I've been at this for ages but I can't make sense of it. Can somebody help me out?Use Ito's Lemma to solve the stochastic differential equation:
X_t=2+\int_{0}^{t}(15-9X_s)ds+7\int_{0}^{t}dB_s
and find:
E(X_t)
Yeah I see I got that backwards.
So when $a$ is positive, paths spiral out from $(0,0)$. When $a$ is negative, paths spiral in toward $(0,0)$, and the limit cycle is unstable?
Problem:
The following two-dimensional system of ODEs possesses a limit-cycle solution for certain values of the parameter$a$. What is the nature of the Hopf bifurcation that occurs at the critical value of $a$ and state what the critical value is.
$\dot{x}=-y+x(a+x^2+(3/2)y^2)$...
I ended up just solving the two equations:
$\dot{r}=r(4-r^2)$ and $\dot{\theta}=-1$
------------------------------
I found three non-equilibrium solutions:
$r=2$ and $\theta=-1$
When $r<2$, $\dot{r}>0$, so $r$ increases and solutions move out toward the $r=2$ circle.
When $r>2$...
I have:
$\dot{x}=4x+y-x(x^2+y^2)$
$\dot{y}=4y-x-y(x^2+y^2)$
And I need to find $\dot{r}$ and $\dot{\theta}$
I got as far as:
$\dot{x}=r(\text{sin}(\theta)-\text{cos}(\theta)(r^2-4))$
$\dot{y}=r(-\text{sin}(\theta)(r^2-4)-\text{cos}(\theta))$
How do I go from here to $\dot{r}$ and...
Well my notes say:
The limiting distribution for a doubly stochastic is the uniform distribution over the state space, i.e.
$\boldsymbol\pi=\left(1/n,...,1/n\right)$ for an $n\times n$ matrix.
So I assume that if both the columns and the rows sum to 1, you don't have to solve any equations...
I can find the stationary distribution vector $\boldsymbol\pi$ for a stochastic matrix $P$ using:
$\boldsymbol\pi P=\boldsymbol\pi$, where $\pi_1+\pi_2+\ldots+\pi_k=1$
However, I can't find a textbook that explains how to do this for a doubly stochastic (bistochastic) matrix. Could somebody...
Consider:
$P=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)$
Show that $P^n$ has no limit, but that: $A_n=\frac{1}{n+1}(I+P+P^2+\ldots+P^n)$ has a limit.
I can see that $P^{EVEN}=\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)$ and $P^{ODD}=\left( \begin{array}{cc} 0...
1) For each $t$, find $P(B_t\neq 1)$ .
2) For any $T>0$, prove $V_t=B_{t+T}-B_T$ is a Weiner process.
...
For 2) should I be looking at something like this:
Let: $\Delta (t+T)=\frac{t}{N}$ for large $N$
$\Rightarrow...
Okay, I think I see.
$B_1\sim\textbf{N}(0,1)$
So: $E({B_1})=E(X)=0$, $E({B_1}^2)=E(X^2)=1$, etc.(a) $E({B_1^4})=E(X^4)=3$
(b) $E({B_1^6})=E(X^6)=15$
(e) $E({5B_1}^4+{6B_1}^2+{5B_1}^3)=5E(X^4)+6E(X^2)+5E(X^3)=5(3)+6(1)+5(0)=21$
I'm absolutely perplexed. Could you work through one of the problems so I get an idea of how to do the other problems? (I promise this isn't a homework assignment; it's an exercise sheet that is... meant to help familiarize me with the "basics.") I'm looking through a zillion stochastic...