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Skorokhod integral
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<h2 id="definition">Definition</h2> <h3>Preliminaries: the Malliavin derivative</h3> <p>Consider a fixed <a href="/facts/Probability_space/dEcv2VrU">probability space</a> ( Ω , Σ , P ) {\displaystyle (\Omega ,\Sigma ,\mathbf {P} )} and a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> H {\displaystyle H} ; E {\displaystyle \mathbf {E} } denotes <a href="/facts/Expected_value/1XV0JKL8">expectation</a> with respect to P {\displaystyle \mathbf {P} } </p><p> E [ X ] := ∫ Ω X ( ω ) d P ( ω ) . {\displaystyle \mathbf {E} [X]:=\int _{\Omega }X(\omega )\,\mathrm {d} \mathbf {P} (\omega ).} </p><p>Intuitively speaking, the Malliavin derivative of a random variable F {\displaystyle F} in L p ( Ω ) {\displaystyle L^{p}(\Omega )} is defined by expanding it in terms of Gaussian random variables that are parametrized by the elements of H {\displaystyle H} and differentiating the expansion formally; the Skorokhod integral is the adjoint operation to the Malliavin derivative. </p><p>Consider a family of R {\displaystyle \mathbb {R} } -valued <a href="/facts/Random_variables/TwTBXnLT">random variables</a> W ( h ) {\displaystyle W(h)} , indexed by the elements h {\displaystyle h} of the Hilbert space H {\displaystyle H} . Assume further that each W ( h ) {\displaystyle W(h)} is a Gaussian (<a href="/facts/Normal_distribution/UapjjPyQ">normal</a>) random variable, that the map taking h {\displaystyle h} to W ( h ) {\displaystyle W(h)} is a <a href="/facts/Linear_map/Q1PBKNVV">linear map</a>, and that the <a href="/facts/Expected_value/1XV0JKL8">mean</a> and <a href="/facts/Covariance/hILocGNH">covariance</a> structure is given by </p><p> E [ W ( h ) ] = 0 , {\displaystyle \mathbf {E} [W(h)]=0,} E [ W ( g ) W ( h ) ] = ⟨ g , h ⟩ H , {\displaystyle \mathbf {E} [W(g)W(h)]=\langle g,h\rangle _{H},} </p><p>for all g {\displaystyle g} and h {\displaystyle h} in H {\displaystyle H} . It can be shown that, given H {\displaystyle H} , there always exists a probability space ( Ω , Σ , P ) {\displaystyle (\Omega ,\Sigma ,\mathbf {P} )} and a family of random variables with the above properties. The Malliavin derivative is essentially defined by formally setting the derivative of the random variable W ( h ) {\displaystyle W(h)} to be h {\displaystyle h} , and then extending this definition to "<a href="/facts/Smooth_function/mffL4ch2">smooth enough</a>" random variables. For a random variable F {\displaystyle F} of the form </p><p> F = f ( W ( h 1 ) , … , W ( h n ) ) , {\displaystyle F=f(W(h_{1}),\ldots ,W(h_{n})),} </p><p>where f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } is smooth, the Malliavin derivative is defined using the earlier "formal definition" and the chain rule: </p><p> D F := ∑ i = 1 n ∂ f ∂ x i ( W ( h 1 ) , … , W ( h n ) ) h i . {\displaystyle \mathrm {D} F:=\sum _{i=1}^{n}{\frac {\partial f}{\partial x_{i}}}(W(h_{1}),\ldots ,W(h_{n}))h_{i}.