OsekowskiAbstract:Let X = (Xt)t≥0 be a nonnegative semimartingale and H = (Ht)t≥0 be a predictable process taking values in [−1, 1]. Let Y denote the stochastic integral of H with respect to X. We show that (i) If X is a supermartingale, then || sup t≥0 Yt||1 ≤ 3|| sup t≥0 Xt||1 and the constant 3 is the best possible. (ii) If X is a submartingale satisfying ||X||∞ ≤ 1, then || sup t≥0 Yt||p ≤ 2Γ(p + 1), 1 ≤ p <∞. The constant 2Γ(p + 1)1/p is the best possible. Department of Mathematics, Informatics and Mechanics University of Warsaw Banacha 2, 02-097 Warsaw Poland email: ados@mimuw.edu.pl
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a weak type inequality for orthogonal Submartingales and subharmonic functions
Bulletin of The Polish Academy of Sciences Mathematics, 2011Co-Authors: Adam OsekowskiAbstract:Let X be a submartingale and Y be a semimartingale which is orthogonal and strongly dierentially subordinate to X. The paper contains the proof of the sharp estimate
Yuhu Feng - One of the best experts on this subject based on the ideXlab platform.
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Semi-order fuzzy supermartingales and Submartingales with continuous time
Fuzzy Sets and Systems, 2002Co-Authors: Yuhu Feng, Xinglong ZhuAbstract:A new suitable semi-order in the fuzzy number space (En, D), the semi-order fuzzy supermartingales and Submartingales with continuous time and their characteristics are introduced. The famous Riesz decomposition and Doob-Meyer decomposition theorems in standard martingale theory are generalized to semi-order fuzzy supermartingales. The proof of Doob-Meyer decomposition is rather technical and so the theory of fuzzy-valued measures generated by increasing fuzzy processes is developed.
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Decomposition theorems for fuzzy supermartingales and Submartingales
Fuzzy Sets and Systems, 2000Co-Authors: Yuhu FengAbstract:Doob decomposition and Riesz decomposition theorems in standard martingale theory are generalized to fuzzy martingales. The concepts of two types of Riesz decomposition for fuzzy supermartingales are given and the necessary and sufficient conditions of that fuzzy supermartingale or submartingale (resp. supermartingale) has Doob decomposition (resp. Riesz decomposition) are discussed in detail.
P.k. Bhattacharya - One of the best experts on this subject based on the ideXlab platform.
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A maximal inequality for nonnegative Submartingales
Statistics & Probability Letters, 2005Co-Authors: P.k. BhattacharyaAbstract:A slightly weaker version of a well-known maximal inequality for martingales is shown to hold for nonnegative Submartingales.
