The experiemently measured trajectories of systmes modeled by ordinary differential equation (ODE) do not always give give good prediction. The ODE looks like

with ,

where is a given, smooth vector field and the solution is the trajectory

In reality, systems behave with some randomness. Hence, it is intuitive or reasonable to extend the ODE in some way to capture the random effects which disturbing the system.

Not finished post.

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**What is Vector Spaces?**

**What is Metric spaces?**

A function defined on pairs of points (x, y) from a set is called a metric if it has properties (i)–(iii) following. In that case, and together form a metric space.

(i) is equality if and only if ;

(ii) ;

(iii) $latex d(x,y)+d(y,z)\geq d(x,z)$.

The function is translation invariance, which means If x and y are two points and we translate them by adding to both, then their distance should not change: that is, .The second is that the metric should scale correctly, that is for a constant , such that .

**What is the Normed spaces & Banach spaces?**

**What is function spaces?**

A function space have functions with fixed domain and range as its elements (continuous function on [-1,1] etc,) and it is a normed space .

The norm of function in is a function space’s way to measure how large is. It is common, though not universal, for the norm to be defined by a simple formula and for the space to consist precisely of those functions for which the resulting definition makes sense and is finite. Thus, the mere fact that a function belongs to a function space can already convey some qualitative information about that function. For example, it may imply some regularity, decay, boundedness, or integrability on the function . The actual value of the norm makes this information quantitative. It may tell us how regular is, how much decay it has, by which constant it is bounded, or how large its integral is.

**Examples**

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There is a website dedicated to it. To promote this great publication, I have no mean to keep its website secret– PCM. Furthermore, You can get into this site with userid Guest and password PCM (at least it works for this moment, I hope it will be forever).

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The It\^o functional I is continuous in the topology of uniform convergence in the case of the control variable is one dimensional,but only continuous in the uniform topology in the special case when the vector field commute. The It\^o functional is defined as

,

,

.

But the It\^o map I is continuous in the topology of convergence in the metric of $latex p$-variation of rough paths even in the vector case.

The classical ODE theory collapse when the path is very rough. The reason is very simple: the differential in the usual sense does not make sense. However, the classical ODE theory can be built by using the increment process of the path to construct approximate solutions, and one may take the limit to obtain the solution without first identifying the differential $latex dX_t$. Therefore, one may regard the whole collection as the ‘differential’ of if the path is of finite variation. However, if $latex t \to X_t$ is very rough, then the increment process is not enough to capture the differential , one needs the higher terms.

The fundamental idea is that the full differential of is the collection of all iterated path integrals, namely

Suppose is a continuous path in, for example, a finite-dimensional vector space such that one is able to solve a differential equation , and therefore we at least hope to define the path integral $latex\int\alpha(X)dX$, for any smooth one-form , and so especially the iterated path integrals

If is a path with finite variation then the first-order iterated path integral is nothing more than its increment process and the higher-order integrals can be obtained from

,

where the limit runs over all finite partition of [s,t] as the mesh size tends to zero. This means that the increment process determines all higher-order iterated path integrals, and hence the integrals of one-forms against the path . This explains why one only need in the classical smoothy-controlled ODE.

The advantage of using tensor product is that one can easily express a basic requirement for any reasonable integration theory. That is the additive property of integrals over different intervals. More precisely, one could set

and regard it as an element in the tensor algebra . Then the additive property exactly means that

The above identity is called Chen’s identity. However, it makes analysis hard if one is working with an infinite sequence , though i is not impossible.

The bounded variation of , that is

yields that the higher-order integrals are determined uniquely by , but those paths in which we are interested rarely satisfy the BV condition. For example, the brownian motion paths does not satisfy the BV condition even for , but they do satisfy the following weaker condition:

for any .

Therefore, if is a such a non-smooth path which satisfies the above condition and if we are able to define its iterated path integrals , then it is reasonable to expect these iterated path integrals to satisfying the scaling control:

.

A rough path, roughly speaking, is such a continuous path for which we have an integration theory, and therefore from which a sequence of iterated path integrals may be constructed.

Given a Banach space with norm together with a sequence of tensor norms on the algebraic tensor products satisfying the following compatibility condition:

For each , define the following (truncated) tensor algebra

: .

