In this paper, we prove two main results. The first one is to give a new condition for the existence of two-parameter $p, q$-variation path integrals. Our condition of locally bounded $p,q$-variation is more natural and easy to verify than those of Young. This result can be easily generalized to multi-parameter case. The second result is to define the integral of local time $\int_{-\infty}^\infty\int_0^t g(s,x)d_{s,x}L_s(x)$ pathwise and then give generalized It$\hat {\rm o}$'s formula when $\nabla^-f(s,x)$ is only of bounded $p,q$-variation in $(s,x)$. In the case that $g(s,x)=\nabla^-f(s,x)$ is of locally bounded variation in $(s,x)$, the integral $\int_{-\infty}^\infty\int_0^t \nabla^-f(s,x)d_{s,x}L_s(x)$ is the Lebesgue-Stieltjes integral and was used in Elworthy, Truman and Zhao \cite{Zhao}. When $g(s,x)=\nabla^-f(s,x)$ is of only locally $p, q$-variation, where $p\geq 1$,$q\geq 1$, and $2q+1>2pq$, the integral is a two-parameter Young integral of $p,q$-variation rather than a Lebesgue-Stieltjes integral. In the special case that $f(s,x)=f(x)$ is independent of $s$, we give a new condition for Meyer's formula and $\int_{-\infty}^\infty L_t(x)d_x\nabla^-f(x)$ is defined pathwise as a Young integral. For this we prove the local time $L_t(x)$ is of $p$-variation in $x$ for each $t\geq 0$, for each $p>2$ almost surely ($p$-variation in the sense of Lyons and Young, i.e. $\sup\limits_{E: \ a finite partition of [-N,N]} \sum\limits_{i=1}^m|L_t(x_i)-L_t(x_{i-1})|^p<\infty$).
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 Itô and of Stratonovich type. We apply it to solve stochastic differential equations in a pathwise manner.
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 cadlag processes. A class of Levy rough paths is introduced and characterized by a sub-ellipticity condition on the left-invariant diffusion vector fields and and a certain integrability property of the Carnot--Caratheodory norm with respect to the Levy measure on the group, using Hunt's framework of Lie group valued Levy processes. Examples of Levy rough paths include standard multi-dimen…
We consider a real-valued path; it is possible to associate a tree to this path, and we explore the relations between the tree, the properties of $p$-variation of the path, and integration with respect to the path. In particular, the fractal dimension of the tree is estimated from the variations of the path, and Young integrals with respect to the path, as well as integrals from the rough paths theory, are written as integrals on the tree. Examples include some stochastic paths such as martingales, Lévy processes and fractional Brownian motions (for which an estimator of the Hurst parameter is…
Integration by parts formulae are given for a class of measures on the space of paths of a smooth manifold $M$ determined by the laws of degenerate diffusions. The mother of such formulae, on the path space of diffeomorphism group of $M$ is shown to arise from a quasi-invariance property of measures determined by stochastic flows. From this the other formulae are derived by filtering out redundant noise using an associated LeJan-Watanabe connection.
