The following approach of deriving Ito’s Formula is in the spirit of Evans handout ‘An introduction to stochastic differential equations’. His approach is intuitive at the cost of some omission of detail and precision.

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

$X^. (t) = b(X(t)) (t>0)$ with $X(0) = x_0$,

where $b: R^n\to R^n$ is a given, smooth vector field and the solution is the trajectory $X(\dot):[0,\infty)\to R^n.$

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.

There are several well defined abstract spaces contrast the familiar Euclidean space, namely Metric spaces, Normed spaces, Function spaces, Hilbert spaces and Banach spaces,  etc. To give a overview of these abstract spaces and potential connections between these spaces. This post is dedicated to that purpose. The post is based on Terrence Tao‘s article ‘Function Spaces’ , Timothy Gowers‘s ‘Normed space and banach spaces’ and ‘Metric spaces’, which are published in Princeton Companion to the Mathematics (PCM). About PCM, I have a short post for it or even much better link for further information. Read the rest of this entry »

This is fabulas book. The editor-in-Chief is  Timothy Gowers who need no introduction and June Barrow-Green is an assistant editor (with particular expertise in the history of mathematics). The central focus of the book will be to describe modern pure mathematics, in all its diversity, in a way that is serious, sometimes quite detailed, but always accessible at the lowest possible level. It is very helpful for person like me in particular, who love math and finding ways to study math seriously, as the book take each topic very seriously, trying to be understandable even to the beginner (fundamental definitions are explained clearly with follow up examples) and explanations are given in considerably detail.

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).

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, Read the rest of this entry »

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

$D = \frac{d}{dx} \,$

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

## Rough paths

bib file of the bibliography below.

Markov SemigroupWe consider the family of convolution operators on $L^{\infty}(\mathbb R^d)$ indexed by $t \geq 0$ and given for every $f \in L^{\infty}(\mathbb R^d)$ by$P_t f(x)=E_x(f(X_t))=\int_{R^d} f(x+y)\mathbb P(X_t\in dy)$ The Semigroup $(P_t, t\geq 0)$ has the Feller property, that is for every $f \in l_0$:

• $P_t f \in l_0$ for every $t\geq 0$,
• $lim_{t\to 0} P_t f= f$ (uniformly.) Read the rest of this entry »

Implication A statement of the form :

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.

The following link is an abstract of the book by A D Aleksandrov on his view on Mathematics. A D Aleksandrov’s view of Mathematics Which introduce the methods, contents and meaning of mathematics.

• How much we know about the related topic which we are going to build new theory on.
• What information are possibly useful for proving or disproving a new idea. Read the rest of this entry »