Dividend policy

In probability theory, for a probability measure P on a Hilbert space H with inner product $⟨\cdot ,\cdot ⟩$, the covariance of P is the bilinear form Cov: H × H → R given by

$\mathrm{C}\mathrm{o}\mathrm{v}(x,y)={\int }_H⟨x,z⟩⟨y,z⟩\mathrm{d}\mathbf{P}(z)$

for all x and y in H. The covariance operator C is then defined by

$\mathrm{C}\mathrm{o}\mathrm{v}(x,y)=⟨Cx,y⟩$

(from the Riesz representation theorem, such operator exists if Cov is bounded). Since Cov is symmetric in its arguments, the covariance operator is self-adjoint (the infinite-dimensional analogy of the transposition symmetry in the finite-dimensional case). When P is a centred Gaussian measure, C is also a nuclear operator. In particular, it is a compact operator of trace class, that is, it has finite trace.

Even more generally, for a probability measure P on a Banach space B, the covariance of P is the bilinear form on the algebraic dual B^#, defined by

$\mathrm{C}\mathrm{o}\mathrm{v}(x,y)={\int }_B⟨x,z⟩⟨y,z⟩\mathrm{d}\mathbf{P}(z)$

where $⟨x,z⟩$ is now the value of the linear functional x on the element z.

Quite similarly, the covariance function of a function-valued random element (in special cases called random process or random field) z is

$\mathrm{C}\mathrm{o}\mathrm{v}(x,y)=\int z(x)z(y)\mathrm{d}\mathbf{P}(z)=E(z(x)z(y))$

where z(x) is now the value of the function z at the point x, i.e., the value of the linear functional $u\mapsto u(x)$ evaluated at z.

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