My Favourite Add-ons for Firefox

Firefox为插件提供了collection服务
可以把平时常用的插件一一收录在一起
一来为新手提供方便
二来重装ff时也不用一个一个插件重新去找了

以下是我的收藏～
https://addons.mozilla.org/collection/fish

Enjoy~

Stochastic Control and HJB Equations

Here is the formulation and rough ideas of solving another optimization problem in Stochastic Control.

For y=(s,x), define

J^u = E^y \left[ \int_0^{\tau_G} f^{u_t}(Y_t)dt + g(Y_{\tau_G})\chi_{\{\tau_G<\infty\}} \right].

The problem is: for each y\in G, to find \Phi(y) and a control u^* = u^*(t,\omega) = u^*(y,t,\omega) such that

\Phi(y) := \sup_{u(t,\omega)}J^u(y) = J^{u^*}(y).

Such a control u^* is called optimal control if it exists.

Here we only consider Markov controls, which means the value of u(t,\omega) at time t only depends on the state of the system at this time. Thus we can identify u(Y)=u(t,X_t).

The idea to solve this problem is very much the same as Variational Inequality. We just deal with J^u w.r.t. some fixed u here. Intuitively the "guess" function \Phi for any fixed v should satisfies

f^v + L^v\Phi \leq 0,

and for some suitable u we wish it will attain its maximum 0. This pretty much summarize the idea of the Hamilton-Jacobi-Bellman equation, which roughly says, under certain conditions,

\sup_{v\in U} \{f^v(y) + (L^v\Phi)(y)\}=0 for all y \in G,

and

\Phi(y) = g(y) for all y \in G.

The supremum is obtained for the best v, i.e., when v=u^*.

To solve for this kind of stochastic control problem, if it can be solved, one can starting with a guess of \phi of corresponding PDE. Find the critical points by taking the partial derivative w.r.t. v in a proper sense, and substitute the best v (u^*) into the original f^v + L^v\phi to find the form of \Phi. The problems is solved.

Unfortunately most HJB equations cannot be solved in an explicit way. Instead, one should turn to the existence and uniqueness of the solution of those HJB equations.

New Renderer for Latex Blogging

The previous latex renderer has some problems:
1) Cannot edit after rendered,
2) The server is unstable: the equations disappear from time to time,
3) Restriction on grammar is strange: even $$\tau\in T$$ can return error.

Fortunately there is a new alternative based on Javascript now (with instruction of installation):

http://doc.yourequations.com/

The disadvantages are
1) Require html code: J^u(y) is obtained by

<code lang="eq.latex">J^u(y)</code>

2) Alignment of text and mathematical symbols is not very nice, the same as for all other online renders.

Currently there is one way to make subtle position adjustments by adding style="vertical-align: 1pt;" inside the code tag, depending how far the current position deviates:

<code lang="eq.latex" style="vertical-align: 1pt;">J^u(y)</code>

Optimal Stopping and Variantional Inequalities

For a fixed domain G in \mathbb{R}^k and an Itō diffusion Y_k in \mathbb{R}^k defined by

dY_t = b(Y_t)dt + \sigma(Y_t)dB_t; ~~ Y_0=y.

Let f be profit function, g be terminal cost, both from \mathbb{R}^k to \mathbb{R}. Let \tau\in T, set of stopping times, define

J^\tau(y) = E^y\left[ \int_0^\tau f(Y_t)dt + g(Y_\tau) \right].

Our problem is to find the optimal stopping time \tau^* \in T such that

\Phi(y) = \sup_{\tao\in T} J^\tau(y) = J^{\tau^*}(y).

Now, suppose we can find some function \phi such that \phi(y)=\Phi(y) holds for all y\in G. How can we construct this \phi?

Step 1) Since \Phi is taking supremum over some set (some collection of stopping times), we begin our construction from this inequality

\Phi(y)\leq\phi(y).

Compare both sides:

\Phi(y) = \sup_\tau E^y\left[ \int_0^\tau f(Y_t)dt + g(Y_\tau) \right]

by definition of \Phi(y), and


\phi(y) = E^y \left[ \int_0^\tau -L\phi(Y_t)dt + \phi(Y_\tau) \right]


by applying Dynkin's formula to \phi(Y_\tau^y), where L is the partial differential operator coinciding with the generator A_Y of Y_t on C_0^2(\mathbb{R}^k).
Two conditions can be derived:

L + L \phi \leq 0 ~~ on ~~ G ~~~~~~~ (A)

g \leq \phi ~~ on ~~ G ~~~~~~~~~~~~~~~ (B)

Step 2) For the reversed inequality

\Phi(y)\geq\phi(y),

there are two cases based on condition (B).

2.1) For y\in G satisfying g(y)=\phi(y). Then

\phi(y) = g(y) = J^0(y) \leq \Phi(y).

In this case, \tau = 0.

2.2) For y\in G satisfying g(y)<\phi(y). Then apply Dynkin's formula again, we have

\phi(y) = E^y\left[ \int_0^\tau -L\phi(Y_t)dt + \phi(Y_{\tau_D}) \right].

Compare this with \Phi(y) above, we need two more conditions:

\phi = g ~~ on ~~ \partial D ~~~~~~~~~~~~~ (C)

f -L\phi \leq 0 ~~ on ~~ D ~~~~~~~ (D)

where D:=\{x\in D; \phi(x)>g(x)\}. Also, notice that (A) and (D) gives

f + L\phi \leq 0 ~~ on ~~ G.

In this case, \tau = \tau_D.

Combining all these (intuitive) conditions together, we may get the rough idea of how Theorem 10.4.1 (Variational Inequalities for Optimal Stopping) [Øksendal, SDE, 2003] formulates. The idea of this article goes to Dr. Yung and Joseph.

Introduction to LaTeX Blogging

Here is an example (with coding $$dX_t = adt + \sigma dBt$$):

Then is how-to:
1) Follow the instruction from this link -
http://wolverinex02.googlepages.com/emoticonsforblogger2

2) Edit greasemonkey script (here is how to edit: http://joyboner.com/how-to-edit-greasemonke-scripts/), change all four texts "http://www.forkosh.dreamhost.com/mathtex.cgi" to "http://www.problem-solving.be/cgi-bin/mathtex.cgi" (without quotes).

Enjoy guys :D

新的blog 新的开端

在此设立新的blog
开始另一段全新的旅程