A system is controlled by a control variable. The control variable could be of finite or infinite dimensions. The roughness of control variable has determine the complexity of the system.The control variable could be very rough.  

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 g^i(.) commute.  The It\^o functional is defined as 

I: X \to Y,

dY_t = f(Y_t)dt+g(Y_t)dX_t ,

Y_0 = a.

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 t \to X_t is very rough. The reason is very simple: the differential dX_t in the usual sense does not make sense. However, the classical ODE theory can be built by using the increment process \{X_t - X_s: 0 \leq s\leq t\} of the path X 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 \{X_t -X_s: 0 \leq s\leq t\}as the ‘differential’ of dX_t if the path t \to X_t is of finite variation. However, if $latex t \to X_t$ is very rough, then the increment process \{X_t -X_s: 0 \leq s\leq t\}is not enough to capture the differential dX_t, one needs the higher terms. 

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

\{\int_{s<t_1<t_2<\cdots<t_k<t} dX_{t_1} \otimes\cdots\otimes dX_{t_k}: 0\leq s\leq t\}

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

X_{s,t}^k \equiv \int_{s<t_1<\cdots<t_k<t} dX_{t_1}\otimes\cdots\otimes dX_{t_k}.

If X is a path with finite variation then the first-order iterated path integral is nothing more than its increment process X_{s,t}^1 \equiv X_t -X_s, and the higher-order integrals can be obtained from

X_{s,t}^k=\lim_{m(D) \to 0}\sum_{l=1}^{m}\sum_{i=1}^{k-1}X_{s,t_{l-1}}^i\otimes X_{t_{l-1},t_l}^{k-i},

where the limit runs over all finite partition D of [s,t] as the mesh size tends to zero. This means that the increment process \{X_{s,t}^1: 0 \leq s\leq t\} determines all higher-order iterated path integrals, and hence the integrals of one-forms against the path X. This explains why one only need \{X_{s,t}^1: 0 \leq s\leq t\} in the classical smoothy-controlled ODE. 

The advantage of using tensor product V^{\otimes k} 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

X_{s,t} =\bigl(1, X_{s,t}^1,\cdots, X_{s,t}^k,\cdots\bigr)

and regard it as an element in the tensor algebra \oplus V^{\otimes k}. Then the additive property exactly means that

X_{s,t} \otimes X_{t,u} = X_{s,u}, 0 \leq s \leq t \leq u.

The above identity is called Chen’s identity.  However, it makes analysis hard if one is working with an infinite sequence X_{s,t} =\bigl(1, X_{s,t}^1,\cdots, X_{s,t}^k,\cdots\bigr), though i is not impossible.

The bounded variation of X, that is

\sup_D \sum_{l}|X_{t_{l-1},t_l}^k|^{1/k} <\infty, k =1,2,\cdots

yields that the higher-order integrals X^k are determined uniquely by X^1, 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 k=1, but they do satisfy the following weaker condition:

\sup_D \sum_{l}|X_{t_{l-1},t_l}^k|^{p} <\infty, for any p>2.

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

\sup_D \sum_{l}|X_{t_{l-1},t_l}^k|^{p/k} <\infty, \quad k =1,2,\cdots.  

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 V with norm |.| together with a sequence of tensor norms |.|_k on the algebraic tensor products V^{\otimes_a k}\equiv V\otimes_a\cdots\otimes_a V satisfying the following compatibility condition:

|\xi\otimes\eta|_{k+l}\leq|\xi|_k|\eta|_l, \quad\forall\xi\in V^{\otimes_a k},\quad\forall\eta\in V^{\otimes_a l}.

For each n\in\mathbb N, define the following (truncated) tensor algebra

T^{(n)}\bigl(V\bigr): T^{(n)}\bigl(V\bigr)=\sum_{k=0}^{n}\oplus V^{\otimes k},V^{\otimes 0} = R

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 \xi =\bigl(\xi^0, \xi^1,\cdots,\xi^n\bigr),\eta =\bigl(\eta^0,\eta^1,\cdots\eta^n\bigr) are two vectors in T^{(n)}\bigl(V\bigr), then

\zeta=\xi\otimes\eta\in T^{(n)}\bigl(V\bigr),

where its kth component is $latex\zeta^k =\sum_{j=0}^{k}\xi^j\otimes\eta^{k-j},\quad k=0,1,\cdots n$. Then norm |.| on T^{(n)}\bigl(V\bigr) is defined by

|\eta|=\sum_{i=0}^{n}|\xi^i|,\quad\hbox{if}\xi =\{\xi^0,\cdots,\xi^n\},

though different, but equivalent. For \xi, \eta\in T^{(n)}\bigl(V\bigr),

|\xi\otimes\eta| = \sum_{k=0}^{n}\biggl|\sum_{j=0}^{k}\xi^j \otimes\eta^{k-j}\biggr|

\leq\sum_{k=0}^{n}\sum_{i+j=k}|\xi^i\otimes\eta^j|

\leq\sum_{k=0}^{n}\sum_{i+j=k}|\xi^i||\eta^j|

=|\xi||\eta|. Use \Delta to denote the simplex \{(s,t): 0\leq s\leq t\leq T\}.

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

\omega(s,t)+\omega(t,u) \leq \omega(s,u),\quad\forall (s,t),(t,u)\in\Delta.

