Abstract Cauchy problems and stochastic integral equations

An (inhomogeneous) abstract Cauchy problem is a type of differential equation of Banach space valued functions. The only derivate in a Cauchy problem is du dt , where u is the sought-after solution. However, because we consider such a general setting, we could for example also formulate some systems of multiple higher order differential equations as a Cauchy problem. In section 3 we consider certain stochastic integral equations. The connection with the (inhomogeneous) abstract Cauchy problem is, that the latter is (almost) a special case of the stochastic integral equations that we study. The idea is that the stochastic element adds the resulting effect of random ’white noise’. This can be useful in all sorts of applications, since it can model the effects of small changes one could not predict. We first consider a stochastic integral equation in a separable Hilbert space, for which an existence and uniqueness proof is known (see [GA]). We would like to generalise this by allowing...Show moreAn (inhomogeneous) abstract Cauchy problem is a type of differential equation of Banach space valued functions. The only derivate in a Cauchy problem is du dt , where u is the sought-after solution. However, because we consider such a general setting, we could for example also formulate some systems of multiple higher order differential equations as a Cauchy problem. In section 3 we consider certain stochastic integral equations. The connection with the (inhomogeneous) abstract Cauchy problem is, that the latter is (almost) a special case of the stochastic integral equations that we study. The idea is that the stochastic element adds the resulting effect of random ’white noise’. This can be useful in all sorts of applications, since it can model the effects of small changes one could not predict. We first consider a stochastic integral equation in a separable Hilbert space, for which an existence and uniqueness proof is known (see [GA]). We would like to generalise this by allowing values in an arbitrary Banach space. This is done in section 3.3. We will also prove a boundedness result there. We finish in section 3.4 with some specific examples of stochastic integral equations. I would like to thank my thesis supervisor Onno van Gaans for the effort he has put into guiding me in the right direction. Thanks to his regular help I have always had the sense of really getting somewhere.Show less

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Abstract Cauchy problems and stochastic integral equations