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. 

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

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). 

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 . 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

.

• 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. 

 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 .