## Glossary

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

# Levy process

A process $X$ with following properties is called a Lévy process
• $X_0=0$
• Independent increments $X_{t+s}-X_t$ independent of $X_t$ for all $s>0$ all $t>0$
• Stationary increments $X_{s+t}-X_t\stackrel{d}{=} X_s$ for all $s>0$ all $t>0$

### General Lévy processes

A general Lévy process can be written as $X(t)=\mu t+\sigma W(t)+Z(t)$ Linear drift $\mu t$,
Brownian motion with variance $\sigma^2$: $\sigma W(t)$.
Pure jump process $Z(t)$

### Lévy-Khintchine representation

The characteristic function of any one-dimensional Lévy process can be written as $\phi(y,t)=E[\exp(iy X(t))]=\exp(tK(y)),$ where $K(y)=i\mu y-y^2\sigma^2/2+K_z(y)$ with $K_z(y)=i\gamma y+\int_{\mathbb{R}}(e^{iyx}-1-iy xI(|x|\leq 1))\nu(dx),$ $\nu$ is called the Lévy measure.

### Interpretation of the Lévy measure

The number $\int_a^b\nu(dx),$ equals the average number of jumps with sizes between a and b per time unit.

### General restriction on $\nu$

$\int_{\mathbb R} \min(x^2,1)\nu(dx)<\infty$ This is equivalent to that all Lévy processes have finite quadratic variation.

### Expectation and variance

Expectation $E[X(t)]=tK'(0)/i=t\left(\mu+\gamma+\int_{|x|>1} x\nu(dx)\right)$ Variance $E[X(t)]=-tK''(0)=t\left(\sigma^2+\int_{\mathbb R} x^2\nu(dx)\right)$ But note that neither the variance nor the expectation needs to be finite!
Moment relations
The expectation $E[|g(X(t))|]$ is finite for all $t>0$ if $\int_{|x|>1} |g(x)|\nu(dx)<\infty,$ provided that $|g(x+y)|\leq c|g(x)g(y)|$ for some $c>0$ and $\forall x,y\in {\mathbb R}$.

### Ito's formula for Lévy processes

\begin{eqnarray*} df(X(t))&=&f'(X(t))\mu dt+f''(X(t))\sigma^2/2dt+\sigma f'(X(t))dW(t)\\ &&+f'(X(t-))dZ(t)\\ &&+f(X(t-)+\Delta Z(t))-f(X(t-))-f'(X(t-))\Delta Z(t), \end{eqnarray*} where $\Delta Z(t)$ is the jump in $Z$.

### Examples of Lévy processes

In financial mathematics we usually choose $\mu$ such that $K(-i)=r$ to obtain the risk neutral dynamics. The Lévy process is then used to decribe the log stock price.
• Wiener process, $K(y)=iy\mu-(y^2/2)\sigma^2,~\mu\in {\mathbb R},\sigma>0$,
• Poisson, $K(y)=iy\mu+\lambda(\exp(icy)-1),\mu\in {\mathbb R},c,\lambda>0$,
• Compound Poisson, $K(y)=iy\mu+\lambda(\phi_J(y)-1)$, where $\phi_J$ is the characteristic function for the jump distribution which we can choose freely (although respecting some moment condtions),
• Merton process = Compound Poisson with Gaussian increments plus a Wiener process with drift [Merton, 1976], $K(y)=iy\mu-(y^2/2)\sigma^2+\lambda(\exp(iy\mu_J-(y^2/2)\sigma_J^2)-1),\mu,\mu_J\in {\mathbb R},c,\lambda,\sigma,\sigma_J>0$,
• Gamma process, $K(y)=iy\mu-\lambda\log(1-iyc),\mu\in {\mathbb R},~\lambda,c>0$,
• Cauchy process, $K(y)=iy\mu-c|y|,\mu\in {\mathbb R},~c>0$,
• Normal Inverse Gaussian (NIG) process [Barndorff-Nielsen, 1997], See also NIG model, $K(y)=iy\mu+\delta(\sqrt{\alpha^2-\beta^2}-\sqrt{\alpha^2-(\beta-iy)^2}),\mu\in{\mathbb R},\alpha,\delta>0,~|\beta|<\alpha$,
• Variance Gamma (VG) process[Madan and Seneta, 1990], $K(y)=iy\mu-1/\kappa\log(1-(iy\theta\kappa-(y^2/2)\sigma^2\kappa)),~\mu,\theta\in {\mathbb R},~\sigma,\kappa>0$,
• Carr Geman Madan Yor (CGMY) process [Carr, Geman, Madan and Yor, 2002], $K(z)=iz\mu+\Gamma(-Y)((M-iz)^Y-M^Y+(G+iz)^Y-G^Y),\mu\in{\mathbb R},Y\in(0,2),~M,G>0$,
• Finite Moment Log Stable (FMLS) process (crash model) [Carr and Wu, 2003], $K(y)=iy\mu+(iyc)^\alpha,~\alpha\in(1,2),~c>0,\mu\in {\mathbb R}$.

### References

Barndorff-Nielsen, O. E., 1997. Processes of Normal Inverse Gaussian Type. Finance and Stochastics 2, 41-68. Article
Cont, R. and Tankov, P. (2004). Finanial Modelling with Jump Processes, Chapman and Hall. Book
Carr, P., Geman, H., Madan, D., Yor, M., 2002. The Fine Structure of Asset Returns: An Empirical Investigation. The Journal of Business 75, 305-332. Article
Carr, P., Wu, L., 2003. The Finite Moment Log stable Process and Option Pricing. Journal of Finance 58, 753-778. Article
Madan, D. B., Seneta, E., 1990. The Variance Gamma (VG) Model for Share Market returns. Journal of Business 63, 511-524. Article
Merton, R. C., 1976. Option Pricing When the Underlying Stock Returns are Discontinuous. Journal of Financial Economics 5, 125-144. Article
Sato, K.I. (1999). Lévy processes and Infinitely Divisble Distributions, Cambridge University Press.

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