Levy processes - Revision history

imported>Luis: /* Connection with linear integro-differential operators */

2016-01-25T21:02:45Z

Connection with linear integro-differential operators

← Older revision		Revision as of 16:02, 25 January 2016
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	== Connection with linear integro-differential operators ==		== Connection with linear integro-differential operators ==

	Any Lévy process $X(t)$ such that $X(0)=0$ ~~almost surely~~ defines a linear semigroup $\{U_t\}_{t\geq0}$ on the space of continuous functions $f:\mathbb{R}^d\to\mathbb{R}^d$ as follows		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

	\[(U_tf)(x)= \mathbb{E}\left [ f(x+X(t)) \right ]\]		\[(U_tf)(x)= \mathbb{E}\left [ f(x+X(t)) \right ]\]

imported>Luis: /* Stochastic control and fully non-linear integro-differential operators */

2012-04-25T17:24:50Z

Stochastic control and fully non-linear integro-differential operators

← Older revision		Revision as of 12:24, 25 April 2012
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	== [[Stochastic control]] and fully non-linear integro-differential operators ==		== [[Stochastic control]] and fully non-linear integro-differential operators ==

	A similar connection holds via the [[Isaacs equation\|Isaacs-Bellman]] equation arising in [[stochastic control]] problems (or more generally in stochastic games). In this case the corresponding semigroup is not linear, and instead one must work in terms of viscosity solutions to build it.		A similar connection holds via the [[Isaacs equation\|Isaacs-Bellman]] equation arising in [[stochastic control]] problems (or more generally in stochastic games). In this case the corresponding semigroup is not linear, and instead one must work in terms of [[viscosity solutions]] to build it.

imported>Luis: /* Connection to linear integro-differential operators */

2012-03-12T21:39:55Z

Connection to linear integro-differential operators

← Older revision		Revision as of 16:39, 12 March 2012
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	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 $\mu(A)$.		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 to linear integro-differential operators ==		== Connection with linear integro-differential operators ==

	Any Lévy process $X(t)$ such that $X(0)=0$ almost surely defines a linear semigroup $\{U_t\}_{t\geq0}$ on the space of continuous functions $f:\mathbb{R}^d\to\mathbb{R}^d$ as follows		Any Lévy process $X(t)$ such that $X(0)=0$ almost surely defines a linear semigroup $\{U_t\}_{t\geq0}$ on the space of continuous functions $f:\mathbb{R}^d\to\mathbb{R}^d$ as follows

imported>Luis: /* Stochastic control and fully non-linear integro-differential operators */

2012-02-18T00:03:12Z

Stochastic control and fully non-linear integro-differential operators

← Older revision		Revision as of 19:03, 17 February 2012
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	\[ L\phi(x) = b \cdot \nabla \phi(x) +\mathrm{tr} \,( A\cdot D^2 \phi )+ \int_{\R^d} (\phi(x+y) - \phi(x) - y \cdot \nabla \phi(x) \chi_{B_1}(y)) \, \mathrm{d} \mu(y) \]		\[ L\phi(x) = b \cdot \nabla \phi(x) +\mathrm{tr} \,( A\cdot D^2 \phi )+ \int_{\R^d} (\phi(x+y) - \phi(x) - y \cdot \nabla \phi(x) \chi_{B_1}(y)) \, \mathrm{d} \mu(y) \]

	== Stochastic control and fully non-linear integro-differential operators ==		== [[Stochastic control]] and fully non-linear integro-differential operators ==

	A similar connection holds via the [[Isaacs equation\|Isaacs-Bellman]] equation arising in [[stochastic control]] problems (or more generally in stochastic games). In this case the corresponding semigroup is not linear, and instead one must work in terms of viscosity solutions to build it.		A similar connection holds via the [[Isaacs equation\|Isaacs-Bellman]] equation arising in [[stochastic control]] problems (or more generally in stochastic games). In this case the corresponding semigroup is not linear, and instead one must work in terms of viscosity solutions to build it.

imported>Luis at 00:01, 18 February 2012

2012-02-18T00:01:34Z

← Older revision		Revision as of 19:01, 17 February 2012
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	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 ==

imported>Luis at 00:00, 18 February 2012

2012-02-18T00:00:10Z

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	\[ \int_{\mathbb{R}^d}\frac{\|y\|^2}{1+\|y\|^2}d\mu(y) <+\infty. \]		\[ \int_{\mathbb{R}^d}\frac{\|y\|^2}{1+\|y\|^2}d\mu(y) <+\infty. \]

