We discuss Itô’s stochastic calculus, which will be applied in later discussions. In Sects. 2.1, 2.2, 2.3, and 2.4, we discuss stochastic calculus related to integrals by Wiener processes and continuous martingales. In Sect. 2.1, we define stochastic integrals based on Wiener processes and continuous martingales. In Sect. 2.2, we establish Itô’s formula. It will be applied for proving Lp-estimates of stochastic integrals, called the Burkholder–Davis–Gundy inequality, and Girsanov’s theorem. The smoothness of the stochastic integral with respect to parameter will be discussed in Sect. 2.3. Fisk–Stratonovitch symmetric integrals will be discussed in Sect. 2.4.
In Sects. 2.5 and 2.6, we discuss stochastic calculus based on Poisson random measures. Stochastic integrals are defined in Sect. 2.5. The chain rules formula for jump processes and Lp-estimates of jump integrals will be discussed in Sect. 2.6. In Sect. 2.7, we discuss the backward processes and backward integrals. These topics are related to dual processes or inverse processes, which will be discussed in Chaps. 3 and 4.
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