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This is taught as part of the MIGSAA CDT programme and offered across SMSTC.
After a brief presentation of the probabilistic framework of the course, we motivate our interest in stochastic PDEs. In particular, we show that to solve basic problems in nonlinear filtering theory we have to investigate certain classes of linear and also nonlinear stochastic PDEs. With this and other applications in mind, we develop first a theory, called L_2 theory, on existence, uniqueness and regularity of the solutions to linear parabolic SPDEs, and then more advanced results from the L_p theory will be presented. Hence we get existence and uniqueness of classical solutions to linear (and semilinear) parabolic SPDEs. Moreover we show that these solutions can also be obtained by the method of random characteristics, which generalises the well known method of characteristics used in solving first order (deterministic) PDEs. Next we generalise the L_2 theory to solve nonlinear stochastic evolution equations. Finally we apply the theory of SPDEs presented in the course to solve the main problems of filtering theory we started with. Moreover, we show some other applications in physics and population genetics.