Stochastic Processes and Applications by Grigorios A. Pavliotis

Author:Grigorios A. Pavliotis
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

5.6 Exercises

1. a.Study the stochastic integral (5.109) and the corresponding SDE (5.108). In particular, derive (5.110).

b.Define this stochastic integral in arbitrary dimensions; write it as an Itô integral and obtain a formula for the correction in the drift.

2.Consider (5.111) and set . a.Use the singular perturbation techniques from Sect. 5.1 to derive (5.112). See also Sect. 6.​5.​1 and Exercise 8 in Chap. 6.

b.Prove (5.113). Analyze this formula when

c.What type of stochastic integral do you obtain when the fluctuation– dissipation theorem, i.e., λ = 1, is satisfied?

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