Math Help - Markov Chains & Transition Probabilities

1. Markov Chains & Transition Probabilities

hi there!

i do not understand how to calculate the transition probabilities for markov chains. can someone explain or provide links that could be useful to understand this better?

thanks

2. Hello,
Originally Posted by lekhabhatt81
hi there!

i do not understand how to calculate the transition probabilities for markov chains. can someone explain or provide links that could be useful to understand this better?

thanks
Well, transition probabilities are the probabilities associated to the changes of state.
You calculate them with the information you're given. And they are (must be ?) conditional probabilities.

The transition matrix is $[a_{i,j}]$, where $a_{i,j}=P(X_{n+1}=j|X_n=i)$
in the case of a non-homogenous Markov chain, the transition matrix varies according to n.
This transition matrix is made of the transition probabilities.

3. Question to calculate TP

Hi there,
this is the question for which I am supposed to calculate the transition probabilities and construct a one step transition probability matrix. Can you explain me how to go about constructing it?

"Simple Insurance Company starts at time 0 with a surplus of 3. At the end of every year, it collects a premium of 2. Every year, it pays a random claim amount as follows:

Claim Amount: 0 1 2 3
Prob. of Claim Amount: 0.15 0.25 0.40 0.20

If at the end of the year, Simple's surplus is more than 3, it pays a dividend equal to the amout of surplus in excess of 3.
If Simple is unable to pay its claims, or if its surplus drops to 0, it goes out of business.
Simple has no administration expenses and interest is equal to 0."

I hope someone can help me soon. I have an exam on this topic on tuesday.
Thanks a lot dear friends!

4. Drawing a diagram will help I think.

Start with state 3 (surplus of 3) and draw where you can go from there and indicate the probabilities, using:

Change in Surplus = 2 - Claim Amount

2 is the premium which the company always recieves at the end of the year.

Now, you should have created another three states - 2,4 & 5

Repeat this process for these new states, but now note that for states 4 & 5 you will have to use:

Change in Surplus = 2 - Claim Amount - Divedend Amount

The Divedend Amount is the current surplus minus 3

Now you should have states 0 through to 5. Just call state 0 "Out of Business" - it is the state that covers having zero surplus as well as having claims the company cannot pay (i.e. a negative surplus).

Repeat the process again for states 1 & 2. State "Out of Business" is recurrent.

Hopefully with that drawn out it will be easier to see how to calculate whatever transition probability you're after

5. Thanks a lot RanDo for your descriptive reply. I still have a little problem. How can you reach to states 2,4 & 5 from state 3? and how do you calculate the probabilities of each of the state? I am so sorry for not being able to understand. But I will be really obliged if you can explain this to me.