# Levy processes

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 Revision as of 18:42, 22 January 2012 (view source)Nestor (Talk | contribs) (Created page with "A Lévy process is an important type of stochastic process (namely, a family of $\mathbb{R}^d$ valued random variables each indexed by a positive number $t\geq 0$). In the c...")← Older edit Latest revision as of 21:02, 25 January 2016 (view source)Luis (Talk | contribs) (→Connection with linear integro-differential operators) (11 intermediate revisions not shown) Line 1: Line 1: A Lévy process is an important type of [[stochastic process]] (namely, a family of $\mathbb{R}^d$ valued random variables each indexed by a positive number $t\geq 0$).  In the context of parabolic integro-differential equations they play the same role that Brownian motion and more general diffusions play in the theory of second order parabolic equations. A Lévy process is an important type of [[stochastic process]] (namely, a family of $\mathbb{R}^d$ valued random variables each indexed by a positive number $t\geq 0$).  In the context of parabolic integro-differential equations they play the same role that Brownian motion and more general diffusions play in the theory of second order parabolic equations. - Informally speaking, a Lévy process is a Brownian motion which may jump, the times, length and direction of the jumps being random variables. A prototypical example would be $X(t)=B(t)+N(t)$ where $B(t)$ is the standard [[Brownian motion]] and $N(t)$ is a [[Compound Poisson process]], the trajectory described by typical sample path of this process would look like the union of several disconnected Brownian motion paths. + Informally speaking, a Lévy process is a random trajectory, generalizing the concept of Brownian motion, which may contain jump discontinuities. A prototypical example would be $X(t)=B(t)+N(t)$ where $B(t)$ is the standard [[Brownian motion]] and $N(t)$ is a [[Compound Poisson process]], the trajectory described by typical sample path of this process would look like the union of several disconnected Brownian motion paths. == Definition == == Definition == - A stochastic process $X(t)$ with values in $\mathbb{R}^d$ is said to be a Lévy process if + A stochastic process $X=\{X(t)\}_{t \geq 0}$ with values in $\mathbb{R}^d$ is said to be a Lévy process if 1.For any sequence $0 \leq t_1 < t_2 <... ## Latest revision as of 21:02, 25 January 2016 A Lévy process is an important type of stochastic process (namely, a family of$\mathbb{R}^d$valued random variables each indexed by a positive number$t\geq 0$). In the context of parabolic integro-differential equations they play the same role that Brownian motion and more general diffusions play in the theory of second order parabolic equations. Informally speaking, a Lévy process is a random trajectory, generalizing the concept of Brownian motion, which may contain jump discontinuities. A prototypical example would be$X(t)=B(t)+N(t)$where$B(t)$is the standard Brownian motion and$N(t)$is a Compound Poisson process, the trajectory described by typical sample path of this process would look like the union of several disconnected Brownian motion paths. ## Contents ## Definition A stochastic process$X=\{X(t)\}_{t \geq 0}$with values in$\mathbb{R}^d$is said to be a Lévy process if 1.For any sequence$0 \leq t_1 < t_2 <...<t_n$the random variables$X(t_0),X(t_1)-X(t_0),...,X(t_n)-X(t_{n-1})$are independent. 2.For any positive times$s\leq t$the random variables$X(t-s)$and$X(t)-X(s)$have the same probability law. 3.Almost surely, the trajectory of$X(t)$is continuous from the right, with limit from the left also known as "càdlàg" for its acronym in french. ## Lévy-Khintchine Formula It follows from the first two properties above that if$X$is a Lévy process and we further assume$X(0)=0$a.s. then for each fixed positive$t$the random variable$X(t)$is infinitely divisible, that is, it can be written as the sum of$n$independent and identically distribued random variables, for all$n\in\mathbb{N}$. Indeed, let$h=\tfrac{t}{n}$, then $X(t) = \left( X(h)-X(0)\right)+\left( X(2h)-X(h)\right)+...+\left( X(t)-X((n-1)h)\right)$ and by the above definition the differences$X(kh)-X((k-1)h)$are independent and distributed the same as$X(h)$. From the infinite divisibility of$X(t)$it follows by a theorem of Lévy and Khintchine that for any$\xi \in \mathbb{R}^d$we have $\mathbb{E} \left [ e^{i\xi\cdot X_t}\right ] = e^{t\eta(\xi)}$ the function$\eta(\xi)$given by $\eta(\xi)=i y\cdot b -\tfrac{1}{2}(A\xi,\xi)+\int_{\mathbb{R}^d} \left ( e^{i \xi\cdot y}-1-i\xi\cdot y \chi_{B_1}(y) \right ) d\mu(y)$ where$b$is a vector,$A$is a positive matrix,$B_1$is the unit ball and$\mu$is a Lévy measure, that is, a Borel measure in$\mathbb{R}^d$such that $\int_{\mathbb{R}^d}\frac{|y|^2}{1+|y|^2}d\mu(y) <+\infty.$ The interpretation of this measure$\mu$is that jumps from some point$x$to$x+y$with$y$in some set$A$occur as a Poisson process with intensity$\mu(A)$. ## Connection with linear integro-differential operators Any Lévy process$X(t)$such that$X(0)=0$defines a linear semigroup$\{U_t\}_{t\geq0}$on the space of continuous functions$f:\mathbb{R}^d\to\mathbb{R}^d$as follows $(U_tf)(x)= \mathbb{E}\left [ f(x+X(t)) \right ]$ Given the initial assumption on$X(0)$it is clear that$U_0$is the identity, and given that$X(t)-X(s)$is distributed as$X(t-s)$it follows that$U_t \circ U_s = U_{t+s}$. As a semigroup,$U_t$has an infinitesimal generator which turns out to be a Linear integro-differential operator. More precisely, if we let$f(x,t):=(U_tf)(x)$, then, assuming that$f(x,t)$has enough regularity it can be checked that $\partial_t f = Lf \;\;\;\mbox{ for all } (x,t)\in\mathbb{R}^d\times \mathbb{R}_+$ where for any smooth function$\phi\$, we have
$L\phi(x) = b \cdot \nabla \phi(x) +\mathrm{tr} \,( A\cdot D^2 \phi )+ \int_{\R^d} (\phi(x+y) - \phi(x) - y \cdot \nabla \phi(x) \chi_{B_1}(y)) \, \mathrm{d} \mu(y)$