# What is martingale: Definition and 20 Discussions

A martingale is any of several designs of tack that are used on horses to control head carriage. Martingales may be seen in a wide variety of equestrian disciplines, both riding and driving. Rules for their use vary widely; in some disciplines they are never used, others allow them for schooling but not in judged performance, and some organizations allow certain designs in competition.
The two most common types of martingale, the standing and the running, are used to control the horse's head height, and to prevent the horse from throwing its head so high that the rider gets hit in the face by the horse's poll or upper neck. When a horse's head gets above a desired height, the martingale places pressure on the head so that it becomes more difficult or impossible to raise it higher.

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1. ### I Expected number of random variables that must be observed

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?
2. ### I Show that ##Y_{\infty}=0 ##

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} =...
3. ### I Martingale, Optional sampling theorem

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...
4. ### Using a Logarithmic Transformation for a Simpler Random Walk Model

Answer to 1. Answer to 2. How would you answer rest of the questions 4 and 5 ?
5. ### A Martingale, calculation of probability - Markov chain needed?

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...
6. ### MHB Finding out the sequence as Martingale

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...
7. ### B $20000 at$0.20 martingale betting

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...
8. ### Showing tha a random variable is a martingale

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) =...
9. ### MHB Brownian Motion: Martingale Property

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...
10. ### How do you prove that this is a Martingale

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)...
11. ### MHB Problem concerning martingale convergence theorem

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 $X_n$...
12. ### Martingale Betting System- expected value

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...
13. ### Martingale problem: help writing and equation using the martingale assumptions

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...
14. ### Stochastic differential of a particular martingale

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...
15. ### Difference between martingale and markov chain

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...
16. ### Understanding Wald's Equation: Exploring the Martingale Property

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!
17. ### Martingale = Independent Increments?

Here's a stupid question: for a Gaussian process, are these two properties equivalent?
18. ### Proving Martingale Property and Stopping Theorem for Probability Homework

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?
19. ### Is Martingale the Key to Success? (Attached File)

question in attached file. thanks in advance
20. ### Is Martingale difference sequence strictly stationary and ergodic?

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