martingale Definition and 20 Threads

In my opinion, answer to (a) is ## \mathbb{E} [N] = p^{-4}q^{-3} + p^{-2}q^{-1} + 2p^{-1} ## In answer to (b), XN is wrong. It should be XN=p-4q-3 - p-3 q-2- p-2 q-1 - p-1. This might be a typographical error. Is my answer to (a) correct?

Let ##X_1, X_2, \dots ##be independent, identically distributed random variables with ##P{X_j = 1} = q, P{X_j = −1} = 1 − q.## Let ##S_0 = 0 ##and for n ≥ 1, ##S_n = X_1 + X_2 + \dots + X_n.## Let ##Y_n = e^{S_n}## Let ##Y_{\infty} = \lim\limits_{n\to\infty} Y_n.## Explain why ##Y_{\infty} =...

In this exercise, we consider simple, nonsymmetric random walk. Suppose 1/2 < q < 1 and ##X_1, X_2, \dots## are independent random variables with ##\mathbb{P}\{X_j = 1\} = 1 − \mathbb{P}\{X_j = −1\} = q.## Let ##S_0 = 0## and ##S_n = X_1 +\dots +X_n.## Let ##F_n## denote the information...

Answer to 1. Answer to 2. How would you answer rest of the questions 4 and 5 ?

I am trying to estimate probability of loosing (probability of bankrupt ##Pb##) using Martingale system in betting. I will ilustrate my problem on the following example: Let: ##p## = probability of NOT getting a draw (in some match) We will use following system for betting: 1) We will bet only...

Consider the sequence $\{X_n\}_{n\geq 1}$ of independent random variables with law $N(0,\sigma^2)$. Define the sequence $Y_n= exp\bigg(a\sum_{i=1}^n X_i-n\sigma^2\bigg),n\geq 1$ for $a$ a real parameter and $Y_0=1.$ Now how to find the values of $a$ such that $\{Y_n\}_{n\geq 1}$ is martingale...

Hello. I'm watching a twitch streamer and he is currently on one of those CSGO betting websites, if you don't know what that is it's fine, it's not important. He is using a martingale system of betting and he started with $20,000 and is betting $0.20 each time. If he loses he doubles his bet...

I'm having a bit of a problem proving the second condition for a martingale, the discrete time branching process Z(n)=X(n)/m^n, where m is the mean number of offspring per individual and X(n) is the size of the nth generation. I have E[z(n)]=E[x(n)]/m^n=m^n/m^n (from definition E[X^n]=m^n) =...

Hi! I need some help at the following exercise... Let B be a typical brownian motion with μ>0 and x ε R. X_{t}:=x+B_{t}+μt, for each t>=0, a brownian motion with velocity μ that starts at x. For r ε R, T_{r}:=inf{s>=0:X_{s}=r} and φ(r):=exp(-2μr). Show that M_{t}:=φ(X_{t}) for t>=0 is...

So the following process involves W(t) which is Brownian Motion, and I need to prove that it is a martingale. Xt=log(1+W(t)2)-∫0t(1-W(s)2)/(1+W(s)2)2ds The problem I am having is the integral. My professor did a lot of integrals w.r.t. W(t), but he didn't do very many integrals where W(t)...

My goal is providing a proof based on martingale convergence theorems for the following fact: Series $S_n:=\sum\limits_{k=1}^n X_k$ of independenet random variables converges in distribution. Prove that $S_n$ converges almost certainly. I suppose these are not sufficent assuptions about $X_n$...

Hi, I have been learning/experimenting with the Martingale betting system recently. I have read a lot about how no "system" works for betting in casinos. However, I want to either prove or disprove the validity of the system by looking at its expected value/payout. I will be using the game of...

Hi, I have been given this problem and the solution however, neither make sense to me. x_t=\int exp(t-s)E(k_s|F_t)ds (the integral is from t to infinity) where k_u=m^d_u-(m^d_u)^*-α(y_u-y_u^*) for all u>0 suppose (m_t, t>0) is a martingale, what is an equation for x_t using...

Hello everyone, I'm studying from Oksendal's book, and I'm stuck at an excercise which asks you to find the differential form of: X(t) = (W(t)^{2}-t)^{2} - 4\int (W(s))^{2}ds where W(t) is a Brownian Motion. I tried several possible functions g(t,W(t)) which could have led to a potential...

What is the difference between martingale and markov chain. As it seems apparently, if a process is a martingale, then the future expected value is dependent on the current value of the process while in markov chain the probability of future value (not the expected value) is dependent on the...

Hi everyone. I was going through a proof of Wald's equation, where it was claimed that if {S_n} is a sequence defined as S_n = \sum_1^{n} Y_i where the Y_i are iid with finite mean \mu, then Z_n = S_n - n \mu is a martingale. But I don't see why... at all! Help!

Here's a stupid question: for a Gaussian process, are these two properties equivalent?

Homework Statement http://img128.imageshack.us/img128/2010/95701129si7.png The Attempt at a Solution If I define Z_n = \frac{X_n}{n+2} with Xn the number of white balls in stage n then how can I prove that it's martingale?

question in attached file. thanks in advance

Is Martingale difference sequence strictly stationary and ergodic? It seems to me that Martingale Difference Sequence is a special case of strictly stationary and ergodic sequences. Also, can somebody give me an example of strict stationarity without independence. Cheers