For any real-valued stochastic process $X$ with cŕdlŕg paths we define non-empty family of processes which have locally finite total variation, have jumps of the same order as the process $X$ and uniformly approximate its paths on compacts. The application of the defined class is the definition of stochastic integral with semimartingale integrand and integrator as a limit of pathwise Lebesgue-Stieltjes integrals. This construction leads to the stochastic integral with some correction term (different from the Stratonovich integral). We compare the obtained result with classical results of Wong-…
We introduce a pathwise integration for Volterra processes driven by Lévy noise or martingale noise. These processes are widely used in applications to turbulence, signal processes, biology, and in environmental finance. Indeed they constitute a very flexible class of models, which include fractional Brownian and Lévy motions and it is part of the so-called ambit fields. A pathwise integration with respect of such Volterra processes aims at producing a framework where modelling is easily understandable from an information perspective. The techniques used are based on fractional calculus and in…
We define two new classes of stochastic processes, called tempered fractional Lévy process of the first and second kinds (TFLP and TFLP $I\!I$, respectively). TFLP and TFLP $I\!I$ make up very broad finite-variance, generally non-Gaussian families of transient anomalous diffusion models that are constructed by exponentially tempering the power law kernel in the moving average representation of a fractional Lévy process. Accordingly, the increment processes of TFLP and TFLP $I\!I$ display semi-long range dependence. We establish the sample path properties of TFLP and TFLP $I\!I$. We further use…
This paper deals with stochastic integrals of form $\int_0^T f(X_u)d Y_u$ in a case where the function $f$ has discontinuities, and hence the process $f(X)$ is usually of unbounded $p$-variation for every $p\geq 1$. Consequently, integration theory introduced by Young or rough path theory introduced by Lyons cannot be applied directly. In this paper we prove the existence of such integrals in a pathwise sense provided that $X$ and $Y$ have suitably regular paths together with some minor additional assumptions. In many cases of interest, our results extend the celebrated results by Young.
The goal of this paper is to define stochastic integrals and to solve stochastic differential equations for typical paths taking values in a possibly infinite dimensional separable Hilbert space without imposing any probabilistic structure. In the spirit of [33, 37] and motivated by the pricing duality result obtained in [4] we introduce an outer measure as a variant of the pathwise minimal superhedging price where agents are allowed to trade not only in $ω$ but also in $\intω\,dω:=ω^2 -\langle ω\rangle$ and where they are allowed to include beliefs in future paths of the price process express…
The paper develops a calculus for a class of real-valued functions having a quadratic variation. The main result is a solution of the representation problem for a class of evolutions having a quadratic variation. The result is applied to build up an asset pricing model. Also in the paper there are some results concerning an extension of the class of all semimartingales.
Two problems are addressed for the path of certain stochastic processes: a) do they define currents? b) are these currents of a classical type? A general answer to question a) is given for processes like semimartingales or with Lyons-Zheng structure. As to question b), it is shown that Hölder continuous paths with exponent $γ> 1/2$ define integral flat chains.
We study a pathwise integral with respect to paths of finite quadratic variation, defined as the limit of non-anticipative Riemann sums for gradient-type integrands. We show that the integral satisfies a pathwise isometry property, analogous to the well-known Ito isometry for stochastic integrals. This property is then used to represent the integral as a continuous map on an appropriately defined vector space of integrands. Finally, we obtain a pathwise 'signal plus noise' decomposition for regular functionals of an irregular path with non-vanishing quadratic variation, as a unique sum of a pa…
We formulate indefinite integration with respect to an irregular function as an algebraic problem and provide a criterion for the existence and uniqueness of a solution. This allows us to define a good notion of integral with respect to irregular paths with Hoelder 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 Hoelder topology.
Rough path analysis can be developed using the concept of controlled paths, and with respect to a topology in which Lévy's area plays a role. For vectors of irregular paths we investigate the relationship between the property of being controlled and the existence of associated Lévy areas. For two paths, one of which is controlled by the other, a pathwise construction of the Lévy area and therefore of mutual stochastic integrals is possible. If the existence of quadratic variation along a sequence of partitions is guaranteed, this leads us to a study of the pathwise change of variable (Itô) for…
In this paper, we prove a finite dimensional approximation scheme for the Wiener measure on closed Riemannian manifolds, establishing a generalization for $L^{1}$-functionals, of the approach followed by Andersson and Driver on [1]. We follow a new approach motived by categorical concept of colimit.
