Initially, I wanted to entitle the textbook “Stochastic Analysis for All ” or “Stochastic Analysis without Tears”, keeping in mind that it will be accessible not only to students of mathematics but also to physicists, chemists, biologists, financiers, actuaries, etc. However, though aiming for as wide a readability as possible, finally, I rejected such titles regarding them as too ambitious.

Most people have an intuitive concept of probability based on their own life experience. However, efforts to precisely define probabilistic notions meet serious difficulties; this is seen looking at the history of probability theory—from elementary combinatorial calculations in hazard games to a rigorous axiomatic theory, having a store of applications in various practical and scientific areas. Possibly, as in no other area of mathematics, in probability theory, there is a huge distance from the beginning and elements to the precise and rigorous theory. This is firstly related to the fact that the “palace” of probability theory is built on the substructure of the rather subtle and abstract measure theory. For example, for a mathematician, a random variable is a real measurable function defined on the space of elementary events, while for practitioners—physicists, chemists, biologists, actuaries, etc.—it is some quantity depending on chance. For a mathematician, the randomness is externalized by a probability measure on the measure space (space of elementary events with a σ-algebra ...

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