Let (X_{t}, t geq 0), be defined as [X_{t}=left{B_{t} mid B_{t} geq 0 ight}, quad forall t>0]

Question:

Let $X_{t}, t \geq 0$, be defined as

\[X_{t}=\left\{B_{t} \mid B_{t} \geq 0\right\}, \quad \forall t>0\]

that is, the process has the paths of the Brownian motion conditioned by the current value being positive.

(a) Show that the pdf of $X_{t}$ is

\[f_{X_{t}}(x)=2 f_{B_{t}}(x), \forall x \geq 0\]

(b) Calculate $\mathbf{E}\left[X_{t}\right]$ and $V\left(X_{t}\right)$.

(c) Is $X_{t}$ a Gaussian process?

(d) Is $X_{t}$ stationary?

(e) Are $X_{t}$ and $\left|B_{t}\right|$ identically distributed?

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

Quantitative Finance

ISBN: 9781118629956

1st Edition

Authors: Maria Cristina Mariani, Ionut Florescu

See More Books

Question Posted: Apr 20, 2024 05:57 AM

See More Questions

(i) Write down the linear program relaxation for the vertex cover problem and solve the linear program. [6 marks] (ii) Based on the solution of the linear program in (b)(i), derive an integer...
Portray in words what transforms you would have to make to your execution to some degree (a) to accomplish this and remark on the benefits and detriments of this thought.You are approached to compose...
answer all questions as instructed below. attend all questions. 4 Computer Vision (a) Explain why such a tiny number of 2D Gabor wavelets as shown in this sequence are so efficient at representing...
The kinetic energy of an electron in a particular Bohr orbit of hydrogen is 1.35 10-19 J. (a) Which Bohr orbit does the electron occupy? (b) Suppose the electron moves away from the nucleus to the...
Anstead Co. is experiencing a decrease in sales and operating income for the fiscal year ending October 31, 2016. Ryan Frazier, controller of Anstead Co., has suggested that all orders received...
Suggest why a patient may be asked to hold his or her breath during a CT scan.
How might the market value of a firm differ from its intrinsic value?
Nelson Fabricators sells a portable EKG machine. The sales manager requires a weekly forecast of the portable EKG machine so that he can schedule production. The manager uses exponential smoothing...
Astro Corporation was started with the issue of 5,100 shares of $9 par stock for cash on January 1, Year 1. The stock was issued at a market price of $19 per share. During Year 1, the company earned...
If $X_{t} \sim N(0, t)$, calculate the distribution of $\left|X_{t} ight|$. Calculate $\mathbf{E}\left|X_{t} ight|$ and $V\left(\left|X_{t} ight| ight)$.
Let $B_{t}$ be a standard Brownian motion started at 0 . Use that for any function $f$ we have: \[\mathbf{E}\left[f\left(B_{t} ight) ight]=\frac{1}{\sqrt{2 \pi t}} \int_{-\infty}^{\infty} f(x)...
Assume that policy makers are using the Taylor rule as a basis for policy changes, as specified in Equation 1. Under each of the following scenarios, show how the real interest rate, output, and...
If you decide not to pay an effective interest rate on a credit card above 15%, what is the maximum APR that you will be ready to pay a card for which the interest is compounded daily? Can someone...
Consider a six-year, $1,000 par value, zero-coupon bond yielding 8 percent. (15 points) 1- What is the duration of this bond? (1 point) 2- If interest rates increase to 9 percent, what is the amount...
Bond Maturity Coupon Price A 2 8% 106.0905 B 2 6% 102.3825 Assume a two-year bond with face value 1000 and coupon rate 5%. Investors expect a 0.4% premium for one year interest and...
Why are negative interest rates used in basic terms? What impact does it have on the economy?
You overhear a friend discuss how Americans are frivolous in their spending and waste lots of money. Use this data on family budgets to argue against his position.
A natural-circulation pillow-block bearing has a journal diameter D of 2.500 in with a unilateral tolerance of 0.001 in. The bushing bore diameter B is 2.504 in with a unilateral tolerance of 0.004...
TRUE-FALSE QUESTIONS 1. In terms of preliminary analytical procedures, assume that the company has introduced a new product with a low price point and significant customer demand. The auditor would...
Consider the regression results given in Table 1.2. below Table 1.2 Wage regression. Dependent Variable: WAGE Method: Least Squares Sample: 1 1289 Included observations: 1289 Coefficient Std. Error...
a) The table below shows the quantity supplied (Y) of a commodity at various prices (X), holding all other factors constant. N 1 2 3 4 5 6 7 8 Y 12 14 10 13 17 12 11 15 X 5 11 7 8 11 7 6 9 i. ii....
Differentiate between consumer energy demand and producer energy demand using their economic foundations (15 marks) QUESTION FIVE a) Explain the distinct characteristic of energy projects (10 marks)...

Let (X_{t}, t geq 0), be defined as [X_{t}=left{B_{t} mid B_{t} geq 0 ight}, quad forall t>0]

Question:

Step by Step Answer:

Quantitative Finance

Students also viewed these Business questions