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Chapter 10: Problem 16

If \(\\{Y(t), t \geqslant 0\\}\) is a Martingale, show that $$ E[Y(t)]=E[Y(0)] $$

Short Answer

Expert verified
By applying the martingale property \(E[Y(t)|\mathcal{F}_0] = Y(0)\) and utilizing the Tower Property (Law of Iterated Expectations), we can conclude that the expected value of a martingale process \(Y(t)\) is equal to the expected value of its initial state \(Y(0)\), as shown by the equation \(E[Y(t)] = E[Y(0)]\).

Step by step solution

01

Definition of a Martingale

Recall that a continuous-time stochastic process \(\{Y(t), t \geqslant 0\}\) is a martingale if it satisfies the following conditions: 1. \(Y(t)\) is adapted to the filtration \(\{\mathcal{F}_t\}\). 2. \(E[|Y(t)|] < \infty \) for all \(t \geq 0\). 3. \(E[Y(t)|\mathcal{F}_s]=Y(s)\) for any \(0 \leq s \leq t\). In this problem, we are specifically interested in the third condition.
02

Apply the Martingale Property

Given that \(\{ Y(t), t \geqslant 0 \}\) is a martingale, we can use the martingale property: \(E[Y(t)|\mathcal{F}_s] = Y(s)\) for any \(0 \leq s \leq t\). We'll set \(s = 0\), since we want to relate \(E[Y(t)]\) with \(E[Y(0)]\). Therefore, we have: $$ E[Y(t)|\mathcal{F}_0] = Y(0) $$
03

Compute the Unconditional Expectation

Now, we'll compute the unconditional expectation, by taking the expectation of the above equation with respect to the \(\sigma\)-algebra generated by the random variable \(Y(0)\): $$ E[E[Y(t)|\mathcal{F}_0]] = E[Y(0)] $$
04

Apply the Tower Property

We can apply the Tower Property (also known as the Law of Iterated Expectations) to the left side of the equation: $$ E[Y(t)] = E[Y(0)] $$ So, we have proven that the expected value of \(Y(t)\) is equal to the expected value of \(Y(0)\) for the martingale \(\{Y(t), t \geqslant 0\}\).

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Most popular questions from this chapter

Show that \(\\{Y(t), t \geqslant 0\\}\) is a Martingale when $$ Y(t)=\exp \left\\{c B(t)-c^{2} t / 2\right\\} $$ where \(c\) is an arbitrary constant. What is \(E[Y(t)] ?\) An important property of a Martingale is that if you continually observe the process and then stop at some time \(T\), then, subject to some technical conditions (which will hold in the problems to be considered), $$ E[Y(T)]=E[Y(0)] $$ The time \(T\) usually depends on the values of the process and is known as a stopping time for the Martingale. This result, that the expected value of the stopped Martingale is equal to its fixed time expectation, is known as the Martingale stopping theorem.

The current price of a stock is 100 . Suppose that the logarithm of the price of the stock changes according to a Brownian motion process with drift coefficient \(\mu=2\) and variance parameter \(\sigma^{2}=1 .\) Give the Black- Scholes cost of an option to buy the stock at time 10 for a cost of (a) 100 per unit. (b) 120 per unit. (c) 80 per unit. Assume that the continuously compounded interest rate is 5 percent. A stochastic process \(\\{Y(t), t \geqslant 0\\}\) is said to be a Martingale process if, for \(s

Show that standard Brownian motion is a Martingale.

Show that \(\\{Y(t), t \geqslant 0\\}\) is a Martingale when $$ Y(t)=B^{2}(t)-t $$ What is \(E[Y(t)] ?\) Hint: First compute \(E[Y(t) \mid B(u), 0 \leqslant u \leqslant s]\).

A stock is presently selling at a price of $$\$ 50$$ per share. After one time period, its selling price will (in present value dollars) be either $$\$ 150$$ or $$\$ 25 .$$ An option to purchase \(y\) units of the stock at time 1 can be purchased at cost \(c y\). (a) What should \(c\) be in order for there to be no sure win? (b) If \(c=4\), explain how you could guarantee a sure win. (c) If \(c=10\), explain how you could guarantee a sure win. (d) Use the arbitrage theorem to verify your answer to part (a).
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