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This chapter is devoted to study existence and uniqueness of solutions of SDEs w.r.t. optional semimartingales as well as their comparison property. A special attention is paid to linear SDEs and their solutions as stochastic exponents and logarithms. It is shown how this technique is exploited for financial modeling and pricing (see Abdelghani and Melnikov, Optional processes: theory and applications, 2020; Abdelghani and Melnikov, Stochastics 92(1):67–89, 2020; Gasparyan, Investigacia Vysshikh Uchebnykh Zavedenii 12:57–60, 1985; Jarni and Ouknine, J Theor Probab 34(4):1811–1830, 2021, and Krasin et al., Ann Financ 14(2):195–209, 2018).

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Optional Stochastic Differential Equations and Their Applications

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  • 发表日期：

    2026年

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  • Alexander Melnikov

摘要

This chapter is devoted to study existence and uniqueness of solutions of SDEs w.r.t. optional semimartingales as well as their comparison property. A special attention is paid to linear SDEs and their solutions as stochastic exponents and logarithms. It is shown how this technique is exploited for financial modeling and pricing (see Abdelghani and Melnikov, Optional processes: theory and applications, 2020; Abdelghani and Melnikov, Stochastics 92(1):67–89, 2020; Gasparyan, Investigacia Vysshikh Uchebnykh Zavedenii 12:57–60, 1985; Jarni and Ouknine, J Theor Probab 34(4):1811–1830, 2021, and Krasin et al., Ann Financ 14(2):195–209, 2018).