Help exercise Exercise 2.4. Toss a coin repeatedly. Assume the probability of head on eachtoss is , as is the probability of tail. Let X; = 1 if the jth toss results ina head and X; = -1 if the jth toss results in a tail. Consider the stochasticprocess Mo, M, M2,.. defined by Mo = 0 andM, =X;, n> 1.j=1This is called a symmetric random walk, with each head, it steps up one, andwith each tail, it steps down one.(i) Using the properties of Theorem 2.3.2, show that Mo, M1, M2, . .. is amartingale.(ii) Let o be a positive constant and, for n > 0, define2Sneo +e-oShow that So, Sı, S2, . .. is a martingale. Note that even though the sym-metric random walk Mn has no tendency to grow, the "geometric sym-metric random walk" eon does have a tendency to grow. This is theresult of putting a martingale into the (convex) exponential function (seeExercise 2.3). In order to again have a martingale, we must "discount" thegeometric symmetric random walk, using the termrate. This term is strictly less than one unless o = 0.2eo te-as the discount

Name: Help exercise Exercise 2.4. Toss a coin repeatedly. Assume the probabi
Uploaded: 2024-03-05T11:52:21.000Z
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Description: Help exercise Exercise 2.4. Toss a coin repeatedly. Assume the probability of head on eachtoss is , as is the probability of tail. Let X; = 1 if the jth toss results ina head and X; = -1 if the jth toss results in a tail. Consider the stochasticproce