Affine Markov processes on a general state space

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Veerman, E.

Publication date

2011

Link to publication

Citation for published version (APA):

Veerman, E. (2011). Affine Markov processes on a general state space. Uitgeverij BOXPress.

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Notation

Let E ⊂ Rp _{be a closed set. Throughout we use the following notation.}

EC _{the class of functions on E with values in C}

B(E) the class of bounded measurable functions in EC

C(E) the class of continuous functions in EC

Cb(E) the class of bounded continuous functions in EC

Cc(E) the class of continuous functions in ECwith compact support

C0(E) the class of continuous functions in ECvanishing at infinity

Ck_(E) _{the class of k-times continuously differentiable functions in E}C

M (E) the class of measurable functions in EC

Rn×m the set of (n × m)-matrices with real-valued coefficients B(Rp₎

the Borel σ-algebra on Rp R+ [0, ∞), likewise R−= (−∞, 0]

R++ (0, ∞), likewise R−−= (−∞, 0)

C+ R++ iR, likewise C−= R−+ iR

Sp _{the set of symmetric matrices in R}p×p

S₊p _{the cone of positive semi-definite matrices in R}p×p

∂x, ∂x+ short hand notation for ∂

∂x and the right-hand side derivative

DE[0, ∞) the class of c`adl`ag functions f : [0, ∞) → E.

I the identity matrix ei the i-th unit vector

|v| Euclidean norm of a vector v

|K|(dz) the variation of a signed measure K(dz) = K+_{(dz) − K}−_(dz),

defined as |K|(dz) = K+_{(dz) + K}−_(dz)

fu the function on E given by x 7→ exp(u>x), for some u ∈ Cp