## The Annals of Applied Probability

### Singular optimal strategies for investment with transaction costs

Ananda P. N. Weerasinghe

#### Abstract

We study an investment decision problem for an investor who has available a risk-free asset (such as a bank account) and a chosen risky asset. It is assumed that the interest rate for the risk-free asset is zero. The amount invested in the risky asset is given by an Itô process with infinitesimal parameters $\mu (\cdot)$ and $\sigma (\cdot)$, which come from a control set. This control set depends on the investor's wealth in the risky asset. The wealth can be transferred between the two assets and there are charges on all transactions equal to a fixed percentage of the amount transacted. The investor's financial goal is to achieve a total wealth of $a > 0$. The objective is to find an optimal strategy to maximize the probability of reaching a total wealth a before bankruptcy. Under certain conditions on the control sets, an optimal strategy is found that consists of an optimal choice of a risky asset and an optimal choice for the allocation of wealth (buying and selling policies) between the two assets.

#### Article information

Source
Ann. Appl. Probab., Volume 8, Number 4 (1998), 1312-1330.

Dates
First available in Project Euclid: 9 August 2002

https://projecteuclid.org/euclid.aoap/1028903383

Digital Object Identifier
doi:10.1214/aoap/1028903383

Mathematical Reviews number (MathSciNet)
MR1661192

Zentralblatt MATH identifier
0967.93096

#### Citation

Weerasinghe, Ananda P. N. Singular optimal strategies for investment with transaction costs. Ann. Appl. Probab. 8 (1998), no. 4, 1312--1330. doi:10.1214/aoap/1028903383. https://projecteuclid.org/euclid.aoap/1028903383

#### References

• [1] Browne, S. (1995). Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin. Math. Oper. Res. 20 937- 958.
• [2] Cvitanic, J. and Karatzas, I. (1996). Hedging and portfolio optimization under transaction costs: a martingale approach. Math. Finance 6 133-165.
• [3] Davis, M. H. A. and Norman, A. R. (1990). Portfolio selection with transaction costs. Math. Oper. Res. 15 676-713.
• [4] Davis, M. H. A., Panas, V. G. and Zariphopolou, T. (1993). European option pricing with transaction costs. SIAM J. Control Optim. 31 470-493.
• [5] Fleming, W. H. and Soner, H. M. (1993). Controlled Markov Processes and Viscosity Solutions. Springer, Berlin.
• [6] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York.
• [7] Lions, P. L. and Sznitman, A.-S. (1984). Stochasatic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 511-537.
• [8] McBeth, D. W. and Weerasinghe, A. (1995). Optimal singular control strategies for controlling a process to a goal. Preprint.
• [9] Meyer, P. A. (1974). Un cours sur les int´egrales Stochastiques. Seminaire de Probabilit´es X. Lecture Notes in Math. 511. Springer, New York.
• [10] Pestien, V. and Sudderth, W. (1985). Continuous-time red and black: how to control a diffusion to a goal. Math. Oper. Res. 13 599-611.
• [11] Protter, M. H. and Weinberger, H. F. (1967). Maximum Principles in Differential Equations, 2nd ed. Springer, New York.
• [12] Protter, P. (1990). Stochastic Integration and Differential Equations. Appl. Math. 21. Springer, New York.
• [13] Shreve, S. E. and Soner, H. M. (1994). Optimal investment and consumption with transaction costs. Ann. Appl. Probab. 4 609-692.
• [14] Sudderth, W. and Weerasinghe, A. (1991). Using fuel to control a process to a goal. Stochastics Stochastics Rep. 34 169-186.
• [15] Sudderth, W. and Weerasinghe, A. (1992). A bang-bang strategy for a finite-fuel stochasatic control problem. Adv. in Appl. Probab. 24 589-603.
• [16] Taksar, M., Klass, J. J. and Assaf, D. (1988). A diffusion model for optimal portfolio selection in the presence of brokerage fees. Math. Oper. Res. 13 277-294.