Let (left(B_{t}ight)_{t in mathbb{R}_{+}})denote a standard Brownian motion. a) Find the probability distribution of the stochastic integral

Question:

Let $\left(B_{t}ight)_{t \in \mathbb{R}_{+}}$denote a standard Brownian motion.

a) Find the probability distribution of the stochastic integral $\int_{0}^{1} t^{2} d B_{t}$.

b) Find the probability distribution of the stochastic integral $\int_{0}^{1} t^{-1 / 2} d B_{t}$.

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

Step by Step Answer:

Related Book For book-img-for-question

Introduction To Stochastic Finance With Market Examples

ISBN: 9781032288277

2nd Edition

Authors: Nicolas Privault

See More Books

Question Posted: Feb 25, 2024 08:00 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...
can someone solve this Modern workstations typically have memory systems that incorporate two or three levels of caching. Explain why they are designed like this. [4 marks] In order to investigate...
(a) Use the following text to derive distributions for rat and chased. Use a five-word window, including open- and closed- class words, ignore case, punctuation and sentence boundaries and weight...
Suppose the United States imposes a tariff on imported sugar. What are the consequences of this on consumers, domestic and foreign producers, and land use?
Evaluate. 1. 62 2. 63 3. 35 4. 112 5. 81
In a worksheet, net income is entered in the following columns: (a) income statement (Dr) and balance sheet (Dr). (b) income statement (Cr) and balance sheet (Dr). (c) income statement (Dr) and...
Assume that $y_{i j}$ are i.i.d and follow a normal distribution with variance $\sigma^{2}$. Under the null hypothesis that a factorial effect 0 , a. Show that \[\frac{\bar{y}_{+}-\bar{y}_{-}}{s / 2}...
Using the data in BE4-6, journalize and post the entry on July 1 and the adjusting entry on December 31 for Craig Insurance Co. Craig uses the accounts Unearned Service Revenue and Service Revenue.
Prepare a condensed cost of goods manufactured schedule The following information is available for Sheridan Company at December 31, 2022: 1. Inventory balance Beginning of Year End of Year Finished...
Given $\left(B_{t}ight)_{t \in \mathbb{R}_{+}}$a standard Brownian motion and $n \geqslant 1$, let the random variable $X_{n}$ be defined as \[ X_{n}:=\int_{0}^{2 \pi} \sin (n t) d B_{t}, \quad...
Let $T>0$. Show that for $f:[0, T] \mapsto \mathbb{R}$ a differentiable function such that $f(T)=0$, we have \[\int_{0}^{T} f(t) d B_{t}=-\int_{0}^{T} f^{\prime}(t) B_{t} d t\] Apply It's...
The binary phase diagram for the silver (Ag) and germanium (Ge) system is shown in Figure 11-34. Figure 11-34 The silver-germanium phase diagram (for Problem 11-35). (a) Schematically draw the phase...
Using the circular doubly linked list shown here, give the expression corresponding to each of the following descriptions: (For example, the expression for the info member of Node 1, referenced from...
The specifications for the Unsorted List ADT state that the item to be deleted is in the list. 1. Rewrite the specification for DeleteItem so that the list is unchanged if the item to be deleted is...
Implement the following specification for a client Boolean function that returns true if two stacks are identical and false otherwise. Boolean Identical(StackType stack1, StackType stack2) You may...
What are the major steps in performing object-oriented design?
Show what is written by the following segments of code, given that item1, item2, and item3 are int variables. 1. 2. Que Type queue; item1 = 1; item2 = 0; item3 = 4; queue. Enqueue (item2); queue....
The laws of refraction and reflection are the same for sound as for light. The speed of sound is 340 m/s in air and 1,510 m/s in water. If a sound wave traveling in air approaches a plane water...
Integration is a vital concept when applied in one?s life. Integrating your life means making ideal choices. Perfect choices on the other go in line with quality decisions. Quality decisions lead to...
Use your knowledge of balance sheets to fill in the missing amounts: Assets Cash..................................................................$10,000 Accounts...
Use your knowledge of balance sheets to fill in the missing amounts: Assets Cash.......................................................................$50,000 Accounts...
Use your knowledge of income statements to fill in the missing items: Sales Cost of goods sold.......................................$575,000 Gross...
Given the following training and test data, what is the precision and recall on the test data (w.rt Class =1) for a i) 1 nearest neighbor algorithm and ii) 3 nearest neighbor algorithm (you can break...
Suppose you are given the following dataset: Example Number X1 X2 Y 1 -1 2 -1 2 -2 -2 +1 3 1 -1 +1 4 -3 1 -1 Perform the Single-Mode Perceptron algorithm on this data. Assume you start with initial...
Can you expound upon the intricacies of enzyme kinetics, exploring how factors such as substrate concentration and enzyme specificity dictate the rate of biochemical reactions in biological systems ?

Let (left(B_{t}ight)_{t in mathbb{R}_{+}})denote a standard Brownian motion. a) Find the probability distribution of the stochastic integral

Question:

Step by Step Answer:

a The stochastic integral int01 t2 d Bt is a centered Gaussian random variable with varia...View the full answer

Introduction To Stochastic Finance With Market Examples

Students also viewed these Business questions