TY - JOUR
T1 - Second-order stochastic target problems with generalized market impact
AU - Bouchard, Bruno
AU - Loeper, Gregoire
AU - Sonera, Halil Mete
AU - Zhou, Chao
N1 - Funding Information:
The frst author was supported by ANR grant CAESARS (ANR-15-CE05-0024). The second author acknowledges support for CQFIS provided by BNP Paribas. The third author was partially supported by the ETH Foundation, the Swiss Finance Institute, and Swiss National Foundation through SNF 200020-172815. The fourth author's research was supported by Singapore MOE AcRF grants R-146-000-219-112 and R-146-000-255-114. The authors would like to thank two anonymous referees and the Associate Editor for their help in improving an earlier version of this paper.
Publisher Copyright:
© 2019 Society for Industrial and Applied Mathematics
PY - 2019
Y1 - 2019
N2 - We extend the study of [B. Bouchard, G. Loeper, and Y. Zou, SIAM J. Control Optim., 55 (2017), pp. 3319-3348; G. Loeper, Ann. Appl. Probab., 28 (2018), pp. 2664-2726] stochastic target problems with general market impacts. Namely, we consider a general abstract model which can be associated to a fully nonlinear parabolic equation. Unlike the earlier articles, the equation is not concave, and the regularization/verifcation approach of our 2017 cannot be applied. We also relax the gamma constraint of the 2017 article. Instead, we need to generalize the a priori estimates of Loeper's article and exhibit smooth solutions from the classical parabolic equations theory. Up to an additional approximating argument, this allows us to show that the superhedging price solves the parabolic equation and that a perfect hedging strategy can be constructed when the coefcients are smooth enough. This representation leads to a general dual formulation. We fnally provide an asymptotic expansion around a model without impact.
AB - We extend the study of [B. Bouchard, G. Loeper, and Y. Zou, SIAM J. Control Optim., 55 (2017), pp. 3319-3348; G. Loeper, Ann. Appl. Probab., 28 (2018), pp. 2664-2726] stochastic target problems with general market impacts. Namely, we consider a general abstract model which can be associated to a fully nonlinear parabolic equation. Unlike the earlier articles, the equation is not concave, and the regularization/verifcation approach of our 2017 cannot be applied. We also relax the gamma constraint of the 2017 article. Instead, we need to generalize the a priori estimates of Loeper's article and exhibit smooth solutions from the classical parabolic equations theory. Up to an additional approximating argument, this allows us to show that the superhedging price solves the parabolic equation and that a perfect hedging strategy can be constructed when the coefcients are smooth enough. This representation leads to a general dual formulation. We fnally provide an asymptotic expansion around a model without impact.
KW - Asymptotic expansion
KW - Fully nonlinear parabolic equation
KW - Generalized market impact
KW - Second-order stochastic target
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U2 - 10.1137/18M1196078
DO - 10.1137/18M1196078
M3 - Article
AN - SCOPUS:85077134632
SN - 0363-0129
VL - 57
SP - 4125
EP - 4149
JO - SIAM Journal on Control and Optimization
JF - SIAM Journal on Control and Optimization
IS - 6
ER -