# What is drift?

1. Oct 24, 2012

### woundedtiger4

In probability, What is drift? I have tried to google but unfortunately I can't find the definition :(
Does it mean a movement from one point to an other point, for example in symmetric random walk: lets say that ω=ω_1ω_2ω_3..... is the infinite sequence of coin tosses (p the pobability of Head "H" on each toss, and q=1-p the probability of Tail "T" on each toss, are both equal to 1/2).

Let,

X_j={+1 if ω_j=H, -1 ifω_j=T

and define M_0=0,

M_K= Ʃ(j=1, k) X_j , k=0,1,2,3,......

The process M_k, k=0,1,2,3,.... is a symmetric random walk.
In this example, can we say that when the position changes from M_1 to M_2 then that change is called as drift?

Edited note:
OK, I found the following definition on wiki:

In probability theory, stochastic drift is the change of the average value of a stochastic (random) process. A related term is the drift rate which is the rate at which the average changes. This is in contrast to the random fluctuations about this average value. For example, the process which counts the number of heads in a series of n coin tosses has a drift rate of 1/2 per toss.

Now I understand that what is drift rate but I am still clueless about stochastic drift (or just "drift").

Last edited: Oct 24, 2012
2. Oct 24, 2012

### chiro

Hey woundedtiger4.

One way of using drift (in financial scenarios) is to describe a changing mean over time for a particular stochastic process.

If the drift term is zero, then the mean over this time series should be relatively stable and un-changing.

Positive drift in this context means that the mean increases in time and a negative drift means it decreases much like a simple gradient.

3. Oct 24, 2012

### woundedtiger4

Thank you very much sir.

4. Oct 25, 2012

### andrewkirk

It is easier to understand in the context of the stochastic equation for the random walk.

Xt = Xt-1 + DR x Δt + V x εt x √Δt

Here Xt is the random variable in the stochastic process, DR is the drift rate and V is the instantaneous volatility.
εt is the random draw at time t that determines the next step
Δt is the time increment

If DR is constant then the drift from time t1 to time t2 is just DR x (t2-t1).
Otherwise, the drift is $\sum_{t=t1}^{t2}DR_t \times \Delta t$

the above formulas are for a discrete process. If it is continuous, they need to be replaced by a Stochastic Differential Equation, and integrals replace sums. It will be a Wiener Process.

5. Oct 26, 2012

### woundedtiger4

thank you sir.

can you please explain it intuitively (for discrete process)?

6. Oct 26, 2012

### ImaLooser

Sure. Drift comes from Brownian motion. If you put small pollen grains on a slide in water, they will jiggle due to random impacts from molecules. That's Brownian motion. Usually there is a steady current in the water due to heat or evaporation, so the grains move smoothly from left to right or whatever. That's drift.

For discrete processes, if there is a higher probability of moving left than right then the process will drift to the left.

7. Oct 27, 2012

### woundedtiger4

In random walk where a movement is unit walk either left or right, lets say that the current position is at M_0 (i.e. (0,0) on graph ) and the future position is M_1 (that could be either (1, 1) or (1, -1) on graph), so when the position was M_0 then at that time the position on y-axis could be 1 or -1 so that is drift?

Edited: Wiki says " stochastic drift is the change of the average value of a stochastic (random) process." so does it mean that drift (I am not talking about drift rate) is just change in position?

8. Oct 27, 2012

### chiro

Think about looking at the gradient relating to the rate of change of the mean over time just like the gradient m in the linear equation y = mx + b.

9. Oct 27, 2012

### alan2

For intuitive purposes I really like the pollen on the slide example given above. You could also think of a drunkard's walk downhill. Without the hill, the process is completely random. It is a random walk in two dimensions with mean position at the origin. Without the alcohol, but with the hill, the process is completely deterministic. Combine the two and you get a random walk which tends downhill because of the deterministic component of gravity (the drift). If you plot position vs. time for a completely random walk it looks like noise about zero. If you plot it with the drift term it looks like random noise about some non-zero function of time. I have a cat with vestibular syndrome (an inner ear abnormality). She always drifts to the right.

10. Oct 27, 2012

### woundedtiger4

excellent explanation

thank you so much sir

11. Oct 27, 2012

### woundedtiger4

very interesting explanation, does your cat have the problem in left & that's why she drifts to the right?

12. Oct 27, 2012

### alan2

Hmm. I'll ask the vet unless someone comes up with a good experimental procedure to determine this.

13. Oct 27, 2012

### woundedtiger4

Sorry, I am not a physicist so please correct me if I am wrong, when you say that "there is a steady current in the water due to heat or evaporation, so the grains move smoothly from left to right or whatever. That's drift."
then what is exactly drift? The steady current is drift or if I see the steady current as a force and if it pushes or pulls the pollens then that movement of pollens is drift?

The last part is much easier for me to understand although I am lil' confused that what if the probability of going right & left is equal then there's no drift, right?

14. Oct 27, 2012

### woundedtiger4

So in a probabilistic sense the drift is some force (mathematically, a function, am I correct?) that converts the stochastic process into a non-stochastic? What is noise? (loud irritating sounds?)

15. Oct 27, 2012

### andrewkirk

Say the current speed is d m/s Southward and in t seconds a pollen grain has moved x metres South.

Then the pollen grain's drift in that time is td Southward and non-drift movement is (x-td) Southward.

In other words, the motion of a grain can be regarded as consisting of two components, one due to drift and the other due to random motion. The two components add to the total movement.

Other terms you may see used for the non-drift movement are 'noise', 'random movement', 'Brownian motion' or 'peculiar motion' .

16. Oct 27, 2012

### alan2

I think that you're still confused and everyone is trying really hard to help here. Motion can be purely random or purely deterministic or it can have both components. The water moving on the slide is deterministic while the the motion of the pollen in the water is random. If you combine the two motions you have water flowing (drift) and in that water is pollen undergoing random motion. So the position of any grain of pollen is a combination of two motions which are additive. The resulting motion is still stochastic because one of its terms is stochastic.

17. Oct 28, 2012

### ImaLooser

The movement of the pollen grain that results from the steady current is drift. It happens because there is a greater constant probability of going right than left or something like that. If the probability is the same then there is zero drift.

18. Oct 28, 2012

### woundedtiger4

I am very thankful to you and all other members who are helping me to understand this topic, God bless all of you :)

19. Oct 28, 2012

### woundedtiger4

Sir, Thank you very much, I am 100% sure that now I understand this topic :)