Its multiplication (also called the tensor product) is the usual multiplication as polynomials, except that the higher-order terms are omitted. In other words, if are two vectors in , then

,

where its th component is $latex\zeta^k =\sum_{j=0}^{k}\xi^j\otimes\eta^{k-j},\quad k=0,1,\cdots n$. Then norm on is defined by

,

though different, but equivalent. For , ,

Use to denote the simplex .

A control is then a continuous, super-additive function on $\delta$ with values in $latex[0,\infty)$ such that . Therefore,

.

**Definition 1** A continuous map from the simplex into a truncated tensor algebra ,and written as

,

is called a multiplicative functional of degree n if (for all ) and , where the tensor product is taken in .

The Chen’s identity represents a basic requirement on any ‘continuous path’ in which has an integration theory. It is equivalent to the additive property of integrals over different intervals. However, Chen’s identity is purely algebraic. If we have an analytic condition would be nice.

**Definition 2** Let be a constant. We say that a map

possesses finite -variation if , for some control .

**Lemma 1 **For ,let

. Then ,where . Let .

Lemma 1 implies that . Applying the maximum principle to , and using the fact that , we may conclude that

. Hence,

. by the fact that ,

we have (**Binomial inequality**)

.

**Definition 3** A multiplicative functional with finite -variation in is called a **rough path** (of roughness p). We say that a rough path in is controlled by if

.

The set of all rough paths with roughness in will be denoted by . *Hence, any rough path with roughness in has a unique, canonical extension to a multiplicative functional in with finite-variation. *

**Almost rough path**

**Definition 4** Let be a constant. A function is called an almost rough path (of roughness p) if it is of finite -variation, and, for some control and some constant , , for all and . Rough path can be established based on almost rough path as shown in Theorem 3.2.1 in Lyons.

**Definition 5** A function is said to have finite total -variation if , where runs over all finite divisions of [0,T] and, denote continuous function from the simplex into the truncated tensor algebra ,with an appropriate norm and with .

The variation metric on is defined by .

A rough path is called a smooth rough path if is a continuous path with finite variation and is the th iterated path integral of the path over the interval [s,t] (for ),that is

.

**Definition 6 **Geometric rough paths with roughness are the rough paths in the closure of smooth rough paths under -variation distance (or, equivalently, under -variation topology). Thus, a rough path is a geometric rough path if there is a sequence of smooth rough paths in such that

.

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**Fractional calculus** is a branch of mathematical analysis that studies the possibility of taking real number powers of the differential operator

and the integration operator *J*. (Usually *J* is used in favor of *I* to avoid confusion with other *I*-like glyphs and identities)

In this context *powers* refer to iterative application or composition, in the same sense that *f*^{2}(x) = f(f(x)).

For example, one may pose the question of interpreting meaningfully

as a square root of the differentiation operator (an operator half iterate), i.e., an expression for some operator that when applied *twice* to a function will have the same effect as differentiation. More generally, one can look at the question of defining

for real-number values of *s* in such a way that when *s* takes an integer value *n*, the usual power of *n*-fold differentiation is recovered for *n* > 0, and the −*n*th power of *J* when *n* < 0.

There are various reasons for looking at this question. One is that in this way the semigroup of powers *D*^{n} in the *discrete* variable *n* is seen inside a *continuous* semigroup (one hopes) with parameter *s* which is a real number. Continuous semigroups are prevalent in mathematics, and have an interesting theory. Notice here that *fraction* is then a misnomer for the exponent, since it need not be rational, but the term*fractional calculus* has become traditional.

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This page is dedicated to gather informations on rough paths theory.

A bib file of the bibliography below.

- Ramification of rough paths. 2006. (arXiv)
- Rough solutions of the periodic Korteweg-de Vries equation. 2006. (arXiv)
- The burkholder-davis-gundy inequality for enhanced martingales. 2006. (arXiv)
- Euler estimates for rough differential equations. 2006.
- A note on the notion of geometric rough paths. 2004.
- On uniformly subelliptic operators and stochastic area. 2006. (arXiv)
- Curvilinear integrals along enriched paths. 2005.

- System Control and Rough Paths. Oxford University Press, 2002.