			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 to linear integro-differential operators ==		== Connection to linear integro-differential operators ==

imported>Milton: Corrected some typos

2012-02-17T18:04:28Z

Corrected some typos

← Older revision		Revision as of 13:04, 17 February 2012
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	the function $\eta(\xi)$ given by		the function $\eta(\xi)$ given by

	\[\eta(\xi)=y\cdot b i -\tfrac{1}{2}(A\xi,\xi)+\int_{\mathbb{R}^d} \left ( e^{\xi\cdot y}-1-i\xi\cdot y \chi_{B_1}(y) \right ) d\mu(y) \]		\[\eta(\xi)=i y\cdot b -\tfrac{1}{2}(A\xi,\xi)+\int_{\mathbb{R}^d} \left ( e^{i \xi\cdot y}-1-i\xi\cdot y \chi_{B_1}(y) \right ) d\mu(y) \]

	where $b$ is a vector, $A$ is a positive matrix and $\mu$ is a Lévy measure, that is, a Borel measure in $\mathbb{R}^d$ such that		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

	\[ \int_{\mathbb{R}^d}\frac{\|y\|^2}{1+\|y\|^2}d\mu(y) <+\infty \]		\[ \int_{\mathbb{R}^d}\frac{\|y\|^2}{1+\|y\|^2}d\mu(y) <+\infty. \]

	== Connection to linear integro-differential operators ==		== Connection to linear integro-differential operators ==
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	Any Lévy process $X(t)$ such that $X(0)=0$ almost surely defines a linear semigroup $\{U_t\}_{t\geq0}$ on the space of continuous functions $f:\mathbb{R}^d\to\mathbb{R}^d$ as follows		Any Lévy process $X(t)$ such that $X(0)=0$ almost surely defines a linear semigroup $\{U_t\}_{t\geq0}$ on the space of continuous functions $f:\mathbb{R}^d\to\mathbb{R}^d$ as follows

	\[(U_tf)(x)= \mathbb{E}\left [ f(x+X(t) \right ]\]		\[(U_tf)(x)= \mathbb{E}\left [ f(x+X(t)) \right ]\]

	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 $U_t \circ U_s = U_{t+s}$.		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 $U_t \circ U_s = U_{t+s}$.

imported>Luis at 05:53, 23 January 2012

2012-01-23T05:53:36Z

← Older revision		Revision as of 00:53, 23 January 2012
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	== Stochastic control and fully non-linear integro-differential operators ==		== Stochastic control and fully non-linear integro-differential operators ==

	A similar connection holds via the [[Isaacs-Bellman]] equation arising in [[stochastic control]] problems (or more generally in stochastic games). In this case the corresponding semigroup is not linear, and instead one must work in terms of viscosity solutions to build it.		A similar connection holds via the [[Isaacs equation\|Isaacs-Bellman]] equation arising in [[stochastic control]] problems (or more generally in stochastic games). In this case the corresponding semigroup is not linear, and instead one must work in terms of viscosity solutions to build it.

imported>Nestor at 19:54, 22 January 2012

2012-01-22T19:54:08Z

← Older revision		Revision as of 14:54, 22 January 2012
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	== Stochastic control and fully non-linear integro-differential operators ==		== Stochastic control and fully non-linear integro-differential operators ==

	A similar connection holds via the [[Isaacs-Bellman]] equation arising in ~~stochastic games/~~stochastic control problems. ~~However, in~~ this case the corresponding semigroup is not linear, ~~however, using~~ viscosity solutions ~~instead of linear analysis one may construct a corresponding semigroup~~.		A similar connection holds via the [[Isaacs-Bellman]] equation arising in [[stochastic control]] problems (or more generally in stochastic games). In this case the corresponding semigroup is not linear, and instead one must work in terms of viscosity solutions to build it.

imported>Nestor at 19:52, 22 January 2012

2012-01-22T19:52:05Z

← Older revision		Revision as of 14:52, 22 January 2012
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	== Connection to linear integro-differential operators ==		== Connection to linear integro-differential operators ==

	Any Lévy process $X(t)$ such that $X(0)=0$ almost surely defines a semigroup $\{U_t\}_{t\geq0}$ on the space of continuous functions $f:\mathbb{R}^d\to\mathbb{R}^d$ as follows		Any Lévy process $X(t)$ such that $X(0)=0$ almost surely defines a linear semigroup $\{U_t\}_{t\geq0}$ on the space of continuous functions $f:\mathbb{R}^d\to\mathbb{R}^d$ as follows

	\[(U_tf)(x)= \mathbb{E}\left [ f(x+X(t) \right ]\]		\[(U_tf)(x)= \mathbb{E}\left [ f(x+X(t) \right ]\]
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	\[ L\phi(x) = b \cdot \nabla \phi(x) +\mathrm{tr} \,( A\cdot D^2 \phi )+ \int_{\R^d} (\phi(x+y) - \phi(x) - y \cdot \nabla \phi(x) \chi_{B_1}(y)) \, \mathrm{d} \mu(y) \]		\[ L\phi(x) = b \cdot \nabla \phi(x) +\mathrm{tr} \,( A\cdot D^2 \phi )+ \int_{\R^d} (\phi(x+y) - \phi(x) - y \cdot \nabla \phi(x) \chi_{B_1}(y)) \, \mathrm{d} \mu(y) \]

			== Stochastic control and fully non-linear integro-differential operators ==

			A similar connection holds via the [[Isaacs-Bellman]] equation arising in stochastic games/stochastic control problems. However, in this case the corresponding semigroup is not linear, however, using viscosity solutions instead of linear analysis one may construct a corresponding semigroup.