اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدBene$check{bf S}$ condition for discontinuous exponential martingale
45
0
0.0
(
0
)
تأليف
R. Liptser
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
It is known the Girsanov exponent $mathfrak{z}_t$, being solution of Doleans-Dade equation $ mathfrak{z_t}=1+int_0^talpha(omega,s)dB_s $ generated by Brownian motion $B_t$ and a random process $alpha(omega,t)$ with $int_0^talpha^2(omega,s)ds<infty$ a.s., is the martingale provided that the Bene${rm check{s}}$ condition $$ |alpha(omega,t)|^2le text{rm const.}big[1+sup_{sin[0,t]}B^2_sbig], forall t>0, $$ holds true. In this paper, we show $B_t$ can be replaced by by a homogeneous purely discontinuous square integrable martingale $M_t$ with independent increments and paths from the Skorokhod space $ mathbb{D}_{[0,infty)} $ having positive jumps $triangle M_t$ with $Esum_{sin[0,t]}(triangle M_s)^3<infty$. A function $alpha(omega,t)$ is assumed to be nonnegative and predictable. Under this setting $mathfrak{z}_t$ is the martingale provided that $$ alpha^2(omega,t)le text{rm const.}big[1+sup_{sin[0,t]}M^2_{s-}big], forall t>0. $$ The method of proof differs from the original Bene${rm check{s}}$ one and is compatible for both setting with $B_t$ and $M_t$.
قيم البحث
اقرأ أيضاً
Let $mathfrak{z}$ be a stochastic exponential, i.e., $mathfrak{z}_t=1+int_0^tmathfrak{z}_{s-}dM_s$, of a local martingale $M$ with jumps $triangle M_t>-1$. Then $mathfrak{z}$ is a nonnegative local martingale with $Emathfrak{z}_tle 1$. If $Emathfrak{
z}_T= 1$, then $mathfrak{z}$ is a martingale on the time interval $[0,T]$. Martingale property plays an important role in many applications. It is therefore of interest to give natural and easy verifiable conditions for the martingale property. In this paper, the property $Emathfrak{z}_{_T}=1$ is verified with the so-called linear growth conditions involved in the definition of parameters of $M$, proposed by Girsanov cite{Girs}. These conditions generalize the Bene^s idea, cite{Benes}, and avoid the technology of piece-wise approximation. These conditions are applicable even if Novikov, cite{Novikov}, and Kazamaki, cite{Kaz}, conditions fail. They are effective for Markov processes that explode, Markov processes with jumps and also non Markov processes. Our approach is different to recently published papers cite{CFY} and cite{MiUr}.
We prove a strong law of large numbers for directed last passage times in an independent but inhomogeneous exponential environment. Rates for the exponential random variables are obtained from a discretisation of a speed function that may be disconti
nuous on a locally finite set of discontinuity curves. The limiting shape is cast as a variational formula that maximises a certain functional over a set of weakly increasing curves. Using this result, we present two examples that allow for partial analytical tractability and show that the shape function may not be strictly concave, and it may exhibit points of non-differentiability, flat segments, and non-uniqueness of the optimisers of the variational formula. Finally, in a specific example, we analyse further the macroscopic optimisers and uncover a phase transition for their behaviour.
A continuous-time particle system on the real line verifying the branching property and an exponential integrability condition is called a branching Levy process, and its law is characterized by a triplet $(sigma^2,a,Lambda)$. We obtain a necessary a
nd sufficient condition for the convergence of the derivative martingale of such a process to a non-trivial limit in terms of $(sigma^2,a,Lambda)$. This extends previously known results on branching Brownian motions and branching random walks. To obtain this result, we rely on the spinal decomposition and establish a novel zero-one law on the perpetual integrals of centred Levy processes conditioned to stay positive.
In this paper, we obtain stability results for martingale representations in a very general framework. More specifically, we consider a sequence of martingales each adapted to its own filtration, and a sequence of random variables measurable with res
pect to those filtrations. We assume that the terminal values of the martingales and the associated filtrations converge in the extended sense, and that the limiting martingale is quasi--left--continuous and admits the predictable representation property. Then, we prove that each component in the martingale representation of the sequence converges to the corresponding component of the martingale representation of the limiting random variable relative to the limiting filtration, under the Skorokhod topology. This extends in several directions earlier contributions in the literature, and has applications to stability results for backward SDEs with jumps and to discretisation schemes for stochastic systems.
We study the problem of bounding path-dependent expectations (within any finite time horizon $d$) over the class of discrete-time martingales whose marginal distributions lie within a prescribed tolerance of a given collection of benchmark marginal d
istributions. This problem is a relaxation of the martingale optimal transport (MOT) problem and is motivated by applications to super-hedging in financial markets. We show that the empirical version of our relaxed MOT problem can be approximated within $Oleft( n^{-1/2}right)$ error where $n$ is the number of samples of each of the individual marginal distributions (generated independently) and using a suitably constructed finite-dimensional linear programming problem.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