- Stochastic differential equations for fractional Brownian motions. C. R. Acad. Sci. Paris Sér. I Math., 331(1):75–80, 2000.
- Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields, 122(1):108–140, 2002.
- Differential equations driven by rough signals: an approach via discrete approximation. 2003.
- The evolution of a random vortex filament. Ann. Probab., 33(5):1825–1855, 2005. (arXiv)
- Random generation of stochastic area integrals. SIAM J. Appl. Math., 54(4):1132–1146, 1994.
- Variable step size control in the numerical solution of stochastic differential equations. SIAM J. Appl. Math., 57(5):1455–1484, 1997.
- Uniqueness for the Signature of a Path of Bounded Variation. 2002.
- Stochastic area for Brownian motion on the Sierpinski gasket. Ann. Probab., 26(1):132–148, 1998.
- Lévy area of Wiener processes in Banach spaces. Ann. Probab., 30(2):546–578, 2002.
- Large deviations and support theorem for diffusion processes via rough paths. 2002.
- Stochastic differential equations driven by a processes generated by divergence form operators. 2002.
- An introduction to rough paths. 2002.
- Smoothness of the Itô map on p-rough path spaces (I): 1 < p < 2. 2002.
- Differential equations driven by rough signals. I. An extension of an inequality of L. C. Young. Math. Res. Lett., 1(4):451–464, 1994.
- The interpretation and solution of ordinary differential equations driven by rough signals. In Stochastic analysis (Ithaca, NY, 1993), pages 115–128. Amer. Math. Soc., Providence, RI, 1995.
- Differential equations driven by rough signals. Rev. Mat. Iberoamericana, 14(2):215–310, 1998.
- On the importance of the Lévy area for systems controlled by converging stochastic processes. application to homogenization. 2002.
- System Control and Rough Paths. Oxford University Press, 2002.
- Calculus for multiplicative functionals, Itô’s formula and differential equations. In Itô’s stochastic calculus and probability theory, pages 233–250. Springer, Tokyo, 1996.
- A class of vector fields on path spaces. J. Funct. Anal., 145(1):205–223, 1997.
- Stochastic Jacobi fields and vector fields induced by varying area on path spaces. Probab. Theory Related Fields, 109(4):539–570, 1997.
- Flow equations on spaces of rough paths. J. Funct. Anal., 149(1):135–159, 1997.
- Calculus of variation for multiplicative functionals. In New trends in stochastic analysis (Charingworth, 1994), pages 348–374. World Sci. Publishing, River Edge, NJ, 1997.
- Flow of diffeomorphisms induced by a geometric multiplicative functional. Probab. Theory Related Fields, 112(1):91–119, 1998.
- On the limit of stochastic integrals of differential forms. In Stochastic processes and related topics (Siegmundsberg, 1994), pages 61–66. Gordon and Breach, Yverdon, 1996.
- The limits of stochastic integrals of differential forms. Ann. Probab., 27(1):1–49, 1999.
- Cubature on Wiener space. 2002.
- Conditional exponential moments for iterated Wiener integrals. Ann. Probab., 27(4):1738–1749, 1999.
- An introduction to rough paths. In Séminaire de Probabilités XXXVII, volume 1832 of Lecture Notes in Math., pages 1–59. Springer, Berlin, 2003.
- Controlling rough paths. J. Funct. Anal., 216(1):86–140, 2004.
- Approximations of the Brownian rough path with applications to stochastic analysis. Ann. Inst. H. Poincaré Probab. Statist., 41(4):703–724, 2005.
- Semi-martingales and rough paths theory. Electron. J. Probab., 10:no. 23, 761–785 (electronic), 2005.
- A pathwise view of solutions of stochastic differential equations. PhD thesis, University of Edinburgh, 1993.
- Diffeomorphic flows driven by Lévy processes. 2000.
- Path-wise solutions of stochastic differential equations driven by Lévy processes. Rev. Mat. Iberoamericana, 17(2):295–329, 2001.
- Solutions of differential equations driven by càdlàg paths of finite p-variation. PhD thesis, Imperial College, London, 1998.
- Young integrals and SPDEs. Pot. Anal., 2006. to appear. (arXiv)

