Abstract
In this chapter we provide the proofs of the most demanding results of Chaps. 12–14. Have you noticed the (∗∗) in the title? It means “very difficult”. Given the dual nature of the proposed audience for the book (scholars and quants), I have provided in the first chapters a lot of background material. Yet I didn’t want to avoid the sometimes difficult mathematical technique that is needed for deep understanding. So, for the convenience of readers, we signal sections that contain advanced material with an asterisk (*) or even a double asterisk (**) for the still more difficult portions.
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Notes
- 1.
Cf. (12.63) for the definition of \(|\widehat{v} - \widehat{v} ^{\prime}| \).
- 2.
Nonnegative function from [0,∞] to itself, continuous and null at 0.
- 3.
Given the continuity of u.
- 4.
Given the continuity of v.
- 5.
See for instance the proof of the comparison principle of Proposition 12.1.10 in [87].
- 6.
Note that the following argument only works at T and cannot be adapted to the case of problem \((\mathcal{V}1)\) on the whole of \(\partial\mathcal{D}\); see the comment at the beginning of the proof.
References
Alvarez, O., & Tourin, A. (1996). Viscosity solutions of nonlinear integro-differential equations. Annales de l’Institut Henri Poincaré (C) Analyse non linéaire, 13(3), 293–317.
Amadori, A. L. (2003). Nonlinear integro-differential evolution problems arising in option pricing: a viscosity solutions approach. Differential and Integral Equations, 16(7), 787–811.
Amadori, A. L. (2007). The obstacle problem for nonlinear integro-differential operators arising in option pricing. Ricerche Di Matematica, 56(1), 1–17.
Barles, G., Buckdahn, R., & Pardoux, E. (1997). Backward stochastic differential equations and integral-partial differential equations. Stochastics & Stochastics Reports, 60, 57–83.
Barles, G., & Souganidis, P. E. (1991). Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Analysis, 4, 271–283.
Crandall, M., Ishii, H., & Lions, P.-L. (1992). User’s guide to viscosity solutions of second order partial differential equations. Bulletin of the American Mathematical Society, 27, 1–67.
Crépey, S., & Matoussi, A. (2008). Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison principle. The Annals of Applied Probability, 18(5), 2041–2069.
Fleming, W., & Soner, H. (2006). Controlled Markov processes and viscosity solutions (2nd ed.). New York: Springer.
Ikeda, N., & Watanabe, S. (1989). Stochastic differential equations and diffusion processes (2nd ed.). Amsterdam: North-Holland.
Jacod, J., & Shiryaev, A. (2003). Limit theorems for stochastic processes. Berlin: Springer.
Pardoux, E., Pradeilles, F., & Rao, Z. (1997). Probabilistic interpretation of systems of semilinear PDEs. Annales de l’Institut Henri Poincaré, série Probabilités–Statistiques, 33, 467–490.
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Crépey, S. (2013). Technical Proofs (∗∗). In: Financial Modeling. Springer Finance(). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37113-4_15
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