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Kavita Ramanan - One of the best experts on this subject based on the ideXlab platform.
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On the Submartingale problem for reflected diffusions in domains with piecewise smooth boundaries
The Annals of Probability, 2017Co-Authors: Weining Kang, Kavita RamananAbstract:Two frameworks that have been used to characterize reflected diffusions include stochastic differential equations with reflection (SDER) and the so-called Submartingale problem. We consider a general formulation of the Submartingale problem for (obliquely) reflected diffusions in domains with piecewise C2C2 boundaries and piecewise continuous reflection vector fields. Under suitable assumptions, we show that well-posedness of the Submartingale problem is equivalent to existence and uniqueness in law of weak solutions to the corresponding SDER. The main step involves showing existence of a weak solution to the SDER given a solution to the Submartingale problem. This generalizes the classical construction, due to Stroock and Varadhan, of a weak solution to an (unconstrained) stochastic differential equation, but requires a completely different approach to deal with the geometry of the domain and directions of reflection and properly identify the local time on the boundary, in the presence of multi-valued directions of reflection at nonsmooth parts of the boundary. In particular, our proof entails the construction of classes of test functions that satisfy certain oblique derivative boundary conditions, which may be of independent interest. Other ingredients of the proof that are used to identify the constraining or local time process include certain random linear functionals, suitably constructed exponential martingales and a dual representation of the cone of directions of reflection. As a corollary of our result, under suitable assumptions, we also establish an equivalence between well-posedness of both the SDER and Submartingale formulations and well-posedness of the constrained martingale problem, which is another framework for defining (semimartingale) reflected diffusions. Many of our intermediate results are also valid for reflected diffusions that are not necessarily semimartingales, and are used in a companion paper [Equivalence of stochastic equations and the Submartingale problem for nonsemimartingale reflected diffusions. Preprint] to extend the equivalence result to a class of nonsemimartingale reflected diffusions.
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On the Submartingale problem for reflected diffusions in domains with piecewise smooth boundaries
arXiv: Probability, 2014Co-Authors: Weining Kang, Kavita RamananAbstract:Two frameworks that have been used to characterize reflected diffusions include stochastic differential equations with reflection and the so-called Submartingale problem. We introduce a general formulation of the Submartingale problem for (obliquely) reflected diffusions in domains with piecewise C^2 boundaries and piecewise continuous reflection vector fields. Under suitable assumptions, we show that well-posedness of the Submartingale problem is equivalent to existence and uniqueness in law of weak solutions to the corresponding stochastic differential equation with reflection. Our result generalizes to the case of reflecting diffusions a classical result due to Stroock and Varadhan on the equivalence of well-posedness of martingale problems and well-posedness of weak solutions of stochastic differential equations in d-dimensional Euclidean space. The analysis in the case of reflected diffusions in domains with non-smooth boundaries is considerably more subtle and requires a careful analysis of the behavior of the reflected diffusion on the boundary of the domain. In particular, the equivalence can fail to hold when our assumptions are not satisfied. The equivalence we establish allows one to transfer results on reflected diffusions characterized by one approach to reflected diffusions analyzed by the other approach. As an application, we provide a characterization of stationary distributions of a large class of reflected diffusions in convex polyhedral domains.
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 $(\Sigma)$
arXiv: Probability, 2009Co-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 -FINITE MEASURES ASSOCIATED WITH SubmartingaleS OF CLASS () (UNE NOUVELLE CONSTRUCTION DES MESURES -FINIES ASSOCI ´ EES AUX SOUS-MARTINGALES DE CLASSE ())
2009Co-Authors: Joseph Najnudel, Ur Mathematik, Ashkan NikeghbaliAbstract:In (7), we prove that for any Submartingale (Xt)t�0 of class (�), defined on a filtered probability space (, F, P,(Ft)t�0), which satisfies some technical conditions, one can construct a �-finite measure Q on (, F), such that for all t � 0, and for all events �t 2 F t: Q(�t,gt) = EP(1�tXt) 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 (�), definie sur un espace de probabilite filtre (, F, P,(Ft)t�0), satisfaisant certaines conditions techniques, on peut construire une mesure �-finie Q sur (, F), telle que pour tout t � 0, et pour tout ´enementt 2 F t: Q(�t,gt) = EP(1�t Xt) ou g est le dernier zero de X. Certains cas particuliers de cette construction sont lies aux penalisations browniennes ou aux mathematiques financieres. Dans cette note, nous donnons une construction plus simple de Q, et nous montrons qu'un analogue de cette mesure peut aussidefini pour des sous-martingalestemps discret.
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.
Adam Osękowski - One of the best experts on this subject based on the ideXlab platform.
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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 OsękowskiAbstract:Let X be a Submartingale and Y be a semimartingale which is orthogonal and strongly differentially subordinate to X. The paper contains the proof of the sharp estimate P ( sup t≥0 |Yt| ≥ 1 ) ≤ 3.375 . . . ||X||1. As an application, a related weak-type inequality for smooth functions on Euclidean domains is established.
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Sharp tail inequalities for nonnegative Submartingales and their strong differential subordinates
Electronic Communications in Probability, 2010Co-Authors: Adam OsękowskiAbstract:Let $f=(f_n)_{n\geq 0}$ be a nonnegative Submartingale starting from $x$ and let $g=(g_n)_{n\geq 0}$ be a sequence starting from $y$ and satisfying $$|dg_n|\leq |df_n|,\quad |\mathbb{E}(dg_n|\mathcal{F}_{n-1})|\leq \mathbb{E}(df_n|\mathcal{F}_{n-1})$$ for $n\geq 1$. We determine the best universal constant $U(x,y)$ such that $$\mathbb{P}(\sup_ng_n\geq 0)\leq ||f||_1+U(x,y).$$ As an application, we deduce a sharp weak type $(1,1)$ inequality for the one-sided maximal function of $g$ and determine, for any $t\in [0,1]$ and $\beta\in\mathbb{R}$, the number $$ L(x,y,t,\beta)=\inf\{||f||_1: \mathbb{P}(\sup_ng_n\geq \beta)\geq t\}.$$ The estimates above yield analogous statements for stochastic integrals in which the integrator is a nonnegative Submartingale. The results extend some earlier work of Burkholder and Choi in the martingale setting.
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Strong Differential Subordination and Sharp Inequalities for Orthogonal Processes
Journal of Theoretical Probability, 2009Co-Authors: Adam OsękowskiAbstract:We introduce a strong differential α -subordination for continuous-time processes, which generalizes this notion from the discrete-time setting, due to Burkholder and Choi. Then we determine the best constants in the L ^ p estimates for a nonnegative Submartingale and its strong α -subordinate under an additional assumption on the orthogonality of these two processes.
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SHARP INEQUALITY FOR BOUNDED SubmartingaleS AND THEIR DIFFERENTIAL SUBORDINATES
Electronic Communications in Probability, 2008Co-Authors: Adam OsękowskiAbstract:Let $\alpha$ be a fixed number from the interval $[0,1]$. We obtain the sharp probability bounds for the maximal function of the process which is $\alpha$-differentially subordinate to a bounded Submartingale. This generalizes the previous results of Burkholder and Hammack.
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.
