Differential equations driven by rough paths with jumps

Friz, Peter K.; Zhang, Huilin

doi:10.1016/j.jde.2018.01.031

Cited by 49 publications

(48 citation statements)

References 41 publications

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“…Here, * stands for one of the different ways to interpret a differential equation in the presence of discontinuities, which in general result in different solutions X . Two common choices (considered in the case q = p by Williams [44] and studied further in [6,7,13,15]) are…”

Section: Differential Equations With Càdlàg Driversmentioning

confidence: 99%

“…The construction of the forward solution for processes with infinitely many discontinuities is more involved, and can be achieved by solving directly the integral equation (3.4). This is done in [15] but is not required here.…”

Section: Remark 33mentioning

confidence: 99%

“…For general , consider the RDE Here, stands for one of the different ways to interpret a differential equation in the presence of discontinuities, which in general result in different solutions X . Two common choices (considered in the case by Williams [ 44 ] and studied further in [ 6 , 7 , 13 , 15 ]) are Geometric (Marcus) RDE. The solution is completely analogous to that of ( 2.2 ): we solve the continuous RDE , where is the linear path function on , and then remove the fictitious time intervals (note that the RDE is well-posed since by Chevyrev [ 6 , Corollary A.6]).…”

Section: Rough Path Formulationmentioning

confidence: 99%

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Superdiffusive limits for deterministic fast–slow dynamical systems

Chevyrev

Friz

Korepanov

et al. 2020

Probab. Theory Relat. Fields

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We consider deterministic fast–slow dynamical systems on $$\mathbb {R}^m\times Y$$ R m × Y of the form $$\begin{aligned} {\left\{ \begin{array}{ll} x_{k+1}^{(n)} = x_k^{(n)} + n^{-1} a\big (x_k^{(n)}\big ) + n^{-1/\alpha } b\big (x_k^{(n)}\big ) v(y_k), \\ y_{k+1} = f(y_k), \end{array}\right. } \end{aligned}$$ x k + 1 ( n ) = x k ( n ) + n - 1 a ( x k ( n ) ) + n - 1 / α b ( x k ( n ) ) v ( y k ) , y k + 1 = f ( y k ) , where $$\alpha \in (1,2)$$ α ∈ ( 1 , 2 ) . Under certain assumptions we prove convergence of the m-dimensional process $$X_n(t)= x_{\lfloor nt \rfloor }^{(n)}$$ X n ( t ) = x ⌊ n t ⌋ ( n ) to the solution of the stochastic differential equation $$\begin{aligned} \mathrm {d} X = a(X)\mathrm {d} t + b(X) \diamond \mathrm {d} L_\alpha , \end{aligned}$$ d X = a ( X ) d t + b ( X ) ⋄ d L α , where $$L_\alpha $$ L α is an $$\alpha $$ α -stable Lévy process and $$\diamond $$ ⋄ indicates that the stochastic integral is in the Marcus sense. In addition, we show that our assumptions are satisfied for intermittent maps f of Pomeau–Manneville type.

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Section: Differential Equations With Càdlàg Driversmentioning

confidence: 99%

Section: Remark 33mentioning

confidence: 99%

Section: Rough Path Formulationmentioning

confidence: 99%

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Superdiffusive limits for deterministic fast–slow dynamical systems

Chevyrev

Friz

Korepanov

et al. 2020

Probab. Theory Relat. Fields

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“…37]), followed by a proper stability analysis of the latter. The study of such non-canonical equations was recently carried out in [21]. Given a path (x, h) in R d+d ′ (smooth), its canonical level-2 rough path lift is given by (…”

Section: Continuity Of the Solution Mapmentioning

confidence: 99%

Canonical RDEs and general semimartingales as rough paths

Chevyrev¹,

Friz²

2019

Ann. Probab.

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110

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In the spirit of Marcus canonical stochastic differential equations, we study a similar notion of rough differential equations (RDEs), notably dropping the assumption of continuity prevalent in the rough path literature. A new metric is exhibited in which the solution map is a continuous function of the driving rough path and a so-called path function, which directly models the effect of the jump on the system. In a second part, we show that general multidimensional semimartingales admit canonically defined rough path lifts. An extension of Lépingle's BDG inequality to this setting is given, and in turn leads to a number of novel limit theorems for semimartingale driven differential equations, both in law and in probability, conveniently phrased via Kurtz-Protter's uniformly-controlled-variations (UCV) condition. A number of examples illustrate the scope of our results.

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“…Such integrals can be understood in the sense of Young [38]. This kind of integrals have been developed and widely used in the theory of differential equations by many authors, see e.g., [8,12,18,19,28,30].…”

Section: Introductionmentioning

confidence: 99%

Selection Properties and Set-Valued Young Integrals of Set-Valued Functions

Michta

Motyl

2020

Results Math

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The paper deals with some selection properties of set-valued functions and different types of set-valued integrals of a Young type. Such integrals are considered for classes of Hölder continuous or with bounded Young p-variation set-valued functions. Two different cases are considered, namely set-valued functions with convex values and without convexity assumptions. The integrals contain as a particular case set-valued stochastic integrals with respect to a fractional Brownian motion, and therefore, their properties are crucial for the investigation of solutions to stochastic differential inclusions driven by a fractional Brownian motion.

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Differential equations driven by rough paths with jumps

Cited by 49 publications

References 41 publications

Superdiffusive limits for deterministic fast–slow dynamical systems

Superdiffusive limits for deterministic fast–slow dynamical systems

Canonical RDEs and general semimartingales as rough paths

Selection Properties and Set-Valued Young Integrals of Set-Valued Functions

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