马尔可夫链期望时间求解：从状态2到破产状态0的期望年数计算

Let's break this down step by step—this is a standard first-passage time problem for Markov chains, and we can solve it using expected value equations.

Step 1: Define our expected values #

Let’s denote:

• $E_0$: Expected years to reach state 0 (bankruptcy) starting from state 0. Since we’re already at the target state, $E_0 = 0$.
• $E_1$: Expected years to reach state 0 starting from state 1 (near bankruptcy).
• $E_2$: Expected years to reach state 0 starting from state 2 (solvent)—this is the value we need to find.

Step 2: Set up the expected value equations #

For any state $i$, the expected time to reach state 0 is equal to 1 year (for the current transition) plus the weighted average of the expected times from each state we might transition to.

Looking at the transition matrix $P= \begin{pmatrix} 1 & 0 & 0 \ 1/2 & 1/4 & 1/4 \ 1/2 & 1/4 & 1/4 \end{pmatrix}$:

• For state 1: From here, we have a 1/2 chance to go directly to state 0, 1/4 chance to stay in state 1, and 1/4 chance to move to state 2. So:
  $$E_1 = 1 + \left(\frac{1}{2} \times E_0\right) + \left(\frac{1}{4} \times E_1\right) + \left(\frac{1}{4} \times E_2\right)$$
• For state 2: Notice the transition probabilities from state 2 are identical to state 1! So its equation is the same:
  $$E_2 = 1 + \left(\frac{1}{2} \times E_0\right) + \left(\frac{1}{4} \times E_1\right) + \left(\frac{1}{4} \times E_2\right)$$

Step 3: Simplify and solve the equations #

First, substitute $E_0 = 0$ into both equations:

1. For $E_1$:
   $$E_1 = 1 + 0 + \frac{1}{4}E_1 + \frac{1}{4}E_2$$
   Rearrange terms to isolate the variables:
   $$E_1 - \frac{1}{4}E_1 - \frac{1}{4}E_2 = 1$$
   $$\frac{3}{4}E_1 - \frac{1}{4}E_2 = 1$$
   Multiply through by 4 to eliminate fractions:
   $$3E_1 - E_2 = 4 \tag{1}$$

2. For $E_2$:
   $$E_2 = 1 + 0 + \frac{1}{4}E_1 + \frac{1}{4}E_2$$
   Rearrange terms:
   $$E_2 - \frac{1}{4}E_2 - \frac{1}{4}E_1 = 1$$
   $$\frac{3}{4}E_2 - \frac{1}{4}E_1 = 1$$
   Multiply through by 4:
   $$3E_2 - E_1 = 4 \tag{2}$$

Now we have a system of two equations. Since the transition probabilities for state 1 and 2 are identical, we can also observe that $E_1 = E_2$—this simplifies things even more. Let's substitute $E_1 = E_2 = E$ into equation (1):
$$3E - E = 4$$
$$2E = 4$$
$$E = 2$$

Final Result #

The expected number of years to reach bankruptcy (state 0) starting from state 2 is 2.

内容的提问来源于stack exchange，提问作者eyesima