Connection between random curves, changes of time, and regenerative times of stochastic processes

Abstract

The product of spaces Φ × D is considered, where Φ is the set of all continuous, nondecreasing functions ϕ:[0,∞)→(0,∞), ϕ(0)=0, ϕ(t)→∞(t→∞), and D is the set of all right continuous functions ξ:(0,∞)→X; here X is some metric space. Two mappings

are defined: the first is the projection q(ϕ,ξ)=ξ, and the second is the change of time U(ϕ,ξ)=ξºϕ. The following equivalence relation is defined on D:

$$\xi _1 \sim \xi _2 \Leftrightarrow \exists _{\varphi _1 , \varphi _1 } \in \Phi :\xi _1 ^\circ \varphi _1 = \xi _2 ^\circ \varphi _2 $$

. Letℳ be the set of all equivalence classes, and let L be the mapping

ξ₄∼ξ₂, Lξ is called the curve corresponding to ξ. The following theorem is proved: two stochastic processes with probability measures P¹ and P² on D possess identical random curves (i.e.,P¹ºL⁻¹=P²ºL⁻¹) if and only if there exist two changes of time (i.e., probability measures Q¹ and Q² on ϕ×D for which P¹=Q¹ºq⁻¹, P²=Q²ºq⁻¹ which take these two processes into a process with measure$\tilde P$(i.e., Q¹ºu⁻¹=Q²ºu⁻¹,=∼P) If (P ¹_x )_x∈X and (P ²_x )_x∈X are two families of probability measures for which P ¹_x ºL⁻¹=P ²_x ºL⁻¹∀x∈X then for each x ε X the corresponding measures Q ¹_X andQ ²_X can be found in the following manner. The set of regenerative times of the family$\left( {\tilde P_x } \right)_{x \in X}$ contains all stopping times which are simultaneously regenerative times of the families (p ¹_x )_x∈X and (P ²_x )_x∈X and possess a certain special property of first intersection.