Recent content by Jason4

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 $r>2$...

Not sure yet, just messing around with the equations at the moment (any tips are always appreciated).

Do I, for example, set: $\dot{r}\cos(\theta)-r\sin(\theta)\, \dot{\theta}=r(\text{sin}(\theta)-\text{cos}(\theta)(r^2-4))$ and solve from there?

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...

Oops, f) should have been: $E[e^{B_2+B_3}]$ Now only ten more questions! Suppose I should start a new thread.

(e) $ E(B_2 B_3)=E({B_2}^2))=(0)^2+2=2$ (c) $ E(e^{B_1})=e^{1/2}$ (f.) $E(e^{B_1+B_2})=e^{B_1}e^{B_2}=e^1e^{3/2}=e^{5/2}$

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...

I need to answer these questions, but I don't have a clue what they mean. Could anybody shed some light? Find: (a) $E({B_1^4})$ (b) $E({B_1^6})$ (c) $E(e^{B_1})$ (e) $E({5B_1}^4+{6B_1}^2+{5B_1}^3)$ (e) $E(B_2 B_3)$ (f.) $E(e^{B_2+B_3})$