} </p><p>In other words, whereas F {\displaystyle F} was a real-valued random variable, its derivative D F {\displaystyle \mathrm {D} F} is an H {\displaystyle H} -valued random variable, an element of the space L p ( Ω ; H ) {\displaystyle L^{p}(\Omega ;H)} . Of course, this procedure only defines D F {\displaystyle \mathrm {D} F} for "smooth" random variables, but an approximation procedure can be employed to define D F {\displaystyle \mathrm {D} F} for F {\displaystyle F} in a large subspace of L p ( Ω ) {\displaystyle L^{p}(\Omega )} ; the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> of D {\displaystyle \mathrm {D} } is the <a href="/facts/Closure_(topology)/9x2wlgUW">closure</a> of the smooth random variables in the <a href="/facts/Seminorm/vMuescKp">seminorm</a> : </p><p> ‖ F ‖ 1 , p := ( E [ | F | p ] + E [ ‖ D F ‖ H p ] ) 1 / p . {\displaystyle \|F\|_{1,p}:={\big (}\mathbf {E} [|F|^{p}]+\mathbf {E} [\|\mathrm {D} F\|_{H}^{p}]{\big )}^{1/p}.} </p><p>This space is denoted by D 1 , p {\displaystyle \mathbf {D} ^{1,p}} and is called the Watanabe–Sobolev space. </p> <h3>The Skorokhod integral</h3> <p>For simplicity, consider now just the case p = 2 {\displaystyle p=2} . The Skorokhod integral δ {\displaystyle \delta } is defined to be the L 2 {\displaystyle L^{2}} -adjoint of the Malliavin derivative D {\displaystyle \mathrm {D} } . Just as D {\displaystyle \mathrm {D} } was not defined on the whole of L 2 ( Ω ) {\displaystyle L^{2}(\Omega )} , δ {\displaystyle \delta } is not defined on the whole of L 2 ( Ω ; H ) {\displaystyle L^{2}(\Omega ;H)} : the domain of δ {\displaystyle \delta } consists of those processes u {\displaystyle u} in L 2 ( Ω ; H ) {\displaystyle L^{2}(\Omega ;H)} for which there exists a constant C ( u ) {\displaystyle C(u)} such that, for all F {\displaystyle F} in D 1 , 2 {\displaystyle \mathbf {D} ^{1,2}} , </p><p> | E [ ⟨ D F , u ⟩ H ] | ≤ C ( u ) ‖ F ‖ L 2 ( Ω ) . {\displaystyle {\big |}\mathbf {E} [\langle \mathrm {D} F,u\rangle _{H}]{\big |}\leq C(u)\|F\|_{L^{2}(\Omega )}.} </p><p>The Skorokhod integral of a process u {\displaystyle u} in L 2 ( Ω ; H ) {\displaystyle L^{2}(\Omega ;H)} is a real-valued random variable δ u {\displaystyle \delta u} in L 2 ( Ω ) {\displaystyle L^{2}(\Omega )} ; if u {\displaystyle u} lies in the domain of δ {\displaystyle \delta } , then δ u {\displaystyle \delta u} is defined by the relation that, for all F ∈ D 1 , 2 {\displaystyle F\in \mathbf {D} ^{1,2}} , </p><p> E [ F δ u ] = E [ ⟨ D F , u ⟩ H ] . {\displaystyle \mathbf {E} [F\,\delta u]=\mathbf {E} [\langle \mathrm {D} F,u\rangle _{H}].} </p><p>Just as the Malliavin derivative D {\displaystyle \mathrm {D} } was first defined on simple, smooth random variables, the Skorokhod integral has a simple expression for "simple processes": if u {\displaystyle u} is given by </p><p> u = ∑ j = 1 n F j h j {\displaystyle u=\sum _{j=1}^{n}F_{j}h_{j}} </p><p>with F j {\displaystyle F_{j}} smooth and h j {\displaystyle h_{j}} in H {\displaystyle H} , then </p><p> δ u = ∑ j = 1 n ( F j W ( h j ) − ⟨ D F j , h j ⟩ H ) . {\displaystyle \delta u=\sum _{j=1}^{n}\left(F_{j}W(h_{j})-\langle \mathrm {D} F_{j},h_{j}\rangle _{H}\right).