- Formule de Taylor stochastique et développement asymptotique d’intégrales de Feynman. In Seminar on Probability, XVI, Supplement, pages 237–285. Springer, Berlin, 1982.
- Extending the Wong-Zakai theorem to reversible Markov processes. 2001.
- Flots et séries de Taylor stochastiques. Probab. Theory Related Fields, 81(1):29–77, 1989.
- Numerical methods for stochastic processes. John Wiley & Sons Inc., New York, 1994.
- Éléments de mathématique. Fasc. XXXVII. Groupes et algèbres de Lie. Chapitre II: Algèbres de Lie libres. Chapitre III: Groupes de Lie. Hermann, Paris, 1972.
- Order conditions of stochastic Runge-Kutta methods by B-series. SIAM J. Numer. Anal., 38(5):1626–1646 (electronic), 2000.
- The Lévy area process for the free Brownian motion. J. Funct. Anal., 179(1):153–169, 2001.
- Asymptotic expansion of stochastic flows. Probab. Theory Related Fields, 96(2):225–239, 1993.
- An efficient approximation method for stochastic differential equations by means of the exponential Lie series. Math. Comput. Simulation, 38(1-3):13–19, 1995.
- The ordinary differential equation approach to asymptotically efficient schemes for solution of stochastic differential equations. Ann. Inst. H. Poincaré Probab. Statist., 32(2):231–250,1996.
- Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula. Ann. of Math. (2), 65:163–178, 1957.
- Integration of paths—a faithful representation of paths by non-commutative formal power series. Trans. Amer. Math. Soc., 89:395–407, 1958.
- On maps of bounded p-variation with p > 1. Positivity, 2(1):19–45, 1998.
- Liens entre équations différentielles stochastiques et ordinaires. C. R. Acad. Sci. Paris Sér. A-B, 283(13):Ai, A939–A942, 1976.
- Liens entre équations différentielles stochastiques et ordinaires. Ann. Inst. H. Poincaré Sect. B (N.S.), 13(2):99–125, 1977.
- An introduction to p-variation and Young integrals – with emphasis on sample functions of stochastic processes. 1998.
- Differentiability of six operators on nonsmooth functions and p-variation, volume 1703 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1999.
- Calcul d’Itô sans probabilités. In Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980) (French), pages 143–150. Springer, Berlin, 1981.
- Série de Taylor stochastique et formule de Campbell-Hausdorff, d’après Ben Arous. In Séminaire de Probabilités, XXVI, volume 1526 of Lecture Notes in Math., pages 579–586. Springer, Berlin, 1992.
- Stochastic differential equations and diffusion processes. North-Holland Publishing Co., Amsterdam, 1981.
- Stochastic differential equations and diffusion processes. North-Holland Publishing Co., Amsterdam, second edition, 1989.
- Numerical solution of stochastic differential equations. Springer-Verlag, Berlin, 1992.
- Numerical solution of SDE through computer experiments. Springer-Verlag, Berlin, 1994.
- On the representation of solutions of stochastic differential equations. In Seminar on Probability, XIV (Paris, 1978/1979) (French), pages 282–304. Springer, Berlin, 1980.
- Wiener’s random function, and other Laplacian random functions. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, pages 171–187, Berkeley and Los Angeles, 1951. University of California Press.
- Processus stochastiques et mouvement brownien. Gauthier-Villars & Cie, Paris, 1965.
- Processus stochastiques et mouvement brownien. Éditions Jacques Gabay, Sceaux, 1992.
- Small perturbation of stochastic parabolic equations: a power series analysis. J. Funct. Anal., 193(1):94–115, 2002.
- Sur deux estimations d’intégrales multiples. In Séminaire de Probabilités, XXV, volume 1485 of Lecture Notes in Math., pages 425–426. Springer, Berlin, 1991.
- An algebraic theory of integration methods. Math. Comp., 26:79–106, 1972.
- Iterated path integrals. Bull. Amer. Math. Soc., 83(5):831–879, 1977.
- Collected papers of K.-T. Chen. Contemporary Mathematicians. Birkhäuser Boston Inc., Boston, MA, 2001.
- A Taylor formula for semimartingales solving a stochastic equation. In Stochastic differential systems (Visegrád, 1980), pages 157–164. Springer, Berlin, 1981.
- On a Taylor formula for a class of Itô processes. Probab. Math. Statist., 3(1):37–51 (1983), 1982.
- Free Lie algebras. The Clarendon Press Oxford University Press, New York, 1993.
- The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations. J. Funct. Anal., 72(2):320–345, 1987.
- On the gap between deterministic and stochastic ordinary differential equations. Ann. Probability, 6(1):19–41, 1978.
- Lie groups, Lie algebras, and their representations, volume 102 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1984.
- Stochastic differential equations and nilpotent Lie algebras. Z. Wahrsch. Verw. Gebiete, 47(2):213–229, 1979.
- An inequality of Hölder type connected with Stieltjes integration. Acta Math., (67):251–282, 1936.

