Difference between martingale and markov chain

[ait.abd]

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 current value only. Are the following true?
1. Martingale is a subset of markov processes because there can be many markov processes whose expected future value is not equal to the current value.
2. If martingale is strictly a markov process then the only difference is that in a markov process we relate the future probability of a value to past observations while in a martingale we relate the future expected value given all past observations.

If I am unable to explain my confusion, please elaborate generally what are the main differences between these two.

Thanks

 

[mathman]

You seem to have the correct idea. A martingale is a special kind of Markov process. As you appear to understand the distribution function of the future of a Markov process is dependent only on the current state, and independent of previous states. Also as you know a martingale includes in its definition that the expectation of future value is the current value.
The first of your two statements is true. The second is a little ambiguous. "Past" in the definition should mean the latest known condition, but not anything before that.

 

[mathman]

I looked at the definition of martingale carefully and it seems to me that it does NOT have to be a Markov process. For example consider a sequence of random variables, each of which has a normal distribution with some mean and variance. To be a martingale, the mean of each variable has to be the value of the previous variable. However the variance could depend on the entire sequence up to that point, so it would not be a Markov process.

 

[BWV]

A martingale can have "memory" you could take a brownian motion with stochastic, autoregressive variance (i.e. a GARCH model) that would not be a Markov process but could still be a martingale

 

[blue_raver22]

for the question! I am happy to provide some clarification on the difference between martingale and markov chain.

First, let's define these two terms. A martingale is a type of stochastic process, meaning it is a random process that evolves over time. However, a key characteristic of a martingale is that its expected future value is equal to its current value, given all past observations. This means that the process has no trend or bias and is essentially unpredictable.

On the other hand, a Markov chain is a type of stochastic process where the probability of transitioning from one state to another is dependent only on the current state and not on any previous states. This means that the future value of the process is not determined solely by the current value, but by the probabilities of transitioning between states.

Now, let's address the statements in the content. The first statement is true - martingale is a subset of Markov processes because there can be many Markov processes whose expected future value is not equal to the current value. This is because a Markov process can have a trend or bias, while a martingale does not.

The second statement is also true - if a martingale is strictly a Markov process, then the only difference is that in a Markov process, we relate the future probability of a value to past observations, while in a martingale, we relate the future expected value given all past observations. Essentially, a martingale is a special case of a Markov process where the expected future value is equal to the current value.

In general, the main differences between these two can be summarized as follows:

1. Expected Future Value: In a martingale, the expected future value is equal to the current value, given all past observations. In a Markov chain, the future value is not necessarily equal to the current value, but is determined by the probabilities of transitioning between states.

2. Dependence on Past Observations: In a martingale, the expected future value is dependent on all past observations, while in a Markov chain, the probability of future values is only dependent on the current state.

3. Bias or Trend: A martingale has no trend or bias, while a Markov chain can have a trend or bias.

I hope this helps to clarify the differences between martingale and Markov chain. Both are important concepts in stochastic processes and have various applications in different