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Expert help when you need it- Q1:1. Prove that if μ, v are invariant measures, then their convex combination àμ + (1-A)v, for λE(0, 1) is also an invariant measure. 2. Continuation of Example L2.45. In order to increase the system availability, it is proposed to add a duplicate repair facility so that both computers can be repaired simultaneously. Write down the transition probability matrix, and find availability of this system. 3. Consider the three-state Markov chain with transition matrix [1/5 4/5 01 P = 0 0 1 0 0 Prove that each state is ergodic. Find the mean recurrence times for each state. 4. Consider the Inventory Chain ExampleL2.47 a. Assume the inventory policy a=2, b=3. Find the long run P&L/day for this case. Consider the policy a=1, b=5. Find the transition probability for this inventory chain. Optional: Find the invariant measure, and the long run P&L/day. b.See Answer
- Q2:Problem L3.1. Consider the setting of Example L3.5. Using Theorem L3.12, prove that M₁ = S-n is a martingale.See Answer
- Q3:Problem L3.2. Let Y₁,..., Y, be iid r.v. such that p(0) := E[e] <∞o. Let S₂ = So + Y₁+...+Y₁₁. Prove that is a martingale wrt Yn. Hint: use Theorem L3.12. M₂ = exp(0Sn) p(0)"See Answer
- Q4:Problem L3.4. Prove the rest of Lemma L.3.7, i.e. the sub/super-martingale cases.See Answer
- Q5:Problem L3.4. Prove the rest of Lemma L3.7, i.e. the sub/super-martingale cases.See Answer
- Q6:Problem L3.5. Prove that if M₂, is a martingale, then, for any 0 ≤ i ≤j≤k<n, we have E[(Mn - Mk)Mj] = 0, and E[(M₂ - Mk) (Mj - M;)] = 0. These properties are known as orthogonality of martingale increments.See Answer
- Q7:Problem L3.6. Using Problem L3.5, prove that, for any martingale Mn, we have E[(M₁ - Mo)²] = ΣE[(Mk - Mk-1)²]. k=0See Answer
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