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- for every ,
- (uniformly.)

Resolvent operator The family of linear operators associated with the L\’evy process, is called resolvent operators. The resolvent operators correspond to the Laplace transform of the semigroup . They are given for every measurable function by It is often more convenient to work with the resolvent operators than with the semigroup, thanks to the smoothing effect of the Laplace transform and to the lack of memory of exponential laws. Fourier Transform Absolute Continuity

Let (*X*, *d*) be a metric space and let *I* be an interval in the real line **R**. A function *f* : *I* → *X* is **absolutely continuous** on *I* if for every positive number , there is a positive number δ so that whenever a sequence of pairwise disjoint sub-intervals [*x*_{k}, *y*_{k}] of *I*, *k* = 1, 2, …, *n* satisfies

then

The collection of all absolutely continuous functions from *I* into *X* is denoted AC(*I*; *X*).

A further generalisation is the space AC^{p}(*I*; *X*) of curves *f* : *I* → *X* such that

for some *m* in the *L*^{p} space *L*^{p}(*I*; **R**). Potential measure It corresponds to the limit case for the -resolvent kernel. Specifically, we put for every , Transience and Recurrence

- We say that a L\’evy process is transient if the potential measures are Radon measures, that is , for every compact set K

- We say that a L\’evy process is recurrent if for every open ball centred at the origin.

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If p, then q.

is called an implication or a conditional statement.

Mathematicians have agreed that an implication will be called false only when the antecedent is true and the consequent is false. That is to say when P is true and q is false.

Function

It is important to note that this common notation is used only when .

Injective

A function is called injective (or one to one) if, for all implies that . An injective function is also referred to as an injection. (not necessary cover rng f).

Surjective

A function is called surjective (or one to one) if . A surjective function is also referred to as a surjection.

Bijective

A function is called bijective or bijection if it is both surjective and injective.

Inverse function

Let is bijective. The inverse function of is the function given by

Suppose that is any function. Then a function is called a

left inverse for if for all

right inverse for if for all

Cardinal numbers

Two sets S and T are called equinumerous, and we write S~T, if there a bijective function from S onto T.

A set S is said to be finite if or if there exists and a bijection . If a set is not finite, it is said to be infinite.

The cardinal number of is n, and if ~, we say that S has n elements. The cardinal number of is taken to be 0. If a cardinal number is not finite, it is called transfinite. (Cardinal number represents the size of a set.)

A set S is said to be denumerable if there exists a bijection If a set is finite or denumerable, it is called countable. If a set is not countable, it is uncountable. The cardinal number of a countable set is denoted by

Power set

Given any set S, let denote the collection of all the subsets of S. The set is called the power set of S.

Triangle inequality

Let , then

It is a direct observation of a triangle with the sum of length of two sides is greater than the remaining side, with inclusion of inequality for the case that when $x, y =0$.

Upper bound, lower bound, maximum and minimum

Let S be a subset of . If there exists a real number such that for all , then is called an upper bound for S. If for all , then is called an lower bound for S.

If an upper bound for S is a member of S, then is called a maximum of S, and we write .

If a lower bound for S is a member of S, then is called a minimum of S, and we write .

Supremum

If S is bounded above, then the least upper bound of S is called its supremum and is denoted by S . Thus S iff

(a)

and

(b) if , then there exists such that .

Completeness axiom

By this axiom, we made the fundamental difference between $latex\mathbb Q$ and . It states that

Every nonempty subset S of that is bounded above has a least upper bound. That is , S exists and is a real number.

Neighborhood

Let and let A neighborhood of is a set of the form

The number is referred to as the radius of

Deleted Neighborhood

Let and let A neighborhood of is a set of the form

Clearly,

Interior point and boundary point

Let . A point is an interior point of S if there exists a neighborhood of such that If for every neighborhood of , and , then is called a boundary point of S.

Closed Sets and Open Sets

Let . If then S is said to be closed. If then S is said to be open.

Cauchy sequence

A sequence is said to be a Cauchy sequence if for each there exists a number such that implies that .

Subsequence

Let be a sequence and let be any sequence of natural numbers such that The sequence is called a subsequence of .

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