We construct a pathwise integration theory, associated with a change of variable formula, for smooth functionals of continuous paths with arbitrary regularity defined in terms of the notion of $p$-th variation along a sequence of time partitions. For paths with finite $p$-th variation along a sequence of time partitions, we derive a change of variable formula for $p$ times continuously differentiable functions and show pointwise convergence of appropriately defined compensated Riemann sums. Results for functions are extended to regular path-dependent functionals using the concept of vertical d…
For any real-valued stochastic process X with càdlàg paths we define non-empty family of processes, which have finite total variation, have jumps of the same order as the process X and uniformly approximate its paths: This allows to decompose any real-valued stochastic process with càdlàg paths and infinite total variation into a sum of uniformly close, finite variation process and an adapted process, with arbitrary small amplitude but infinite total variation. Another application of the defined class is the definition of the stochastic integral with respect to the process X as a limit of path…
We prove that typical (in the model-free finance setting) price paths with jumps may be uniformly approximated with accuracy $c>0$ by paths whose total variation is of order $1/c.$ A more precise result is obtained for semimartingales with jumps.
An integration by parts formula is the foundation for stochastic analysis on path spaces over a (finite dimensional) Riemannian manifold or over $R^n$, from which we may deduce the operator $d$ is closable and define the Laplacian operator on path spaces. A useful formula on the Riemannian manifold is $$dP_tf(v)=(1/t)E f(x_t) \int_0^t \langle d\{x_s\}, v_s\rangle ,$$ for $P_t$ the heat semi-group, $x_t$ the BM, $v_t$ the derivative flow or its conditional expectation (which is a damped parallel translation), $d\{x_s\}$ is the martingale part of $x_t$. As a meta theorem, this leads to the Clark…
This article gives an account on various aspects of stochastic calculus in the plane. Specifically, our aim is 3-fold: (i) Derive a pathwise change of variable formula for a path indexed by a square, satisfying some Hölder regularity conditions with a Hölder exponent greater than 1/3. (ii) Get some Skorohod change of variable formulas for a large class of Gaussian processes defined on the suqare. (iii) Compare the bidimensional integrals obtained with those two methods, computing explicit correction terms whenever possible. As a byproduct, we also give explicit forms of corrections in the resp…
We develop the rough path counterpart of Itô stochastic integration and - differential equations driven by general semimartingales. This significantly enlarges the classes of (Itô / forward) stochastic differential equations treatable with pathwise methods. A number of applications are discussed.
Using Vovk's outer measure, which corresponds to a minimal superhedging price, the existence of quadratic variation is shown for "typical price paths" in the space of càdlàg functions possessing a mild restriction on the jumps directed downwards. In particular, this result includes the existence of quadratic variation of "typical price paths" in the space of non-negative càdlàg paths and implies the existence of quadratic variation in the sense of Föllmer quasi surely under all martingale measures. Based on the robust existence of the quadratic variation, a model-free Itô integration is develo…
We consider a Markov process $X$ associated to a nonnecessarily symmetric Dirichlet form $\mathcal{E}$. We define a stochastic integral with respect to a class of additive functionals of zero quadratic variation and then we obtain an Itô formula for the process $u(X)$, when $u$ is locally in the domain of $\mathcal{E}$.
We develop two methods to reconstruct a path of bounded variation from its signature. The first method gives a simple and explicit expression of any axis path in terms of its signature, but it does not apply directlty to more general ones. The second method, based on an approximation scheme, recovers any tree-reduced path from its signature as the limit of a uniformly convergent sequence of lattice paths.
We prove that the model-free typical (in the sense of Vovk) càdlàg price paths with mildly restricted downward jumps possess quadratic variation which does not depend on the specific sequence of partitions as long as these partitions are obtained from stopping times such that the oscillations of a path on the consecutive (half-open on the right) intervals of these partitions tend (in a specified sense) to 0. Finally, we also define quasi-explicit, partition independent quantities which tend to this quadratic variation.
In this paper, we build the foundation for a theory of controlled rough paths on manifolds. A number of natural candidates for the definition of manifold valued controlled rough paths are developed and shown to be equivalent. The theory of controlled rough one-forms along such a controlled path and their resulting integrals are then defined. This general integration theory does require the introduction of an additional geometric structure on the manifold which we refer to as a "parallelism." The transformation properties of the theory under change of parallelisms is explored. Using these trans…
