1 Complete and Incomplete Markets 
In a complete and arbitrage-free market, every contingent claim can be perfectly replicated by a self-financing trading strategy, which implies the existence of a unique equivalent martingale measure. In contrast, incomplete markets lack this property, often due to limitations in tradable assets or to the presence of additional stochastic factors, like stochastic volatility, leading to multiple martingale measures and non-unique prices. We will introduce the concept of incomplete markets and derive conditions that allow us to identify the market nature. Moreover, we discuss the implications of incompleteness of markets to relevant problems in quantitative finance, namely, pricing and hedging of financial instruments.

2 Quadratic hedging methods 
When markets are incomplete, perfect replication of contingent claims is no longer feasible. Instead, pricing and hedging must rely on optimality principles, such as mean-variance hedging or local risk minimization. These methods yield strategies that minimize quadratic hedging errors under a chosen risk criterion.

3 Basics of Nonlinear Filtering
Nonlinear filtering deals with estimating unobservable components of a stochastic system based on noisy observations. Key tools include the Zakai and Kushner- Stratonovich equations, which describe the evolution of the conditional distribution of the hidden state . This framework is fundamental in partially observed financial models. We discuss the methodology for deriving nonlinear filtering equations and a few special cases.

4 Applications to Financial Problems 
Under Partial Information In many financial applications, key variables such as drift or volatility are not directly observable. Filtering techniques are used to estimate these hidden variables and optimize decisions under uncertainty. In the last part of the course, we discuss a couple of key examples taken from recent literature where nonlinear filtering is applied to relevant financial problems.

References:
• Bain, A., & Crisan, D. (2009). Fundamentals of Stochastic Filtering. Springer.
• Björk, T. (2005). Arbitrage Theory in Continuous Time. Oxford University Press.
• Björk, T., Davis, M. H., & Landén, C. (2010). Optimal investment under partial information. Mathematical Methods of Operations Research, 71, 371-399.
• Cochrane, J. H., & Saa-Requejo, J. (2000). Beyond arbitrage: Good-deal asset price bounds in incomplete markets. Journal of political economy, 108(1), 79-119.
• Föllmer, H., & Sondermann, D. (1986). Hedging of Non-redundant Contingent Claims. In Contributions to Mathematical Economics, pp. 205–223.
• Rieder, U., & Bäuerle, N. (2005). Portfolio optimization with unobservable Markov-modulated drift process. Journal of Applied Probability, 42(2), 362-378.
• Schweizer, M. (2001). A Guided Tour through Quadratic Hedging Approaches. In Option Pricing, Interest Rates and Risk Management, pp. 538–574.

Full attendance is compulsory. This means that students should attend at least 80% of all lectures, at most one lecture can be missed.

Please log in with your WU account to use all functionalities of read!t. For off-campus access to our licensed electronic resources, remember to activate your VPN connection connection. In case you encounter any technical problems or have questions regarding read!t, please feel free to contact the library at readinglists@wu.ac.at.