Weak approximation of stochastic delay [Elektronische Ressource] : differential equations with bounded memory by discrete time series / von Robert Lorenz

Weak Approximation of Stochastic DelayDifferential Equations with Bounded Memory byDiscrete Time SeriesDISSERTATIONzur Erlangung des akademischen Gradesdoctor rerum naturalium(Dr. rer. nat.)im Fach Mathematikeingereicht an derMathematisch-Naturwissenschaftlichen Fakultät IIder Humboldt-Universität zu BerlinvonHerrn Dipl.-Math.RobertLorenzgeboren am 9. Juni 1973 in Schwedt/OderPräsident der Humboldt-Universität zu Berlin:Prof. Dr. Hans Jürgen PrömelDekan der Mathematisch-Naturwissenschaftlichen Fakultät II:Prof. Dr. Uwe KüchlerGutachter:1. Prof. Dr. Uwe Küchler2. Prof. Dr. Evelyn Buckwar3. Prof. Dr. Hans-Michael Dietzeingereicht am: 25. Oktober 2005Tag der mündlichen Prüfung: 20. März 2006AbstractConsider the stochastic delay differential equation (SDDE) with length of memory rdX(t) =b(X )dt+σ(X )dB(t),t twhich has a unique weak solution. Here B is a Brownian motion, b and σ are con-tinuous, locally bounded functions defined on the space C[−r,0], and X denotes thetsegment of the values of X(u) for time points u in the interval [t,t−r]. Our aimh his to construct a sequence of discrete time series X of higher order, such that Xconverges weakly to the solution X of the stochastic differential delay equation as htends to zero.On the other hand we shall establish under which conditions a given sequence ofhtime seriesX of higher order converges weakly to the weak solutionX of a stochasticdifferential delay equation.