Related papers: Martingale Representation in Progressively Enlarge…
In this paper we extend the notion of ``filtration-consistent nonlinear expectation" (or "${\cal F}$-consistent nonlinear expectation") to the case when it is allowed to be dominated by a $g$-expectation that may have a quadratic growth. We…
For a general Multidimensional L\'{e}vy process (satisfying some moment conditions), we introduce the Multidimensional power jump processes and the related Multidimensional Teugels martingales. Furthermore, we orthogonalize the…
Our main result is the martingale representations for Markov additive processes where the modulator is a Levy process. These processes have three parts: the modulator, the jumps of the ordinate triggered by the modulator, and the…
We derive the explicit form of the martingale representation for square-integrable processes that are martingales with respect to the natural filtration of the super-Brownian motion. This is done by using a weak extension of the Dupire…
We show that a discrete time martingale with respect to a filtration with atomless innovations is the (infinite) sum of martingales with independent increments. For the continuous time filtration coming from Brownian Motion filtration, we…
We study class of L\'{e}vy processes having distributions being indentifiable by moments. We define system of polynomial martingales \newline $\left\{ M_{n}(X_{t},t),\mathcal{F}_{\leq t}\right\} _{n\geq 1},$ where $% \mathcal{F}_{\leq t}$…
Our financial setting consists of a market model with two flows of information. The smallest flow F is the "public" flow of information which is available to all agents, while the larger flow G has additional information about the…
In this paper we prove that every random variable of the form $F(M_T)$ with $F:\real^d \to\real$ a Borelian map and $M$ a $d$-dimensional continuous Markov martingale with respect to a Markov filtration $\mathcal{F}$ admits an exact…
The paper studies thin times which are random times whose graph is contained in a countable union of the graphs of stopping times with respect to a reference filtration $\mathbb F$. We show that a generic random time can be decomposed into…
We develop the mathematics of a filtration shrinkage model that has recently been considered in the credit risk modeling literature. Given a finite collection of points $x_1<...<x_N$ in $\mathbb{R}$, the region indicator function $R(x)$…
In this paper we explore the fundamentals of the Martingale Representation Theorem (MRT) and a closely related result, the Clark-Ocone formula. We also investigate how far these theorems can be taken, notably beyond the regular Sobolev…
We show that, under certain smoothness conditions, a Brownian martingale at a fixed time can be represented as an exponential of its value at a later time. The time-dependent generator of this exponential operator is equal to one half times…
In this paper we study progressive filtration expansions with random times. We show how semimartingale decompositions in the expanded filtration can be obtained using a natural link between progressive and initial expansions. The link is,…
Let $(W,H,\mu)$ be the classical Wiener space on $\R^d$. Assume that $X=(X_t)$ is a diffusion process satisfying the stochastic differential equation $dX_t=\sigma(t,X)dB_t+b(t,X)dt$, where $\sigma:[0,1]\times C([0,1],\R^n)\to \R^n\otimes…
We study Fourier multipliers resulting from martingale transforms of general L\'evy processes.
By making use of martingale representations, we derive the asymptotic normality of particle filters in hidden Markov models and a relatively simple formula for their asymptotic variances. Although repeated resamplings result in complicated…
This paper considers the nonlinear theory of G-martingales as introduced by Peng. A martingale representation theorem for this theory is proved by using the techniques and the results established in an accompanying paper for the second…
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\wedge\tau)+Z_tdW_t^{\tau}+dM_t-dK_t,\quad Y_{\tau}=\xi, Y\geq…
This paper constructs a class of martingale transforms based on L\'evy processes on Lie groups. From these, a natural class of bounded linear operators on the $L^p$-spaces of the group (with respect to Haar measure) for $1<p<\infty$, are…
The paper considers the martingale theory in the $G$-framework. A form of Doob's optional sampling is established, which allows to prove the exact analogue of the classical maximal inequality. The obtained results are used to improve the…