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Lévy Processes

This quiz covers fundamental concepts of Lévy processes including definitions, properties, characteristic functions, Lévy measures, generators, and semigroups.

14 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

What is the defining characteristic of a Lévy process?

  1. Independent and stationary increments
  2. Continuous sample paths
  3. Gaussian distribution
  4. Finite variance
Question 2 Multiple Choice (Single Answer)

Which of the following is an example of a Lévy process?

  1. Brownian motion
  2. Poisson process
  3. Geometric Brownian motion
  4. Ornstein-Uhlenbeck process
Question 3 Multiple Choice (Single Answer)

What is the characteristic function of a Lévy process?

  1. $$\phi(u) = \exp\left(\int_\mathbb{R} \left(e^{iux} - 1 - iux\right)\,\nu(dx)\right)$$
  2. $$\phi(u) = \exp\left(\int_\mathbb{R} \left(e^{iux} - 1\right)\,\nu(dx)\right)$$
  3. $$\phi(u) = \exp\left(\int_\mathbb{R} \left(e^{iux} - 1 + iux\right)\,\nu(dx)\right)$$
  4. $$\phi(u) = \exp\left(\int_\mathbb{R} \left(e^{iux} + 1 - iux\right)\,\nu(dx)\right)$$
Question 4 Multiple Choice (Single Answer)

What is the relationship between a Lévy process and its Lévy measure?

  1. The Lévy measure is the distribution of the jumps of the process.
  2. The Lévy measure is the distribution of the increments of the process.
  3. The Lévy measure is the distribution of the sample paths of the process.
  4. The Lévy measure is the distribution of the characteristic function of the process.
Question 5 Multiple Choice (Single Answer)

Which of the following properties is not satisfied by a Lévy process?

  1. Independent increments
  2. Stationary increments
  3. Gaussian distribution
  4. Infinitely divisible distribution
Question 6 Multiple Choice (Single Answer)

What is the relationship between a Lévy process and its drift and diffusion coefficients?

  1. The drift coefficient is the mean of the increments of the process.
  2. The diffusion coefficient is the variance of the increments of the process.
  3. The drift coefficient is the rate of change of the mean of the process.
  4. The diffusion coefficient is the rate of change of the variance of the process.
Question 7 Multiple Choice (Single Answer)

Which of the following is an application of Lévy processes?

  1. Modeling financial asset prices
  2. Modeling the arrival of customers in a queue
  3. Modeling the spread of diseases
  4. All of the above
Question 8 Multiple Choice (Single Answer)

What is the relationship between a Lévy process and a Wiener process?

  1. A Wiener process is a special case of a Lévy process.
  2. A Lévy process is a special case of a Wiener process.
  3. A Wiener process and a Lévy process are independent.
  4. A Wiener process and a Lévy process are mutually exclusive.
Question 9 Multiple Choice (Single Answer)

What is the characteristic function of a compound Poisson process?

  1. $$\phi(u) = \exp\left(\int_\mathbb{R} \left(e^{iux} - 1\right)\,\nu(dx)\right)$$
  2. $$\phi(u) = \exp\left(\int_\mathbb{R} \left(e^{iux} - 1 - iux\right)\,\nu(dx)\right)$$
  3. $$\phi(u) = \exp\left(\int_\mathbb{R} \left(e^{iux} + 1 - iux\right)\,\nu(dx)\right)$$
  4. $$\phi(u) = \exp\left(\int_\mathbb{R} \left(e^{iux} + 1 + iux\right)\,\nu(dx)\right)$$
Question 10 Multiple Choice (Single Answer)

What is the relationship between a Lévy process and its generator?

  1. The generator is the infinitesimal generator of the process.
  2. The generator is the characteristic function of the process.
  3. The generator is the Lévy measure of the process.
  4. The generator is the distribution of the process.
Question 11 Multiple Choice (Single Answer)

Which of the following is a property of the generator of a Lévy process?

  1. It is a linear operator.
  2. It is a bounded operator.
  3. It is a positive operator.
  4. All of the above
Question 12 Multiple Choice (Single Answer)

What is the relationship between a Lévy process and its semigroup?

  1. The semigroup is the collection of transition probabilities of the process.
  2. The semigroup is the collection of characteristic functions of the process.
  3. The semigroup is the collection of generators of the process.
  4. The semigroup is the collection of Lévy measures of the process.
Question 13 Multiple Choice (Single Answer)

Which of the following is a property of the semigroup of a Lévy process?

  1. It is a strongly continuous semigroup.
  2. It is a Markov semigroup.
  3. It is a Feller semigroup.
  4. All of the above
Question 14 Multiple Choice (Single Answer)

What is the relationship between a Lévy process and its associated measure?

  1. The associated measure is the distribution of the process.
  2. The associated measure is the characteristic function of the process.
  3. The associated measure is the Lévy measure of the process.
  4. The associated measure is the generator of the process.
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