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Paracontrolled distribution approach to stochastic Volterra equations
Abstract
Based on the notion of paracontrolled distributions, we provide existence and uniqueness results for rough Volterra equations of convolution type with potentially singular kernels and driven by the newly introduced class of convolutional rough paths. The existence of such rough paths above a wide class of stochastic processes including the fractional Brownian motion is shown. As applications we consider various types of rough and stochastic (partial) differential equations such as rough differential equations with delay, stochastic Volterra equations driven by Gaussian processes and moving average equations driven by Lévy processes.
... In [11] and [12] Deya and Tindel use ideas of Rough Path theory for the treatment of non-singular Volterra equations. Furthermore, in their recent work [28] Prömel and Trabs treat the first order case by use of paracontrolled calculus. ...
We extend the recently developed rough path theory for Volterra equations from [F. Harang and S. Tindel, Volterra equations driven by rough signals, Stoch. Process. Appl. 142 (2021) 34–78] to the case of more rough noise and/or more singular Volterra kernels. It was already observed in [F. Harang and S. Tindel, Volterra equations driven by rough signals, Stoch. Process. Appl. 142 (2021) 34–78] that the Volterra rough path introduced there did not satisfy any geometric relation, similar to that observed in classical rough path theory. Thus, an extension of the theory to more irregular driving signals requires a deeper understanding of the specific algebraic structure arising in the Volterra rough path. Inspired by the elements of “non-geometric rough paths” developed in [M. Gubinelli, Ramification of rough paths, J. Differential Equations 248 (2010) 693–721; M. Hairer and D. Kelly, Geometric versus non-geometric rough path, Ann. Inst. Henri Poincaré-Probab. Stat. 51 (2015) 207–251], we provide a simple description of the Volterra rough path and the controlled Volterra process in terms of rooted trees, and with this description we are able to solve rough Volterra equations driven by more irregular signals.
For stochastic evolution equations with fractional derivatives, classical solutions exist when the order of the time derivative of the unknown function is not too small compared to the order of the time derivative of the noise; otherwise, there can be a generalized solution in suitable weighted chaos spaces. Presence of fractional derivatives in both time and space leads to various modifications of the stochastic parabolicity condition. Interesting new effects appear when the order of the time derivative in the noise term is less than or equal to one-half.
A new paradigm recently emerged in financial modelling: rough (stochastic) volatility, first observed by Gatheral et al. in high-frequency data, subsequently derived within market microstructure models, also turned out to capture parsimoniously key stylized facts of the entire implied volatility surface, including extreme skews that were thought to be outside the scope of stochastic volatility. On the mathematical side, Markovianity and, partially, semi-martingality are lost. In this paper we show that Hairer's regularity structures, a major extension of rough path theory, which caused a revolution in the field of stochastic partial differential equations, also provides a new and powerful tool to analyze rough volatility models.
We introduce affine Volterra processes, defined as solutions of certain stochastic convolution equations with affine coefficients. Classical affine diffusions constitute a special case, but affine Volterra processes are neither semimartingales, nor Markov processes in general. We provide explicit exponential-affine representations of the Fourier-Laplace functional in terms of the solution of an associated system of deterministic integral equations, extending well-known formulas for classical affine diffusions. For specific state spaces, we prove existence, uniqueness, and invariance properties of solutions of the corresponding stochastic convolution equations. Our arguments avoid infinite-dimensional stochastic analysis as well as stochastic integration with respect to non-semimartingales, relying instead on tools from the theory of finite-dimensional deterministic convolution equations. Our findings generalize and simplify recent results in the literature on rough volatility models in finance.
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 L\'epingle'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.
