Path dependent equations driven by Hölder processes (original) (raw)
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Nonlinear Young integrals and differential systems in Hölder media
Transactions of the American Mathematical Society, 2015
For Hölder continuous random field W ( t , x ) W(t,x) and stochastic process φ t \varphi _t , we define nonlinear integral ∫ a b W ( d t , φ t ) \int _a^b W(dt, \varphi _t) in various senses, including pathwise and Itô-Skorohod. We study their properties and relations. The stochastic flow in a time dependent rough vector field associated with φ ˙ t = ( ∂ t W ) ( t , φ t ) \dot \varphi _t=(\partial _tW)(t, \varphi _t) is also studied, and its applications to the transport equation ∂ t u ( t , x ) − ∂ t W ( t , x ) ∇ u ( t , x ) = 0 \partial _t u(t,x)-\partial _t W(t,x)\nabla u(t,x)=0 in rough media are given. The Feynman-Kac solution to the stochastic partial differential equation with random coefficients ∂ t u ( t , x ) + L u ( t , x ) + u ( t , x ) ∂ t W ( t , x ) = 0 \partial _t u(t,x)+Lu(t,x) +u(t,x) \partial _t W(t,x)=0 is given, where L L is a second order elliptic differential operator with random coefficients (dependent on W W ). To establish such a formula the main difficult...
On the Hölder Regularity of a Linear Stochastic Partial-Integro-Differential Equation with Memory
Journal of Fourier Analysis and Applications
In light of recent work on particles fluctuating in linear viscoelastic fluids, we study a linear stochastic partial-integro-differential equation with memory that is driven by a stationary noise on a bounded, smooth domain. Using the framework of generalized stationary solutions introduced in McKinley and Nguyen (SIAM J Math Anal 50(5):5119–5160, 2018), we provide conditions on the differential operator and the noise to obtain the existence as well as Hölder regularity of the stationary solutions for the concerned equation. As an application of the regularity results, we compare to analogous classical results for the stochastic heat equation. When the 1d stochastic heat equation is driven by white noise, solutions are continuous with space and time regularity that is Hölder (1/2-\epsilon )(1/2−ϵ)and( 1 / 2 - ϵ ) and(1/2−ϵ)and(1/4-\epsilon )$$ ( 1 / 4 - ϵ ) respectively. When driven by colored-in-space noise, solutions can have a range of regularity properties depending on the structure of the ...
Infinite Dimensional Analysis, Quantum Probability and Related Topics
Functional Itô calculus was introduced in order to expand a functional [Formula: see text] depending on time [Formula: see text], past and present values of the process [Formula: see text]. Another possibility to expand [Formula: see text] consists in considering the path [Formula: see text] as an element of the Banach space of continuous functions on [Formula: see text] and to use Banach space stochastic calculus. The aim of this paper is threefold. (1) To reformulate functional Itô calculus, separating time and past, making use of the regularization procedures which match more naturally the notion of horizontal derivative which is one of the tools of that calculus. (2) To exploit this reformulation in order to discuss the (not obvious) relation between the functional and the Banach space approaches. (3) To study existence and uniqueness of smooth solutions to path-dependent partial differential equations which naturally arise in the study of functional Itô calculus. More precisely...
2015
In this paper, we establish the existence of a stochastic flow of Sobolev diffeomorphisms R^d∋ x ϕ_s,t(x)∈R^d, s,t∈R for a stochastic differential equation (SDE) of the form dX_t=b(t,X_t) dt+dB_t, s,t∈R,X_s=x∈R^d. The above SDE is driven by a bounded measurable drift coefficient b:R×R^d→R^d and a d-dimensional Brownian motion B. More specifically, we show that the stochastic flow ϕ_s,t(·) of the SDE lives in the space L^2(Ω;W^1,p(R^d,w)) for all s,t and all p∈ (1,∞), where W^1,p(R^d,w) denotes a weighted Sobolev space with weight w possessing a pth moment with respect to Lebesgue measure on R^d. From the viewpoint of stochastic (and deterministic) dynamical systems, this is a striking result, since the dominant "culture" in these dynamical systems is that the flow "inherits" its spatial regularity from that of the driving vector fields. The spatial regularity of the stochastic flow yields existence and uniqueness of a Sobolev differentiable weak solution of the (...