Question: Let {Wt : t > 0} be a Brownian motion, and let { Ft : t 2 0}, be a filtration for this Brownian motion.

Let {Wt : t > 0} be a Brownian motion, and

let { Ft : t 2 0}, be a filtration for this

Let {Wt : t > 0} be a Brownian motion, and let { Ft : t 2 0}, be a filtration for this Brownian motion. Show that: (a) Xt = W2 - t is a martingale. (b) St = exp (oWt + (a - ,)t ) is a martingale for a = 0, a submartingale for a > 0 and a supermartingale for a

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer
Step: 1 Unlock blur-text-image
Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock
Step: 3 Unlock

Students Have Also Explored These Related Mathematics Questions!

Q:

Question 5 [20 Marks| NOTE: any general results you use in this question should be stated but not proved, with the ezception of part (a)(iii), where a derivation is required if the solution is to be...

Q:

Want to know the detail of those question how to use Brownian and probability space to solve the black scholes model 2. Let {Wt : t > 0} be a Brownian motion on a probability space (0, F, P) with a...

Q:

Let W = {Wt : t 0} be a Brownian motion on F = P) I Show that W is an F-martingale. 2 Show that for every a R. the process xta exp(aWt *02t) is an F-martingale. (20% of total points) (20% of total...

Q:

Let W = {Wt : t 0} be a Brownian motion on F = P) I Show that W is an F-martingale. 2 Show that for every a R. the process xta exp(aWt *02t) is an F-martingale. (20% of total points) (20% of total...

Q:

Let (12, F,P) be a probability space and let {W(1),t> 0} be a Brownian motion process with filtration F(t),t> 0. a. Is W*(t) a martingale? b. Suppose the price s(t) of a risky asset at time t follows...

Q:

Let (W(t), t > 0) be a standard Brownian motion on (N2, F, P) and dX(t) = W(t)dt + dW(t), with X(0) = 0. Under what probability measure on (12, F) is (X(t), t > 0) a standard Brownian motion? Let...

Q:

Let {Wt: t2 0} be a standard Brownian motion. We call the solution {Xt: t2 0} to the Langevin equation Xt = Wt+/ Xsds the Ornstein-Uhlenbeck process. In particular, it is used for modeling short-term...

Q:

EXERCISE 3 Let {Wt: t2 0} be a standard Brownian motion. We call the solution {Xt : t2 0} to the Langevin equation Xt = Wt+ Xsds the Ornstein-Uhlenbeck process. In particular, it is used for modeling...

Q:

4. (5 marks) Ito calculus. (a) Let X(t) and Y(t) be two stochastic processes, such that dx MX(X(t), t)dt +ox(X(t),t)dZx dY = wy(Y(t), t)dt +oy(Y(t), t)dZy with Zx(t), Zy(t) being Brownian motions....

Q:

Let \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)denote a standard Brownian motion generating the filtration \(\left(\mathcal{F}_{t}ight)_{t \in \mathbb{R}_{+}}\). a) Letting \(X_{t}:=\sigma B_{t}+u t,...

Q:

Do the previous problem for nitrogen, N2, gas. A rigid container has 2 kg of carbon dioxide gas at 100 kPa, 1200 K that is heated to 1400 K. Solve for the heat transfer using a. the heat capacity...

Q:

Suppose that the Treasury bond futures price is 101-12. Which of the following four bonds is cheapest to deliver? Price 125-05 142-15 115-31 144-02 Bond Conversion Factor 1.2131 1.3792 1.1149 1.4026

Q:

True or false: A portfolio manager can be both long or short a SOFR strip. True False

Q:

Several organizational strategies exist for developing effective presentations, depending on your purpose and the nature of your content. List and explain five of these strategies and give an...

Q:

The background knowledge of the interpreter

Q:

The expert rule: We have more confidence in information presented by someone who is an expert in the field than by someone who is not.

Q:

How easy the information is to remember