Definition 1 A continuous map X from the simplex \Delta into a truncated tensor algebra T^{(n)}\biggl(V \biggr),and written as

X_{s,t}=\biggl(X_{s,t}^0, X_{s,t}^1,\cdots,X_{s,t}^n\biggr), \quad\hbox{with} X_{s,t}^k\in V^{\otimes k},\quad\hbox{for any} (s,t)\in\Delta,

is called a multiplicative functional of degree n if X_{s,t}^0 \equiv 1 (for all (s,t)\in\Delta) and X_{s,t}\otimes X_{t,u}=X_{s,u},\quad\forall (s,t),(t,u)\in\Delta, where the tensor product \otimes is taken in T^{(n)}\bigl(V \bigr).

The Chen’s identity represents a basic requirement on any ‘continuous path’ in T^{(n)}\bigl(V \bigr) 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 p\geq 1 be a constant. We say that a map

X: \Delta\to T^{(n)}\bigl(V \bigr) possesses finite p-variation if  |X_{s,t}^i|\leq \omega(s,t)^{i/p},\quad\forall i=1,2,\cdots,n,\quad\forall (s,t)\in\Delta, for some control \omega.

 

Lemma 1 For \theta\in [0,1],let

F_{\theta}(t,x)=\frac{1}{t}\frac{A_\theta(t,x)}{\int_{0}^{1}A_\theta(t,u)du}, \quad t\geq\frac{1}{n}\quad\hbox{and}\quad x\in [0,1]. Then (L-\partial_t)F_\theta\geq 0,where L=\partial_x\bigl(x(1-x)\partial_x). Let F=\sum_{i=0}^{n} F_{i/n}(t,x).

Lemma 1 implies that  (L-\partial_t)F\geq 0. Applying the maximum principle to F, and using the fact that F(1/n,x)=n(n+1), we may conclude that

 F(p/n,x)=n(n+1),\quad\hbox{for all} p\geq 1. Hence,

\frac{n}{p}\sum_{i=0}^{n}\frac{x^{i/p}(1-x)^{(n-i)/p}}{\int_{0}^{1}x^{i/p}(1-x)^{(n-i)/p}dx}\leq n(n+1). by the fact that \int_{0}^{1}u^{\lambda-1}(1-u)^{\mu-1}du=\frac{\Gamma(\lambda)\Gamma(\mu)}{\Gamma(\lambda+\mu)},\quad\forall \lambda,\mu\in (0,\infty),

we have (Binomial inequality)

\sum_{i=0}^{n}\frac{(n/p)!x^{i/p}(1-x)^{(n-i)/p}}{(i/p)!\bigl((n-i)/p\bigr)!}\leq\frac{n+1}{n+p} p^2\leq p^2.

Definition 3 A multiplicative functional with finite p-variation in T^{([p])}(V) is called a rough path (of roughness p). We say that a rough path X in T^{([p])}(V) is controlled by \omega if 

|X_{s,t}^i|\leq\omega(s,t)^{i/p},\quad\forall i=1,\cdots,[p]\quad\hbox{and}\forall(s,t)\in\Delta.

The set of all rough paths with roughness p in T^{([p])}(V)will be denoted by \Omega_p(V). Hence, any rough path with roughness p in T^{([p])}(V) has a unique, canonical extension to a multiplicative functional in T^{(\infty)}(V) with finitep-variation. 

Almost rough path

Definition 4 Let p\geq 1 be a constant. A function X:\Delta\to T^{([p])}(V) is called an almost rough path (of roughness p) if it is of finite p-variation, X_{s,t}^0 =1 and, for some control \omega and some constant \theta>1, |\bigl(X_{s,t}\otimes X_{t,u}\bigr)^i -X_{s,u}^i|\leq \omega(s,u)^\theta, for all (s,t),(t,u)\in\Delta and i=1,\cdots,[p]. Rough path can be established based on almost rough path as shown in Theorem 3.2.1 in Lyons.

 

Definition 5 A function X\in C_0\bigl(\Delta, T^{(n)}(V)\bigr) is said to have finite total p-variation if  \sup_D\sum_l |X_{t_{l-1},t_l}|^{p/i}<\infty, \quad i=1,2,\cdots,n, where \sup_D runs over all finite divisions of [0,T] and, C_0\bigl(\Delta, T^{(n)}(V)\bigr) denote continuous function from the simplex \Delta into the truncated tensor algebra T^{(n)}(V),with an appropriate norm and with X_{s,t}^0\equiv 1.

The p variation metric d_p on C_{0,p}\bigl(\Delta, T^{([p])}(V)\bigr) is defined by d_p(X,Y)=\max_{1\leq i\leq [p]}\sup_D\Bigl(\sum_l|X_{t_{l-1},t_l}^i -X_{t_{l-1},t_l}^i|\Bigr)^{i/p}.

A rough path X\in\Omega_p(V) is called a smooth rough path if t\to X_t\equiv X_{0,t}^1 is a continuous path with finite variation and X_{s,t}^i is the ith iterated path integral of the path X_t over the interval [s,t] (for i=1,2,\cdots,[p]),that is

X_{s,t}^i=\int_{s<t_1<\cdots<t_i<t}dX_{t_1}\otimes\cdots\otimes dX_{t_i},\quad\forall (s,t)\in\Delta .

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

d_p\bigl(X(n),X\bigr)\to 0, \quad\hbox{as}\quad n\to\infty.

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