Exercise 2.4. Toss a coin repeatedly. Assume the probability of head on each toss is , as is the probability of tail. Let X; = 1 if the jth toss results in a head and X; = -1 if the jth toss results in a tail. Consider the stochastic process Mo, M, M2,. defined by Mo = 0 and %3D ... M, =X;, n21. j=1

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Exercise 2.4. Toss a coin repeatedly. Assume the probability of head on each
toss is , as is the probability of tail. Let X; = 1 if the jth toss results in
a head and X; = -1 if the jth toss results in a tail. Consider the stochastic
process Mo, M, M2,.. defined by Mo = 0 and
M, =X;, n> 1.
j=1
This is called a symmetric random walk, with each head, it steps up one, and
with each tail, it steps down one.
(i) Using the properties of Theorem 2.3.2, show that Mo, M1, M2, . .. is a
martingale.
(ii) Let o be a positive constant and, for n > 0, define
2
Sn
eo +e-o
Show 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 the
result of putting a martingale into the (convex) exponential function (see
Exercise 2.3). In order to again have a martingale, we must "discount" the
geometric symmetric random walk, using the term
rate. This term is strictly less than one unless o = 0.
2
eo te-
as the discount
Transcribed Image Text:Exercise 2.4. Toss a coin repeatedly. Assume the probability of head on each toss is , as is the probability of tail. Let X; = 1 if the jth toss results in a head and X; = -1 if the jth toss results in a tail. Consider the stochastic process Mo, M, M2,.. defined by Mo = 0 and M, =X;, n> 1. j=1 This is called a symmetric random walk, with each head, it steps up one, and with each tail, it steps down one. (i) Using the properties of Theorem 2.3.2, show that Mo, M1, M2, . .. is a martingale. (ii) Let o be a positive constant and, for n > 0, define 2 Sn eo +e-o Show 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 the result of putting a martingale into the (convex) exponential function (see Exercise 2.3). In order to again have a martingale, we must "discount" the geometric symmetric random walk, using the term rate. This term is strictly less than one unless o = 0. 2 eo te- as the discount
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