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Kumar Muthuraman and Sunil Kumar (2008), "Solving Free-boundary Problems with Applications in Finance", Foundations and TrendsĀ® in Stochastic Systems: Vol. 1: No. 4, pp 259-341. http://dx.doi.org/10.1561/0900000006

© 2008 K. Muthuraman and S. Kumar

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**In this article:**

1 Introduction

2 Portfolio Optimization with One Stock and Transaction Costs

3 American Option Pricing

4 Portfolio Optimization with Two Stocks and Transaction Costs

5 Computing the Solution of the Fixed-boundary PDE

6 Portfolio Optimization with Many Stocks

References

Stochastic control problems in which there are no bounds on the rate of control reduce to so-called free-boundary problems in partial differential equations (PDEs). In a free-boundary problem the solution of the PDE and the domain over which the PDE must be solved need to be determined simultaneously. Examples of such stochastic control problems are singular control, optimal stopping, and impulse control problems. Application areas of these problems are diverse and include finance, economics, queuing, healthcare, and public policy. In most cases, the free-boundary problem needs to be solved numerically.

In this survey, we present a recent computational method that solves these free-boundary problems. The method finds the free-boundary by solving a sequence of fixed-boundary problems. These fixed-boundary problems are relatively easy to solve numerically. We summarize and unify recent results on this *moving boundary* method, illustrating its application on a set of classical problems, of increasing difficulty, in finance. This survey is intended for those are primarily interested in computing numerical solutions to these problems. To this end, we include actual Matlab code for one of the problems studied, namely, American option pricing.

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1: Introduction

2: Portfolio Optimization with One Stock and Transaction Costs

3: American Option Pricing

4: Portfolio Optimization with Two Stocks and Transaction Costs

5: Computing the Solution of the Fixed Boundary PDE

6: Portfolio Optimization with Many Stocks

References

Stochastic control problems in which there are no bounds on the rate of control reduce to so-called free-boundary problems in partial differential equations (PDEs). In a free-boundary problem the solution of the PDE and the domain over which the PDE must be solved need to be determined simultaneously. Examples of such stochastic control problems are singular control, optimal stopping and impulse control problems. Application areas of these problems are diverse and include finance, economics, queuing, healthcare and public policy. In most cases, the free-boundary problem needs to be solved numerically. This volume presents a recent computational method that solves these free-boundary problems. The method finds the free boundary by solving a sequence of fixed-boundary problems. These fixed boundary problems are relatively easy to solve numerically. Recent results on this moving boundary method are summarized and unified, illustrating its application on a set of classical problems, of increasing difficulty, in finance. This volume is intended for those who are primarily interested in computing numerical solutions to these problems. To this end, actual Matlab code is included for one of the problems studied, namely, American option pricing.