We develop in this work a general version of paracontrolled calculus that allows to treat analytically within this paradigm a whole class of singular partial differential equations with the same efficiency as regularity structures. This work deals with the analytic side of the story and offers a toolkit for the study of such equations, under the form of a number of continuity results for some operators, while emphasizing the simple and systematic mechanics of computations within paracontrolled calculus, via the introduction of two model operations $\mathsf{E}$ and $\mathsf{F}$ . We illustrate the efficiency of this elementary approach on the example of the generalized parabolic Anderson model equation $$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+L)u=f(u)\unicode[STIX]{x1D701},\end{eqnarray}$$
on a 3-dimensional closed manifold, and the generalized KPZ equation $$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+L)u=f(u)\unicode[STIX]{x1D701}+g(u)(\unicode[STIX]{x2202}u)^{2},\end{eqnarray}$$
driven by a $(1+1)$ -dimensional space/time white noise.
Rough differential equations are solved for signals in general Besov spaces
unifying in particular the known results in H\"older and p-variation topology.
To this end the paracontrolled distribution approach, which has been introduced
by Gubinelli, Imkeller and Perkowski ["Paracontrolled distribution and singular
PDEs", Forum of Mathematics, Pi (2015)] to analyze singular stochastic PDEs, is
extended from H\"older to Besov spaces. As an application we solve stochastic
differential equations driven by random functions in Besov spaces and Gaussian
processes in a pathwise sense.
We investigate the asymptotic behavior as time goes to infinity of Hawkes
processes whose regression kernel has $L^1$ norm close to one and power law
tail of the form $x^{-(1+\alpha)}$, with $\alpha\in(0,1)$. We in particular
prove that when $\alpha\in(1/2,1)$, after suitable rescaling, their law
converges to that of a kind of integrated fractional Cox-Ingersoll-Ross
process, with associated Hurst parameter $H=\alpha-1/2$. This result is in
contrast to the case of a regression kernel with light tail, where a classical
Brownian CIR process is obtained at the limit. Interestingly, it shows that
persistence properties in the point process can lead to an irregular behavior
of the limiting process. This theoretical result enables us to give an
agent-based foundation to some recent findings about the rough nature of
volatility in financial markets.
Motivated by potential applications to fractional Brownian motion ([3]), we study Volterra stochastic differential of the form
$$
{X_t} = x + \int_0^t {K(t,s)} b(s,{X_S})ds + \int_0^t {K(t,s)} \sigma (s,{X_s})d{B_S}
$$ (E)
where (B
s
, s ∊ [0, 1]) is a one-dimensional standard Brownian motion and (K(t,s), t,s ∊ [0, 1]) is a deterministic kernel whose properties will be made precise below but for which we do not assume any boundedness property.
We develop a Fourier approach to rough path integration, based on the series
decomposition of continuous functions in terms of Schauder functions. Our
approach is rather elementary, the main ingredient being a simple commutator
estimate, and it leads to recursive algorithms for the calculation of pathwise
stochastic integrals, both of It\^o and of Stratonovich type. We apply it to
solve stochastic differential equations in a pathwise manner.
These are short notes from a series of lectures given at the University of
Rennes in June 2013, at the University of Bonn in July 2013, at the XVIIth
Brazilian School of Probability in Mambucaba in August 2013, and at ETH Zurich
in September 2013. We give a concise overview of the theory of regularity
structures as exposed in the article \cite{Regular}. In order to allow to focus
on the conceptual aspects of the theory, many proofs are omitted and statements
are simplified. We focus on applying the theory to the problem of giving a
solution theory to the stochastic quantisation equations for the Euclidean
$\Phi^4_3$ quantum field theory.
We extend the work of T. Lyons [T.J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (2) (1998) 215–310] and T. Lyons and Z. Qian [T. Lyons, Z. Qian, System Control and Rough Paths, Oxford Math. Monogr. Oxford Univ. Press, Oxford, 2002] to define integrals and solutions of differential equations along product of p and q rough paths, with 1/p+1/q>11/p+1/q>1. We use this to write an Itô formula at the level of rough paths, and to see that any rough path can always be interpreted as a product of a p-geometric rough path and a p/2p/2-geometric rough path.
