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 derivative for functionals of superprocesses.
In this paper, we study rough path properties of stochastic integrals of It^{o}s type and Stratonovichs type with respect to $G$-Brownian motion. The roughness of $G$-Brownian Motion is estimated and then the pathwise Norris lemma in $G$-framework is obtained.
In this paper, we study the $frac{1}{H}$-variation of stochastic divergence integrals $X_t = int_0^t u_s {delta}B_s$ with respect to a fractional Brownian motion $B$ with Hurst parameter $H < frac{1}{2}$. Under suitable assumptions on the process u, we prove that the $frac{1}{H}$-variation of $X$ exists in $L^1({Omega})$ and is equal to $e_H int_0^T|u_s|^H ds$, where $e_H = mathbb{E}|B_1|^H$. In the second part of the paper, we establish an integral representation for the fractional Bessel Process $|B_t|$, where $B_t$ is a $d$-dimensional fractional Brownian motion with Hurst parameter $H < frac{1}{2}$. Using a multidimensional version of the result on the $frac{1}{H}$-variation of divergence integrals, we prove that if $2dH^2 > 1$, then the divergence integral in the integral representation of the fractional Bessel process has a $frac{1}{H}$-variation equals to a multiple of the Lebesgue measure.
In this paper we prove that every random variable of the form $F(M_T)$ with $F:real^d toreal$ a Borelian map and $M$ a $d$-dimensional continuous Markov martingale with respect to a Markov filtration $mathcal{F}$ admits an exact integral representation with respect to $M$, that is, without any orthogonal component. This representation holds true regardless any regularity assumption on $F$. We extend this result to Markovian quadratic growth BSDEs driven by $M$ and show they can be solved without an orthogonal component. To this end, we extend first existence results for such BSDEs under a general filtration and then obtain regularity properties such as differentiability for the solution process.
In this paper, we obtain L{e}vys martingale characterization of $G$-Brownian motion without the nondegenerate condition. Base on this characterization, we prove the reflection principle of $G$-Brownian motion. Furthermore, we use Krylovs estimate to get the reflection principle of $tilde{G}$-Brownian motion.
In this paper, we establish representation theorems for generators of backward stochastic differential equations (BSDEs in short) in probability spaces with general filtration from the perspective of transposition solutions of BSDEs. As applications, we give a converse comparison theorem for generators of BSDEs and also some characterizations to positive homogeneity, independence of y, subadditivity and convexity of generators of BSDEs. Then, we extend concepts of g-expectations and conditional g-expectations to the probability spaces with general filtration and investigate their properties.