A study on stochastic differential equations

The derivation of the Ito ̂ formula for the squared norm of a semi-martingel whose components all take values in a single Hilbert space differs little from the finite-dimensional case. However, the jump in complexity from this case to the case described above is like the jump in complexity from bounded to unbounded operators. This paper demonstrates that stochastic equations with unbounded operators cannot be treated in a single space, necessitating the separation of the domain of the operators (the space V) from the interval of the operators (the space V' or H). This issue was first addressed by Pardoux. In this thesis, using completely different methods, Pardoux's result is generalized to non-reflective and non-separable spaces, and a new proof is given.