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| In [[mathematical analysis]], the '''Russo–Vallois integral''' is an extension to [[stochastic process]]es of the classical [[Riemann–Stieltjes integral]]
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| :<math>\int f \, dg=\int fg' \, ds</math>
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| for suitable functions <math>f</math> and <math>g</math>. The idea is to replace the [[derivative]] <math>g'</math> by the difference quotient
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| :<math>g(s+\varepsilon)-g(s)\over\varepsilon</math> and to pull the limit out of the integral. In addition one changes the type of convergence.
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| ==Definitions==
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| '''Definition:''' A sequence <math>H_n</math> of [[stochastic process]]es [[Convergence of random variables|converges]] uniformly on [[compact set]]s in probability to a process <math>H,</math>
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| :<math>H=\text{ucp-}\lim_{n\rightarrow\infty}H_n,</math> | |
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| if, for every <math>\varepsilon>0</math> and <math>T>0,</math>
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| :<math>\lim_{n\rightarrow\infty}\mathbb{P}(\sup_{0\leq t\leq T}|H_n(t)-H(t)|>\varepsilon)=0.</math> | |
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| On sets:
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| :<math>I^-(\varepsilon,t,f,dg)={1\over\varepsilon}\int_0^tf(s)(g(s+\varepsilon)-g(s))\,ds</math> | |
| :<math>I^+(\varepsilon,t,f,dg)={1\over\varepsilon}\int_0^t f(s)(g(s)-g(s-\varepsilon)) \, ds</math>
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| and
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| :<math>[f,g]_\varepsilon (t)={1\over \varepsilon}\int_0^t(f(s+\varepsilon)-f(s))(g(s+\varepsilon)-g(s))\,ds.</math>
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| '''Definition:''' The forward integral is defined as the ucp-limit of
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| :<math>I^-</math>: <math>\int_0^t fd^-g=\text{ucp-}\lim_{\varepsilon\rightarrow\infty}I^-(\varepsilon,t,f,dg).</math>
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| '''Definition:''' The backward integral is defined as the ucp-limit of
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| :<math>I^+</math>: <math>\int_0^t f \, d^+g = \text{ucp-}\lim_{\varepsilon\rightarrow\infty}I^+(\varepsilon,t,f,dg).</math>
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| '''Definition:''' The generalized bracket is defined as the ucp-limit of
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| :<math>[f,g]_\varepsilon</math>: <math>[f,g]_\varepsilon=\text{ucp-}\lim_{\varepsilon\rightarrow\infty}[f,g]_\varepsilon (t).</math>
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| For continuous [[semimartingale]]s <math>X,Y</math> and a [[cadlag function]] H, the Russo–Vallois integral coincidences with the usual [[Ito integral]]:
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| :<math>\int_0^t H_s \, dX_s=\int_0^t H \, d^-X.</math>
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| In this case the generalised bracket is equal to the classical covariation. In the special case, this means that the process
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| :<math>[X]:=[X,X] \, </math>
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| is equal to the [[quadratic variation process]].
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| Also for the Russo-Vallios-Integral an [[Ito formula]] holds: If <math>X</math> is a continuous semimartingale and
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| :<math>f\in C_2(\mathbb{R}),</math>
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| then
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| :<math>f(X_t)=f(X_0)+\int_0^t f'(X_s) \, dX_s + {1\over 2}\int_0^t f''(X_s) \, d[X]_s.</math>
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| By a duality result of [[Triebel]] one can provide optimal classes of [[Besov space]]s, where the Russo–Vallois integral can be defined. The norm in the Besov space
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| :<math>B_{p,q}^\lambda(\mathbb{R}^N)</math>
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| is given by
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| :<math>||f||_{p,q}^\lambda=||f||_{L_p} + \left(\int_0^\infty {1\over |h|^{1+\lambda q}}(||f(x+h)-f(x)||_{L_p})^q \, dh\right)^{1/q}</math>
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| with the well known modification for <math>q=\infty</math>. Then the following theorem holds: | |
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| '''Theorem:''' Suppose
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| :<math>f\in B_{p,q}^\lambda,</math>
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| :<math>g\in B_{p',q'}^{1-\lambda},</math>
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| :<math>1/p+1/p'=1\text{ and }1/q+1/q'=1.</math> | |
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| Then the Russo–Vallois integral
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| :<math>\int f \, dg</math>
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| exists and for some constant <math>c</math> one has
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| :<math>\left| \int f \, dg \right| \leq c ||f||_{p,q}^\alpha ||g||_{p',q'}^{1-\alpha}.</math>
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| Notice that in this case the Russo–Vallois integral coincides with the [[Riemann–Stieltjes integral]] and with the [[Young integra]]l for functions with [[finite p-variation]].
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| {{no footnotes|date=January 2012}}
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| ==References==
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| *Russo, Vallois: Forward, backward and symmetric integrals, Prob. Th. and rel. fields 97 (1993)
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| *Russo, Vallois: The generalized covariation process and Ito-formula, Stoch. Proc. and Appl. 59 (1995)
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| *Zähle; Forward Integrals and SDE, Progress in Prob. Vol. 52 (2002)
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| *Fournier, Adams: Sobolev Spaces, Elsevier, second edition (2003)
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| {{DEFAULTSORT:Russo-Vallois integral}}
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| [[Category:Definitions of mathematical integration]]
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| [[Category:Stochastic processes]]
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They contact me Emilia. California is our beginning place. One of the issues she loves most is to do aerobics and now she is trying to make money with it. Hiring is his profession.
my web blog :: http://www.associazioneitalianafotografi.it/