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Register a new userGeneral path integrals and stable SDEs
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The theory of one-dimensional stochastic differential equations driven by Brownian motion is classical and has been largely understood for several decades. For stochastic differential equations with jumps the picture is still incomplete, and even some of the most basic questions are only partially understood. In the present article we study existence and uniqueness of weak solutions to [ {rm d}Z_t=sigma(Z_{t-}){rm d} X_t ]driven by a (symmetric) $alpha$-stable Levy process, in the spirit of the classical Engelbert-Schmidt time-change approach. Extending and completing results of Zanzotto we derive a complete characterisation for existence und uniqueness of weak solutions for $alphain(0,1)$. Our approach is not based on classical stochastic calculus arguments but on the general theory of Markov processes. We proof integral tests for finiteness of path integrals under minimal assumptions.
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In this paper, we study (strong and weak) existence and uniqueness of a class of non-Markovian SDEs whose drift contains the derivative in the sense of distributionsof a continuous function.
We consider a stable driven degenerate stochastic differential equation, whose coefficients satisfy a kind of weak H{o}rmander condition. Under mild smoothness assumptions we prove the uniqueness of the martingale problem for the associated generator under some dimension constraints. Also, when the driving noise is scalar and tempered, we establish density bounds reflecting the multi-scale behavior of the process.
In this paper, we first prove that the existence of a solution of SDEs under the assumptions that the drift coefficient is of linear growth and path--dependent, and diffusion coefficient is bounded, uniformly elliptic and Holder continuous. We apply Gaussian upper bound for a probability density function of a solution of SDE without drift coefficient and local Novikov condition, in order to use Maruyama--Girsanov transformation. The aim of this paper is to prove the existence with explicit representations (under linear/super--linear growth condition), Gaussian two--sided bound and Holder continuity (under sub--linear growth condition) of a probability density function of a solution of SDEs with path--dependent drift coefficient. As an application of explicit representation, we provide the rate of convergence for an Euler--Maruyama (type) approximation, and an unbiased simulation scheme.
The path independence of additive functionals for SDEs driven by the G-Brownian motion is characterized by nonlinear PDEs. The main result generalizes the existing ones for SDEs driven by the standard Brownian motion.
We establish the existence of smooth densities for solutions to a broad class of path-dependent SDEs under a Hormander-type condition. The classical scheme based on the reduced Malliavin matrix turns out to be unavailable in the path-dependent context. We approach the problem by lifting the given $n$-dimensional path-dependent SDE into a suitable $L_p$-type Banach space in such a way that the lifted Banach-space-valued equation becomes a state-dependent reformulation of the original SDE. We then formulate Hormanders bracket condition in $mathbb R^n$ for non-anticipative SDE coefficients defining the Lie brackets in terms of vertical derivatives in the sense of the functional It^o calculus. Our pathway to the main result engages an interplay between the analysis of SDEs in Banach spaces, Malliavin calculus, and rough path techniques.
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