Independent and stationary increments

Continuous sample paths

Gaussian distribution

Finite variance

Brownian motion

Poisson process

Geometric Brownian motion

Ornstein-Uhlenbeck process

$$\phi(u) = \exp\left(\int_\mathbb{R} \left(e^{iux} - 1 - iux\right)\,\nu(dx)\right)$$

$$\phi(u) = \exp\left(\int_\mathbb{R} \left(e^{iux} - 1\right)\,\nu(dx)\right)$$

$$\phi(u) = \exp\left(\int_\mathbb{R} \left(e^{iux} - 1 + iux\right)\,\nu(dx)\right)$$

$$\phi(u) = \exp\left(\int_\mathbb{R} \left(e^{iux} + 1 - iux\right)\,\nu(dx)\right)$$

The Lévy measure is the distribution of the jumps of the process.

The Lévy measure is the distribution of the increments of the process.

The Lévy measure is the distribution of the sample paths of the process.

The Lévy measure is the distribution of the characteristic function of the process.

Independent increments

Stationary increments

Gaussian distribution

Infinitely divisible distribution

The drift coefficient is the mean of the increments of the process.

The diffusion coefficient is the variance of the increments of the process.

The drift coefficient is the rate of change of the mean of the process.

The diffusion coefficient is the rate of change of the variance of the process.

Modeling financial asset prices

Modeling the arrival of customers in a queue

Modeling the spread of diseases

All of the above

A Wiener process is a special case of a Lévy process.

A Lévy process is a special case of a Wiener process.

A Wiener process and a Lévy process are independent.

A Wiener process and a Lévy process are mutually exclusive.

$$\phi(u) = \exp\left(\int_\mathbb{R} \left(e^{iux} - 1\right)\,\nu(dx)\right)$$

$$\phi(u) = \exp\left(\int_\mathbb{R} \left(e^{iux} - 1 - iux\right)\,\nu(dx)\right)$$

$$\phi(u) = \exp\left(\int_\mathbb{R} \left(e^{iux} + 1 - iux\right)\,\nu(dx)\right)$$

$$\phi(u) = \exp\left(\int_\mathbb{R} \left(e^{iux} + 1 + iux\right)\,\nu(dx)\right)$$

The generator is the infinitesimal generator of the process.

The generator is the characteristic function of the process.

The generator is the Lévy measure of the process.

The generator is the distribution of the process.

It is a linear operator.

It is a bounded operator.

It is a positive operator.

All of the above

The semigroup is the collection of transition probabilities of the process.

The semigroup is the collection of characteristic functions of the process.

The semigroup is the collection of generators of the process.

The semigroup is the collection of Lévy measures of the process.

It is a strongly continuous semigroup.

It is a Markov semigroup.

It is a Feller semigroup.

All of the above

The associated measure is the distribution of the process.

The associated measure is the characteristic function of the process.

The associated measure is the Lévy measure of the process.

The associated measure is the generator of the process.

It is a probability measure.

It is a tight measure.

It is a Feller measure.

All of the above