This paper introduces the class of volatility modulated L\'{e}vy-driven
Volterra (VMLV) processes and their important subclass of L\'{e}vy
semistationary (LSS) processes as a new framework for modelling energy spot
prices. The main modelling idea consists of four principles: First,
deseasonalised spot prices can be modelled directly in stationarity. Second,
stochastic volatility is regarded as a key factor for modelling energy spot
prices. Third, the model allows for the possibility of jumps and extreme spikes
and, lastly, it features great flexibility in terms of modelling the
autocorrelation structure and the Samuelson effect. We provide a detailed
analysis of the probabilistic properties of VMLV processes and show how they
can capture many stylised facts of energy markets. Further, we derive forward
prices based on our new spot price models and discuss option pricing. An
empirical example based on electricity spot prices from the European Energy
Exchange confirms the practical relevance of our new modelling framework.
We introduce a new notion of "regularity structure" that provides an
algebraic framework allowing to describe functions and / or distributions via a
kind of "jet" or local Taylor expansion around each point. The main novel idea
is to replace the classical polynomial model which is suitable for describing
smooth functions by arbitrary models that are purpose-built for the problem at
hand. In particular, this allows to describe the local behaviour not only of
functions but also of large classes of distributions.
We then build a calculus allowing to perform the various operations
(multiplication, composition with smooth functions, integration against
singular kernels) necessary to formulate fixed point equations for a very large
class of semilinear PDEs driven by some very singular (typically random) input.
This allows, for the first time, to give a mathematically rigorous meaning to
many interesting stochastic PDEs arising in physics. The theory comes with
convergence results that allow to interpret the solutions obtained in this way
as limits of classical solutions to regularised problems, possibly modified by
the addition of diverging counterterms. These counterterms arise naturally
through the action of a "renormalisation group" which is defined canonically in
terms of the regularity structure associated to the given class of PDEs.
As an example of a novel application, we solve the long-standing problem of
building a natural Markov process that is symmetric with respect to the (finite
volume) measure describing the \Phi^4_3 Euclidean quantum field theory. It is
natural to conjecture that the Markov process built in this way describes the
Glauber dynamic of 3-dimensional ferromagnets near their critical temperature.
We introduce an approach to study certain singular PDEs which is based on
techniques from paradifferential calculus and on ideas from the theory of
controlled rough paths. We illustrate its applicability on some model problems
like differential equations driven by fractional Brownian motion, a fractional
Burgers type SPDE driven by space-time white noise, and a non-linear version of
the parabolic Anderson model with a white noise potential.
Ambit stochastics is the name for the theory and applications of ambit fields
and ambit processes and constitutes a new research area in stochastics for
tempo-spatial phenomena. This paper gives an overview of the main findings in
ambit stochastics up to date and establishes new results on general properties
of ambit fields. Moreover, it develops the concept of tempo-spatial stochastic
volatility/intermittency within ambit fields. Various types of volatility
modulation ranging from stochastic scaling of the amplitude, to stochastic time
change and extended subordination of random measures and to probability and
L\'{e}vy mixing of volatility/intensity parameters will be developed. Important
examples for concrete model specifications within the class of ambit fields are
given.
In this paper we study a class of stochastic differential equations with
additive noise that contains a fractional Brownian motion and a Poisson point
process of class (QL). The differential equation of this kind is motivated by
the reserve processes in a general insurance model, in which the long term
dependence between the claim payment and the past history of liability becomes
the main focus. We establish some new fractional calculus on the fractional
Wiener-Poisson space, from which we define the weak solution of the SDE and
prove its existence and uniqueness. Using a extended form of Krylov-type
estimate for the combined noise of fBM and compound Poisson, we prove the
existence of the strong solution, along the lines of Gy\"ongy and Pardoux
(1993). Our result in particular extends a recent work of Mishura-Nualart
(2004).
