martingale & filtration

[Lun]

I'm studying stochastic calculus by myself

However, after reading many materials, even I have learnt Ito lemma, I still can't master what martingale & filtration are about

Can any one explain it ? I hope that the answer will not include any equation, what I need is sth intuitive. For equation, I have many books and many copy of the equations, but I can't master the concept behind, that's why I hope that someone can explain it to me with plain text instead of posting equations again at here. An illustrative example would be appreciated ! Thanks !

 

[igram]

I will try to explain just using words. This is a basic definition-
A martingale is a stochastic process with sequential variables where expected value of each variable is the same and independent of previous observations.
It was originally developed to demonstrate fairness of gambling games, example: if you want to prove that remembering the cards that had already been played in the previous hands of a card game wouldn't impact your expected winnings.

do you understand what a 'filtration' is/ definition of a 'measure' /'stopping times'? These concepts are the first things to learn when try to study martingales. Don't get bogged down by theorems try solving some examples.

 

[Sanket Patel]

How did you mange to learn Ito's Lemma without first understanding the concept of a martingale?

 

[bob]

The simplest way to think of a filtration is just as the accumulation of information. For a random variable that evolves in continuous time, a filtration adapted to that evolution is one where the entire history of the evolution, up to and including the present time, is known.

This sounds like a trivial thing, but imagine retrieving historical data on a stock price, for example. If you're only observing the stock price at the close each day, your filtration does not contain all the information that might conceivably be relevant, since it collapses all possible paths from one day to the next into a single event. This is an example of a subfiltration of the (theoretical) filtration adapted to the stock price process.

Subfiltrations like this are not always a handicap. In modeling the stochastic evolution of credit spreads, for example, it's useful to be able to define a subfiltration that has a "hole" in it at the time of any default event.

 

[Aditya Chitral]

Did you try to read Shreve?

 

[Lun]

Thank you for you all

I haven't been here for more than 2 years, people are still very helpful, and it seems that here has more members to interact.

With an equation, you can get the 1st derivative by applying the formula, but it doesn't mean that you have to understand what limit/delta is. THat's why I can learn Ito's Lemma without first understanding the concept of a martingale.

It's like driving a car, but without understanding how the engine works.

I know that it's very risky, that's why I'm now spending time on it.

I will spend some time on understanding the feedback from you all, and see if I can master the concept, thanks !!

 

[Lun]

I will try to explain just using words. This is a basic definition-
A martingale is a stochastic process with sequential variables where expected value of each variable is the same and independent of previous observations.
It was originally developed to demonstrate fairness of gambling games, example: if you want to prove that remembering the cards that had already been played in the previous hands of a card game wouldn't impact your expected winnings.

do you understand what a 'filtration' is/ definition of a 'measure' /'stopping times'? These concepts are the first things to learn when try to study martingales. Don't get bogged down by theorems try solving some examples.

what's the difference from a sequence of iid ? iid is independent form the past. iid is identical, so the expected value is the same

---------- Post added at 11:15 AM ---------- Previous post was at 11:09 AM ----------

The simplest way to think of a filtration is just as the accumulation of information. For a random variable that evolves in continuous time, a filtration adapted to that evolution is one where the entire history of the evolution, up to and including the present time, is known.

This sounds like a trivial thing, but imagine retrieving historical data on a stock price, for example. If you're only observing the stock price at the close each day, your filtration does not contain all the information that might conceivably be relevant, since it collapses all possible paths from one day to the next into a single event. This is an example of a subfiltration of the (theoretical) filtration adapted to the stock price process.

Subfiltrations like this are not always a handicap. In modeling the stochastic evolution of credit spreads, for example, it's useful to be able to define a subfiltration that has a "hole" in it at the time of any default event.

if "the entire history of the evolution" is the "filtration adapted to that evolution", then what's the application ?

say, I know all historical data on a stock price, then what can I do ?
or in this way, if I don't know all, what can't I do then ?

 

[Roger Trimble]

if "the entire history of the evolution" is the "filtration adapted to that evolution", then what's the application ?

say, I know all historical data on a stock price, then what can I do ?
or in this way, if I don't know all, what can't I do then ?

Here's one application:
Let's say you want to construct an optimal portfolio that incorporates views of the market. Let's also say your investment horizon is one month.
In order to develop views you build a factor model that estimates where the assets in your portfolio will be in one month. If you use historical monthly data dating back to 1960 you will obtain different estimators than if you use historical daily data for the past 2 years ( or less ). In this case you might obtain a better estimator if you only use more recent data.

 

[bob]

if "the entire history of the evolution" is the "filtration adapted to that evolution", then what's the application ?

say, I know all historical data on a stock price, then what can I do ?
or in this way, if I don't know all, what can't I do then ?

The question is wrongheaded--like demanding to know what the application of a compact vector space is. A filtration is part of the foundation required in order to discuss stochastic processes and be sure you're not screwing it up. I offered the stock price example as an illustration of the concept, not a justification for its utility.

As it turns out, there are a number of pitfalls in analysis of historical data that can be seen as using a filtration that's not adapted to the process in question.

A common example is survivorship bias in backtesting of trading algorithms. Suppose you wish to consider the performance of a given strategy that takes its universe as the stocks in the S&P 500. Any backtest that takes today's membership and then goes back in time to simulate the strategy using only those issues is invalid, since it is using future information (the membership of the S&P 500 today, which is conditioned on survival up to today) in testing past performance.

 

[igram]

what's the difference from a sequence of iid ? iid is independent form the past. iid is identical, so the expected value is the same

{Xn} is an iid sequence
Suppose {F(n)} is a filtration and {Xn} is "adapted" to {Fn}
then {Xn} would be a martingale if
E[Xn] < infinity and
E[xn+1/Fn] = Xn

So being an iid is more likely a prerequisite for a martingale, you can clearly see its a necessary but not a sufficient condition.