Related papers: Martingale representation for degenerate diffusion…
We propose a novel method for drift estimation of multiscale diffusion processes when a sequence of discrete observations is given. For the Langevin dynamics in a two-scale potential, our approach relies on the eigenvalues and the…
We introduce polynomial processes taking values in an arbitrary Banach space $B$ via their infinitesimal generator $L$ and the associated martingale problem. We obtain two representations of the (conditional) moments in terms of solutions…
Let $v:[0,T]\times \R^d \to \R$ be the solution of the parabolic backward equation $ \partial_t v + (1/2) \sum_{i,l} [\sigma \sigma^\perp]_{il} \partial_{x_i \partial_{x_l} v + \sum_{i} b_i \partial_{x_i}v + kv =0$ with terminal condition…
This paper studies the weak and strong solutions to the stochastic differential equation $ dX(t)=-\frac12 \dot W(X(t))dt+d\mathcal{B}(t)$, where $(\mathcal{B}(t), t\ge 0)$ is a standard Brownian motion and $W(x)$ is a two sided Brownian…
A new technique for proving uniqueness of martingale problems is introduced. The method is illustrated in the context of elliptic diffusions in $R^d$.
In this paper we introduce the concept of conic martingales}. This class refers to stochastic processes having the martingale property, but that evolve within given (possibly time-dependent) boundaries. We first review some results about…
In this paper we consider the Stratonovich reflected stochastic differential equation $dX_t=\sigma(X_t)\circ dW_t+b(X_t)dt+dL_t$ in a bounded domain $\O$ which satisfies conditions, introduced by Lions and Sznitman, which are specified…
We investigate aspects of semimartingale decompositions, approximation and the martingale representation for multidimensional correlated Markov processes. A new interpretation of the dependence among processes is given using the martingale…
We demonstrate that backward stochastic differential equations (BSDE) may be reformulated as ordinary functional differential equations on certain path spaces. In this framework, neither It\^{o}'s integrals nor martingale representation…
In this paper we show that the weak representation property of a semimartingale $X$ with respect to a filtration $\mathbb{F}$ is preserved in the progressive enlargement $\mathbb{G}$ by a random time $\tau$ avoiding $\mathbb{F}$-stopping…
The constructive martingale representation theorem of functional It\^o calculus is extended, from the space of square integrable martingales, to the space of local martingales. The setting is that of an augmented filtration generated by a…
In this note we define and study the stochastic process $X$ in link with a parabolic transmission operator $(A,D(A))$ in divergence form. The transmission operator involves a diffraction condition along a transmission boundary. To that aim…
This paper develops an approach for solving perpetual discounted optimal stopping problems for multidimensional diffusions, with special emphasis on the $d$-dimensional Wiener process. We first obtain some verification theorems for…
A geometric reformulation of the martingale problem associated with a set of diffusion processes is proposed. This formulation, based on second order geometry and Ito integration on manifolds, allows us to give a natural and effective…
Let $X_t$ solve the multidimensional It\^o's stochastic differential equations on $\R^d$ $$dX_t=b(t,X_t)dt+\sigma(t,X_t)dB_t$$ where $b:[0,\infty)\times\R^d\to\R^d$ is smooth in its two arguments,…
We obtain Calder{\'o}n-Zygmund estimates for some degenerate equations of Kolmogorov type with inhomogeneous coefficients. We then derive the well-posedness of the martingale problem associated to related degenerate operators, and therefore…
We introduce a variational theory for processes adapted to the multi-dimensional Brownian motion filtration that provides a differential structure allowing to describe infinitesimal evolution of Wiener functionals at very small scales. The…
In this paper we study the Stratonovich stochastic differential equation $\mathrm{d} X=|X|^{\alpha}\circ\mathrm{d} B$, $\alpha\in(-1,1)$, which has been introduced by Cherstvy et al. [New Journal of Physics 15:083039 (2013)] in the context…
Determinantal process is a dynamical extension of a determinantal point process such that any spatio-temporal correlation function is given by a determinant specified by a single continuous function called the correlation kernel.…
For a real Borel measurable function b, which satisfies certain integrability conditions, it is possible to define a stochastic integral of the process b(Y) with respect to a Brownian motion W, where Y is a diffusion driven by W. It is well…