} </p> <h2 id="properties">Properties</h2> <ul><li>The <a href="/facts/Isometry/K5wX8ah6">isometry</a> property: for any process u {\displaystyle u} in D 1 , p {\displaystyle \mathbf {D} ^{1,p}} that lies in the domain of δ {\displaystyle \delta } , E [ ( δ u ) 2 ] = E ∫ | u t | 2 d t + E ∫ D s u t D t u s d s d t . {\displaystyle \mathbf {E} {\big [}(\delta u)^{2}{\big ]}=\mathbf {E} \int |u_{t}|^{2}dt+\mathbf {E} \int D_{s}u_{t}\,D_{t}u_{s}\,ds\,dt.} If u {\displaystyle u} is an adapted process, then D s u t = 0 {\displaystyle D_{s}u_{t}=0} for s > t {\displaystyle s>t} , so the second term on the right-hand side vanishes. The Skorokhod and Itô integrals coincide in that case, and the above equation becomes the <a href="/facts/It%25C3%25B4_isometry/Ln4AQcBR">Itô isometry</a>.</li> <li>The derivative of a Skorokhod integral is given by the formula D h ( δ u ) = ⟨ u , h ⟩ H + δ ( D h u ) , {\displaystyle \mathrm {D} _{h}(\delta u)=\langle u,h\rangle _{H}+\delta (\mathrm {D} _{h}u),} where D h X {\displaystyle \mathrm {D} _{h}X} stands for ( D X ) ( h ) {\displaystyle (\mathrm {D} X)(h)} , the random variable that is the value of the process D X {\displaystyle \mathrm {D} X} at "time" h {\displaystyle h} in H {\displaystyle H} .</li> <li>The Skorokhod integral of the product of a random variable F {\displaystyle F} in D 1 , 2 {\displaystyle \mathbf {D} ^{1,2}} and a process u {\displaystyle u} in dom ( δ ) {\displaystyle \operatorname {dom} (\delta )} is given by the formula δ ( F u ) = F δ u − ⟨ D F , u ⟩ H . {\displaystyle \delta (Fu)=F\,\delta u-\langle \mathrm {D} F,u\rangle _{H}.} </li></ul> <h2 id="alternatives">Alternatives</h2> <p>An alternative to the Skorokhod integral is the <a href="/facts/Ogawa_integral/yvEbKjGP">Ogawa integral</a>. </p> <ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Skorokhod_integral">"Skorokhod integral"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li> <li>Ocone, Daniel L. (1988). "A guide to the stochastic calculus of variations". <i>Stochastic analysis and related topics (Silivri, 1986)</i>. Lecture Notes in Math. 1316. Berlin: Springer. pp. 1–79. <a href="/facts/MR_(identifier)/uP137L11">MR</a><a href="https://mathscinet.ams.org/mathscinet-getitem?mr=953793">953793</a></li> <li><a href="/facts/Marta_Sanz-Sol%25C3%25A9/PWbRqKp3">Sanz-Solé, Marta</a> (2008). <a href="http://www.ma.ic.ac.uk/~dcrisan/lecturenotes-london.pdf">"Applications of Malliavin Calculus to Stochastic Partial Differential Equations (Lectures given at Imperial College London, 7–11 July 2008)"</a> (PDF). Retrieved 2008-07-09.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Hitsuda, Masuyuki (1972). "Formula for Brownian partial derivatives". Second Japan-USSR Symp. Probab. Th.2.: 111–114. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Kuo, Hui-Hsiung (2014). "The Itô calculus and white noise theory: a brief survey toward general stochastic integration". Communications on Stochastic Analysis. 8 (1). doi:10.31390/cosa.8.1.07. <a href="https://repository.lsu.edu/cgi/viewcontent.cgi?article=1325&context=cosa" target="_blank">https://repository.lsu.edu/cgi/viewcontent.cgi?article=1325&context=cosa</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>