The renormalized projection operator technique for linear stochastic differential equations Abstract. A backward stochastic differential equation (BSDE) is an Ito stochastic differential equation (SDE) for which a random terminal condition on the state has been specified. The paper deals with estimation problems for partly observed stochastic processes described by linear SDEs with uncertain disturbances. The disturbances and unknown initial states are supposed to be constrained by the inequality including mathematical expectation of the integral quadratic cost. We consider our equations as BSDEs, and construct at given instant the random information set of all possible states which are compatible with the measurements and the constraints. The center of this set represents the best estimation of the process' state. The evolutionary equations for the random information set and for the best estimation are given. Some examples and applications are considered.
In the field of Complex Analysis, it is acknowledged that Cauchy’s Residue Theorem plays an essential role, which allows the calculation of complex integrals by adding up the residues of singularities in the complex plane. Many mathematicians have developed various theorems out of Cauchy’s Residue Theorem and have solved numerous problems using Cauchy’s Residue Theorem, but there are still a lot more studies needed. Thus, this paper focuses on examining Residue Theorem deeply by introducing singularity point and residue, combining Laurent series and complex integral, then deducting Cauchy’s Residue Theorem. This paper then concentrates on solving four unique complex integrals to illustrate Cauchy’s Residue Theorem by analyzing the graph of integration, reformatting the integrals, applying theorems or tricks, integrating the reformatted integrals, and simplifying the results. As a result, this paper not only presented a deeper analysis of the deduction of Cauchy’s Residue Theorem, but also presented the solutions towards four previously unsolved complex integrals. The deduction of Cauchy’s Residue Theorem and the four complex analysis problems have important applications for dealing with integrals associated with hyperbolic functions and lead to future research in other areas of mathematics and physics.
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