We show how to generalize Lyons' rough paths theory in order to give a pathwise meaning to some nonlinear infinite-dimensional evolution equations associated to an analytic semigroup and driven by an irregular noise. As an illustration, we apply the theory to a class of 1d SPDEs driven by a space-time fractional Brownian motion. oui
We define and solve Volterra equations driven by an irregular signal, by means of a variant of the rough path theory called algebraic integration. In the Young case, that is for a driving signal with H\"older exponent greater than 1/2, we obtain a global solution, and are able to handle the case of a singular Volterra coefficient. In case of a driving signal with H\"older exponent in (1/3,1/2], we get a local existence and uniqueness theorem. The results are easily applied to the fractional Brownian motion with Hurst coefficient H>1/3.
In this article, we illustrate the flexibility of the algebraic integration formalism introduced by M. Gubinelli (2004), by establishing an existence and uniqueness result for delay equations driven by rough paths. We then apply our results to the case where the driving path is a fractional Brownian motion with Hurst parameter H>1/3.
We exhibit a fundamental link between Hairer's theory of regularity structures [Hai14] and the paracontrolled calculus of [GIP15]. By using paraproducts we provide a Littlewood-Paley description of the spaces of modelled distributions in regularity structures that is similar to the Besov description of classical Hölder spaces.
This is an introduction of Besov spaces and Triebel--Lizorkin spaces.
Other related function spaces are included.
We consider rough paths with jumps. In particular, the analogue of Lyons' extension theorem and rough integration are established in a jump setting, offering a pathwise view on stochastic integration against càdlàg processes. A class of Lévy rough paths is introduced and characterized by a sub-ellipticity condition on the left-invariant diffusion vector fields and a certain integrability property of the Carnot-Caratheodory norm with respect to the Lévy measure on the group, using Hunt's framework of Lie group valued Lévy processes. Examples of Lévy rough paths include a standard multidimensional Lévy process enhanced with a stochastic area as constructed by D. Williams, the pure area Poisson process and Brownian motion in a magnetic field. An explicit formula for the expected signature is given.
It has been recently shown that rough volatility models, where the volatility is driven by a fractional Brownian motion with small Hurst parameter, provide very relevant dynamics in order to reproduce the behavior of both historical and implied volatilities. However, due to the non-Markovian nature of the fractional Brownian motion, they raise new issues when it comes to derivatives pricing. Using an original link between nearly unstable Hawkes processes and fractional volatility models, we compute the characteristic function of the log-price in rough Heston models. In the classical Heston model, the characteristic function is expressed in terms of the solution of a Riccati equation. Here we show that rough Heston models exhibit quite a similar structure, the Riccati equation being replaced by a fractional Riccati equation.
Stochastic Processes for Insurance and Finance offers a thorough yet accessible reference for researchers and practitioners of insurance mathematics. Building on recent and rapid developments in applied probability, the authors describe in general terms models based on Markov processes, martingales and various types of point processes. Discussing frequently asked insurance questions, the authors present a coherent overview of the subject and specifically address: The principal concepts from insurance and finance; Practical examples with real life data; Numerical and algorithmic procedures essential for modern insurance practices; Assuming competence in probability calculus, this book will provide a fairly rigorous treatment of insurance risk theory recommended for researchers and students interested in applied probability as well as practitioners of actuarial sciences.
We discuss stochastic calculus for large classes of Gaussian processes, based on rough path analysis. Our key condition is a covariance measure structure combined with a classical criterion due to Jain and Monrad [Ann. Probab. 11 (1983) 46–57]. This condition is verified in many examples, even in absence of explicit expressions for the covariance or Volterra kernels. Of special interest are random Fourier series, with covariance given as Fourier series itself, and we formulate conditions directly in terms of the Fourier coefficients. We also establish convergence and rates of convergence in rough path metrics of approximations to such random Fourier series. An application to SPDE is given. Our criterion also leads to an embedding result for Cameron–Martin paths and complementary Young regularity (CYR) of the Cameron–Martin space and Gaussian sample paths. CYR is known to imply Malliavin regularity and also Ito-like probabilistic estimates for stochastic integrals (resp., stochastic differential equations) despite their (rough) pathwise construction. At last, we give an application in the context of non-Markovian Hormander theory.
