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Convexity bounds for BSDE solutions, with applications to indifference valuation. (English) Zbl 1227.60073

From the author’s abstract: “We consider backward stochastic differential equations (BSDEs) with a particular quadratic generator and study the behaviour of their solutions when the probability measure is changed, the filtration is shrunk or the underlying probability space is transformed. Our main results are upper bounds for the solutions of the original BSDEs in terms of solutions to other BSDEs which are easier to solve. We illustrate our results by applying them to exponential utility indifference valuation in a multidimensional Itô process setting.”

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
91G80 Financial applications of other theories
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[1] Alvino, A.; Lions, P.-L.; Trombetti, G., Comparison results for elliptic and parabolic equations via symmetrization: a new approach, Differ. Integral Equ., 4, 25-50 (1991) · Zbl 0735.35003
[2] Ankirchner, S., Imkeller, P., Reis, G.: Pricing and hedging of derivatives based on non-tradable underlyings. Math. Finance (to appear). http://wahrscheinlichkeitstheorie.hu-berlin.de/ · Zbl 1217.91178
[3] Azoff, E. A., Borel measurability in linear algebra, Proc. Am. Math. Soc., 42, 346-350 (1974) · Zbl 0286.15006 · doi:10.1090/S0002-9939-1974-0327799-1
[4] Barrieu, P., El Karoui, N.: Pricing, hedging, and designing derivatives with risk measures. In: Carmona, R. (ed.) Indifference Pricing: Theory and Applications, pp. 77-146. Princeton University Press (2009) · Zbl 1189.91200
[5] Briand, P.; Hu, Y., Quadratic BSDEs with convex generators and unbounded terminal conditions, Probab. Theory Relat. Fields, 141, 543-567 (2008) · Zbl 1141.60037 · doi:10.1007/s00440-007-0093-y
[6] El Karoui, N.; Peng, S.; Quenez, M. C., Backward stochastic differential equations in finance, Math. Finance, 7, 1-71 (1997) · Zbl 0884.90035 · doi:10.1111/1467-9965.00022
[7] Frei, C.; Schweizer, M., Exponential utility indifference valuation in two Brownian settings with stochastic correlation, Adv. Appl. Probab., 40, 401-423 (2008) · Zbl 1154.91024 · doi:10.1239/aap/1214950210
[8] Frei, C.; Schweizer, M.; Delbaen, F.; Rásonyi, M.; Stricker, C., Exponential utility indifference valuation in a general semimartingale model, Optimality and Risk—Modern Trends in Mathematical Finance. The Kabanov Festschrift, 49-86 (2009), Berlin: Springer, Berlin · Zbl 1188.91216 · doi:10.1007/978-3-642-02608-9_4
[9] Hu, Y.; Imkeller, P.; Müller, M., Utility maximization in incomplete markets, Ann. Appl. Probab., 15, 1691-1712 (2005) · Zbl 1083.60048 · doi:10.1214/105051605000000188
[10] Karatzas, I.; Shreve, S., Methods of Mathematical Finance. Applications of Mathematics, vol. 39 (1998), New York: Springer, New York · Zbl 0941.91032
[11] Kazamaki, N., Continuous Exponential Martingales and BMO. Lecture Notes in Mathematics, vol. 1579 (1994), New York: Springer, New York · Zbl 0806.60033
[12] Kobylanski, M., Stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab., 28, 558-602 (2000) · Zbl 1044.60045 · doi:10.1214/aop/1019160253
[13] Mania, M.; Schweizer, M., Dynamic exponential utility indifference valuation, Ann. Appl. Probab., 15, 2113-2143 (2005) · Zbl 1134.91449 · doi:10.1214/105051605000000395
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