Small time path behavior of double stochastic integrals and applications to stochastic control

Patrick Cheridito; H. Mete Soner; Nizar Touzi

doi:10.1214/105051605000000557

November 2005 Small time path behavior of double stochastic integrals and applications to stochastic control

Patrick Cheridito, H. Mete Soner, Nizar Touzi

Ann. Appl. Probab. 15(4): 2472-2495 (November 2005). DOI: 10.1214/105051605000000557

Abstract

We study the small time path behavior of double stochastic integrals of the form ∫₀^t(∫₀^rb(u) dW(u))^T dW(r), where W is a d-dimensional Brownian motion and b is an integrable progressively measurable stochastic process taking values in the set of d×d-matrices. We prove a law of the iterated logarithm that holds for all bounded progressively measurable b and give additional results under continuity assumptions on b. As an application, we discuss a stochastic control problem that arises in the study of the super-replication of a contingent claim under gamma constraints.

References

1.

Bank, P. and Baum, D. (2004). Hedging and portfolio optimization in financial markets with a large trader. Math. Finance 14 1–18. MR2030833 1119.91040 10.1111/j.0960-1627.2004.00179.x Bank, P. and Baum, D. (2004). Hedging and portfolio optimization in financial markets with a large trader. Math. Finance 14 1–18. MR2030833 1119.91040 10.1111/j.0960-1627.2004.00179.x

2.

Cheridito, P., Soner, H. M. and Touzi, N. (2005). The multi-dimensional super-replication problem under gamma constraints. Ann. Inst. H. Poincaré (C) Non Linear Analysis 22 633–666. Cheridito, P., Soner, H. M. and Touzi, N. (2005). The multi-dimensional super-replication problem under gamma constraints. Ann. Inst. H. Poincaré (C) Non Linear Analysis 22 633–666.

3.

Levental, S. and Skorohod, A. V. (1997). On the possibility of hedging options in the presence of transaction costs. Ann. Appl. Probab. 7 410–443. MR1442320 0883.90018 10.1214/aoap/1034625338 euclid.aoap/1034625338 Levental, S. and Skorohod, A. V. (1997). On the possibility of hedging options in the presence of transaction costs. Ann. Appl. Probab. 7 410–443. MR1442320 0883.90018 10.1214/aoap/1034625338 euclid.aoap/1034625338

4.

Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Springer, Berlin. MR1725357 0917.60006 Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Springer, Berlin. MR1725357 0917.60006

5.

Shiryaev, A. N. (1990). Probability, 2nd ed. Springer, New York. Shiryaev, A. N. (1990). Probability, 2nd ed. Springer, New York.

6.

Soner, H. M. and Touzi, N. (2000). Superreplication under gamma constraints. SIAM J. Control Optim. 39 73–96. MR1780909 0960.91036 10.1137/S0363012998348991 Soner, H. M. and Touzi, N. (2000). Superreplication under gamma constraints. SIAM J. Control Optim. 39 73–96. MR1780909 0960.91036 10.1137/S0363012998348991

7.

Soner, H. M. and Touzi, N. (2002). Stochastic target problems, dynamic programming, and viscosity solutions. SIAM J. Control Optim. 41 404–424. MR1920265 1011.49019 10.1137/S0363012900378863 Soner, H. M. and Touzi, N. (2002). Stochastic target problems, dynamic programming, and viscosity solutions. SIAM J. Control Optim. 41 404–424. MR1920265 1011.49019 10.1137/S0363012900378863

8.

Soner, H. M. and Touzi, N. (2002). Dynamic programming for stochastic target problems and geometric flows. J. European Math. Soc. 4 201–236. MR1924400 1003.49003 10.1007/s100970100039 Soner, H. M. and Touzi, N. (2002). Dynamic programming for stochastic target problems and geometric flows. J. European Math. Soc. 4 201–236. MR1924400 1003.49003 10.1007/s100970100039

Citation Download Citation

Patrick Cheridito, H. Mete Soner, and Nizar Touzi "Small time path behavior of double stochastic integrals and applications to stochastic control," The Annals of Applied Probability 15(4), 2472-2495, (November 2005). https://doi.org/10.1214/105051605000000557

Published: November 2005

Access the abstract

JOURNAL ARTICLE
24 PAGES

DOWNLOAD PDF + SAVE TO MY LIBRARY