Existence, uniqueness and continuity properties of solutions of stochastic Volterra equations with singular integral kernels (driven by Brownian motion) are proven.
Let B be a fractional Brownian motion with Hurst index H(0,1). Denote by the positive, real zeros of the Bessel function J–H of the first kind of order –H, and let be the positive zeros of J1–H. In this paper we prove the series representation where X1,X2,... and Y1,Y2,... are independent, Gaussian random variables with mean zero and and the constant cH2 is defined by cH2=–1(1+2H) sin H. We show that with probability 1, both random series converge absolutely and uniformly in t[0,1], and we investigate the rate of convergence.
In this paper, we study the existence-uniqueness and large deviation estimate for stochastic Volterra integral equations with singular kernels in 2-smooth Banach spaces. Then we apply them to a large class of semilinear stochastic partial differential equations (SPDE), and obtain the existence of unique maximal strong solutions (in the sense of SDE and PDE) under local Lipschitz conditions. Moreover, stochastic Navier–Stokes equations are also investigated.
We formulate indefinite integration with respect to an irregular function as an algebraic problem which has a unique solution under some analytic constraints. This allows us to define a good notion of integral with respect to irregular paths with Hölder exponent greater than 1/3 (e.g. samples of Brownian motion) and study the problem of the existence, uniqueness and continuity of solution of differential equations driven by such paths. We recover Young's theory of integration and the main results of Lyons’ theory of rough paths in Hölder topology.
We consider the Cauchy problem for a stochastic delay differential equation driven by a fractional Brownian motion with Hurst parameter H>½. We prove an existence and uniqueness result for this problem, when the coefficients are sufficiently regular. Furthermore, if the diffusion coefficient is bounded away from zero and the coefficients are smooth functions with bounded derivatives of all orders, we prove that the law of the solution admits a smooth density with respect to Lebesgue measure on R.
Existence and uniqueness of solutions is established for stochastic Volterra integral equations driven by right continuous semimartingales. This resolves (in the affirmative) a conjecture of M. Berger and V. Mizel.
Stochastic Volterra equations are studied where the coefficients $F(t, s, x)$ are random and adapted to $\mathscr{F}_{s\vee t}$ rather than the customary $\mathscr{F}_{s\wedge t}$. Such a hypothesis, which is natural in several applications, leads to stochastic integrals with anticipating integrands. We interpret these as Skorohod integrals, which generalize Ito's integrals to the case where the integrand anticipates the future of the Wiener integrator. We shall nevertheless construct an adapted solution, which is even a semimartingale if the coefficients are smooth enough.
This paper aims to provide a systematic approach to the treatment of differential equations of the type dyt = Si fi(yt) dxti where the driving signal xt is a rough path. Such equations are very common and occur particularly frequently in probability where the driving signal might be a vector valued Brownian motion, semi-martingale or similar process. However, our approach is deterministic, is totally independent of probability and permits much rougher paths than the Brownian paths usually discussed. The results here are strong enough to treat the main probabilistic examples and significantly widen the class of stochastic processes which can be used to drive stochastic differential equations. (For a simple example see [10], [1]). We hope our results will have an influence on infinite dimensional analysis on path spaces, loop groups, etc. as well as in more applied situations. Variable step size algorithms for the numerical integration of stochastic differential equations [8] have been constructed as a consequence of these results.
Based on the notion of first order dyadic p-variation, we give a new characterization of Besov spaces for 0<s<1, 1<=p,q<=+[infinity] and s>1/p. We also give results in the case where p<1. Hence we provide simple tools that enable us to derive new regularity properties for the trajectories of various continuous time stochastic processes.
We prove the existence and uniqueness as well as the continuity of the solution to stochastic Volterra equations with singular kernels and non-Lipschitz coefficients. As application, we then study SDEs with fractional integrals.
We define and solve Volterra equations driven by an irregular signal, by means of a variant of the rough path theory allowing to handle generalized integrals weighted by an exponential coefficient. The results are applied to the fractional Brownian motion with Hurst coefficient greater than 1/3