Shiqi Song
We characterize the random times $ρ$ whose Azema supermartingales $Z^ρ$ take the form $Z^ρ=U/U^*$ for some non negative local martingales $U$ starting from 1 vanishing at infinity, where $U^*$ denotes the running maximum process of $U$.
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We provide a general Doob-Meyer decomposition for $g$-supermartingale systems, which does not require any right-continuity on the system. In particular, it generalizes the Doob-Meyer decomposition of Mertens (1972) for classical supermartingales, as well as Peng's (1999) version for right-continuous $g$-supermartingales. As examples of application, we prove an optional decomposition theorem for $g$-supermartingale systems, and also obtain a general version of the well-known dual formation for BSDEs with constraint on the gains-process, using very simple arguments.
We prove results on the existence of Doléans-Dade measures and of the Doob-Meyer decomposition for supermartingales indexed by a general index set
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In the paper, we introduce the notion of a local regular supermartingale relative to a convex set of equivalent measures and prove for it an optional Doob decomposition in the discrete case. This Theorem is a generalization of the famous Doob decomposition onto the case of supermartingales relative to a convex set of equivalent measures.
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In the paper, we introduce the notion of a local regular supermartingale relative to a convex set of equivalent measures and prove for it the necessary and sufficient conditions of optional Doob decomposition in the discrete case. This Theorem is a generalization of the famous Doob decomposition onto the case of supermartingales relative to a convex set of equivalent measures. The description of all local regular supermartingales relative to a convex set of equivalent measures is presented. A notion of complete set of equivalent measures is introduced. We prove that every non negative bounded …
Azéma associated with an honest time L the supermartingale $Z_{t}^{L}=\mathbb{P}[L>t|\mathcal{F}_{t}]$ and established some of its important properties. This supermartingale plays a central role in the general theory of stochastic processes and in particular in the theory of progressive enlargements of filtrations. In this paper, we shall give an additive characterization for these supermartingales, which in turn will naturally provide many examples of enlargements of filtrations. In particular, we use this characterization to establish some path decomposition results, closely related to or re…
Given a finite honest time, we first show that the associated Azéma optional supermartingale can be expressed as the drawdown and the relative drawdown of some local optional supermartingales with continuous running supremum. The relative drawdown representation then allows us to provide a characterisation of finite honest times using a family of non-negative local optional supermartingales with continuous running supremum which converges to zero at infinity. Then we extend the notion of semimartingales of class-$(Σ)$ by allowing for jumps in its finite variation part of the semimartingale dec…
Optional Stopping for Superhedging Supermartingales
本文利用非格积分理论，对超对冲超鞅建立了停时的塔不等式、Doob可选停止定理和Dubins上穿不等式，所有证明不依赖经典测度论工具。
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 $τ$ is an arbitrary random (default) time such that its Azéma's supermartingale $Z_t^τ=¶(τ>t|\F_t)$ is continuous, then $τ$ avoids stopping times. We then disprove a conjecture about the equality between the hazard process and the martingale hazard process, which first appeared in \cite{jenbrutk1}, and we show how it should be modified to become a theorem. The pseudo-stopping times, introduced in \cite{AshkanYor}, appear as the most general class of random ti…
This paper extends results of Mortimer and Williams (1991) about changes of probability measure up to a random time under the assumptions that all martingales are continuous and that the random time avoids stopping times. We consider locally absolutely continuous measure changes up to a random time, changes of probability measure up to and after an honest time, and changes of probability measure up to a pseudo-stopping time. Moreover, we apply our results to construct a change of probability measure that is equivalent to the enlargement formula and to build for a certain class of pseudo-stoppi…
American Options with Last Exit Times: A Free-Boundary Approach
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This paper addresses reflected backward stochastic differential equations (RBSDE hereafter) that take the form of \begin{eqnarray*} \begin{cases} dY_t=f(t,Y_t, Z_t)d(t\wedgeτ)+Z_tdW_t^τ+dM_t-dK_t,\quad Y_τ=ξ, Y\geq S\quad\mbox{on}\quad \Lbrack0,τ\Lbrack,\quad \displaystyle\int_0^τ(Y_{s-}-S_{s-})dK_s=0\quad P\mbox{-a.s..}\end{cases} \end{eqnarray*} Here $τ$ is an arbitrary random time that might not be a stopping time for the filtration $\mathbb F$ generated by the Brownian motion $W$. We consider the filtration $\mathbb G$ resulting from the progressive enlargement of $\mathbb F$ with $τ$ wher…
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