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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== | |
| '''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>
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| :<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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