Nonlocal fully nonlinear parabolic differential equations arising in time-inconsistent problems

Nonlocality, nonlinearity, and time inconsistency in stochastic differential games

“…for some constant 𝐶, which is generic in this paper and its value could vary in different inequalities below. Then, similar to (Lei and Pun, 2023 Here, 𝐶 is a constant depending on 𝐴 and 𝐵. Therefore, it follows that 𝑽 ∈ 𝚯 (2𝑟+𝛼) [0,𝛿] .…”

“…Then, we revisit the definition of "parabolic" Hölder spaces, which is commonly adopted in the studies of local parabolic equations, including Ėǐdel'man (1969), Lei and Pun (2023). Let 𝐶 𝑙 2𝑟…”

“…However, their work has converted the key open problem in Björk et al (2017) to another open problem in PDE or SDE theory. Recently, Lei and Pun (2023) has shown the existence and uniqueness of a nonlocal fully nonlinear parabolic PDE in a small-time setting.…”

“…The diagonal dependence, referring to that $$(\partial _I\bm {u})_{|I|\le 2r}$$ of $$\bm {F}$$ are evaluated not only at $$(t,s,y)$$ but also at $$(s,s,y)$$, directly results in nonlocality. When $$r=1$$ and $$m=1$$, the system (1) or (2) is reduced to nonlocal fully nonlinear parabolic partial differential equations (PDEs) studied in Lei and Pun (2023).…”

“…then the family of random fields (𝑿(⋅), 𝒀(⋅, ⋅), 𝒁(⋅, ⋅), 𝚪(⋅, ⋅), 𝑨(⋅, ⋅)) is an adapted solution of the following flow of 2FBSDEs: We make three important observations about the stochastic system (67): (I) When the generator 𝔽 is independent of diagonal terms, that is, 𝒀(𝜏, 𝜏), 𝒁(𝜏, 𝜏), and 𝚪(𝜏, 𝜏), the flow of FBSDEs ( 67) is reduced to a family of 2FBSDEs parameterized by 𝑡, which is exactly the 2FBSDEs in Kong et al (2015) and equivalent to the ones in Cheridito et al (2007) for any fixed 𝑡; (II) Equation ( 67) is more general than the systems in Wang and Yong (2019), Wang (2020), Hamaguchi (2021b), Lei and Pun (2023) since it allows for a nonlinearity of (𝒀(𝑡, 𝜏), 𝒁(𝑡, 𝜏), 𝚪(𝑡, 𝜏)) by increasing the dimensions and/or introducing an additional SDE of (𝚪, 𝑨) as well as diagonal terms (𝒀(𝜏, 𝜏), 𝒁(𝜏, 𝜏), 𝚪(𝜏, 𝜏)) in almost arbitrary way; (III) Theorem 5.2 shows how to solve the flow of multidimensional 2FBS-DEs (67) from the perspective of nonlocal systems. Inspired by Cheridito et al (2007), Soner et al (2011), the opposite implication of solutions (from 2FBSDEs to PDE) is likely valid by establishing the well-posedness of Equation (67) in the theoretical framework of SDEs.…”