Malliavin derivative

In mathematics, the Malliavin derivative is a notion of derivative in the Malliavin calculus. Intuitively, it is the notion of derivative appropriate to paths in classical Wiener space, which are 'usually' not differentiable in the usual sense. In mathematics, the Malliavin derivative is a notion of derivative in the Malliavin calculus. Intuitively, it is the notion of derivative appropriate to paths in classical Wiener space, which are 'usually' not differentiable in the usual sense. Let H {displaystyle H} be the Cameron–Martin space, and C 0 {displaystyle C_{0}} denote classical Wiener space: By the Sobolev embedding theorem, H ⊂ C 0 {displaystyle Hsubset C_{0}} . Let denote the inclusion map. Suppose that F : C 0 → R {displaystyle F:C_{0} o mathbb {R} } is Fréchet differentiable. Then the Fréchet derivative is a map i.e., for paths σ ∈ C 0 {displaystyle sigma in C_{0}} , D F ( σ ) {displaystyle mathrm {D} F(sigma );} is an element of C 0 ∗ {displaystyle C_{0}^{*}} , the dual space to C 0 {displaystyle C_{0};} . Denote by D H F ( σ ) {displaystyle mathrm {D} _{H}F(sigma );} the continuous linear map H → R {displaystyle H o mathbb {R} } defined by sometimes known as the H-derivative. Now define ∇ H F : C 0 → H {displaystyle abla _{H}F:C_{0} o H} to be the adjoint of D H F {displaystyle mathrm {D} _{H}F;} in the sense that Then the Malliavin derivative D t {displaystyle mathrm {D} _{t}} is defined by The domain of D t {displaystyle mathrm {D} _{t}} is the set F {displaystyle mathbf {F} } of all Fréchet differentiable real-valued functions on C 0 {displaystyle C_{0};} ; the codomain is L 2 ( [ 0 , T ] ; R n ) {displaystyle L^{2}(;mathbb {R} ^{n})} .