Difference between martingale and markov chain

Click For Summary

Discussion Overview

The discussion focuses on the differences between martingales and Markov chains, exploring their definitions, properties, and relationships. Participants examine whether martingales are a subset of Markov processes and clarify the implications of their definitions in terms of expected values and dependencies on past observations.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant states that a martingale is a special kind of Markov process, noting that the future distribution of a Markov process depends only on the current state.
  • Another participant challenges this by suggesting that a martingale does not necessarily have to be a Markov process, providing an example of a sequence of random variables with normal distributions where variance could depend on the entire sequence.
  • A later reply mentions that a martingale can exhibit "memory," using the example of Brownian motion with stochastic, autoregressive variance (GARCH model), which would not qualify as a Markov process.
  • One participant agrees with the first statement regarding martingales being a subset of Markov processes but finds the second statement about the relationship between past observations and future expected values to be ambiguous.

Areas of Agreement / Disagreement

Participants express differing views on whether martingales are a subset of Markov processes, with some supporting this idea and others contesting it. The discussion remains unresolved regarding the precise relationship between the two concepts.

Contextual Notes

Participants highlight that the definitions of martingales and Markov processes may involve nuances, such as the role of past observations and the nature of dependencies, which could affect their classification.

ait.abd
Messages
24
Reaction score
0
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
 
Physics news on Phys.org
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.
 
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.
 
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
 

Similar threads

Graduate Martingale, calculation of probability - Markov chain needed?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad A seemingly simple problem about probability
  • · Replies 13 ·
Replies
13
Views
3K
Graduate Is this a correct observation about Markov-chains and stoc.processes?
  • · Replies 14 ·
Replies
14
Views
2K
Graduate Markov Chain aggregation method
  • · Replies 1 ·
Replies
1
Views
2K
Markov processes (Weight Suspended equally by n Cables)
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Markov Random Topological Spaces
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Bell vs Kolmogorov: Unravelling Probability Theory Limits
  • · Replies 93 ·
4
Replies
93
Views
8K
Graduate Poisson Martingales and Gambler's Ruin
  • · Replies 9 ·
Replies
9
Views
5K
Graduate Math Modeling with markov chains
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Role of future measurements on Bayes Filter
  • · Replies 3 ·
Replies
3
Views
1K