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A simulation of integral and derivative of the solution of a stochastici integral equation

Nguyen Quy Hy, Nguyen Thi Minh (1992)

Annales Polonici Mathematici

A stochastic integral equation corresponding to a probability space ( Ω , Σ ω , P ω ) is considered. This equation plays the role of a dynamical system in many problems of stochastic control with the control variable u ( · ) : ℝ 1 → ℝ m . One constructs stochastic processes η ( 1 ) ( t ) , η ( 2 ) ( t ) connected with a Markov chain and with the space ( Ω , Σ ω , P ω ) . The expected values of η ( i ) ( t ) (i = 1,2) are respectively the expected value of an integral representation of a solution x(t) of the equation and that of its derivative x u ' ( t ) .

Brownian penalisations related to excursion lengths, VII

B. Roynette, P. Vallois, M. Yor (2009)

Annales de l'I.H.P. Probabilités et statistiques

Limiting laws, as t→∞, for brownian motion penalised by the longest length of excursions up to t, or up to the last zero before t, or again, up to the first zero after t, are shown to exist, and are characterized.

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