Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Stochastic equations in infinite dimensions

(1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

https://link.springer.com/content/pdf/10.1057%2Fjors.1987.184.pdf

Applied Probability and Queues

Levy processes are the natural continuous-time analogue of random walks and form a rich class of stochastic processes around which a robust mathematical theory exists. Their mathematical significance is justified by their application in many areas of classical and modern stochastic models.

This textbook forms the basis of a graduate course on the theory and applications of Levy processes, from the perspective of their path fluctuations. Central to the presentation are decompositions of the paths of Levy processes in terms of their local maxima and an understanding of their short- and long-term behaviour.

The book aims to be mathematically rigorous while still providing an intuitive feel for underlying principles. The results and applications often focus on the case of Levy processes with jumps in only one direction, for which recent theoretical advances have yielded a higher degree of mathematical transparency and explicitness. Each chapter has a comprehensive set of exercises with complete solutions.

Introductory Lectures on Fluctuations of Lévy Processes with Applications

1. Stochastic processes and random fields 2. Continuous semimartingales and stochastic integrals 3. Semimartingales with spatial parameter and stochastic integrals 4. Stochastic flows 5. Convergence of stochastic flows 6. Stochastic partial differential equations.

Stochastic flows and stochastic differential equations

In this paper the neutral valuation approach is applied to American and game options in incomplete markets. Neutral prices occur if investors are utility maximizers and if derivative supply and demand are balanced. Game contingent claims are derivative contracts that can be terminated by both counterparties at any time before expiration. They generalize American options where this right is limited to the buyer of the claim. It turns out that as in the complete case, the price process of American and game contingent claims corresponds to a Snell envelope or to the value of a Dynkin game, respectively. On the technical level, an important role is played by $\sigma$
 -sub- and $\sigma$
 -supermartingales. We characterize these processes in terms of semimartingale characteristics.

Pricing derivatives of American and game type in incomplete markets

Fractional Brownian motion as a weak limit of Poisson shot noise processes - with applications to finance.

Introduced by Kifer (2000), game options function in the same way as American options with the added feature that the writer may also choose to exercise, at which time they must pay out the intrinsic option value of that moment plus a penalty. In Kyprianou (2004) an explicit formula was obtained for the value function of the perpetual put option of this type. Crucial to the calculations which lead to the aforementioned formula was the perpetual nature of the option. In this paper we address how to characterize the value function of the finite expiry version of this option via mixtures of other exotic options by using mainly martingale arguments.

/pdf/callable-puts-as-composite-exotic-options-86hmwha0xb.pdf

Callable puts as composite exotic options

A convertible bond is a security that the holder can convert into a specified number of underlying shared. In addition, very often the issuer can recall the bond, paying some compensation, or force the holder to convert it immediately. Therefore, the pricing problem has also a game-theoretic aspect. When modelling convertible (callable) bonds within the framework of a firm value model, they can be consideres as an example of a standard game contingent claim as long as no dividends are distributed to the equity holders. This article reviews the classical as well as some recend literature in this field. Furthermore, we introduce a mathematically riorous concept of no-arbitrage price processes for this kind of derivatives, which explicitly incorporates the feature that the contract can be terminated by both counterparties prematurely. We compare this dynamic conception to price derivatives with the static one by Karatzas and Kou [18].

Convertible Bonds: Financial Derivatives of Game Type

We study Merton’s portfolio optimization problem in a limit order market. An investor trading in a limit order market has the choice between market orders that allow immediate transactions and limit orders that trade at more favorable prices but are executed only when another market participant places a corresponding market order. Assuming Poisson arrivals of market orders from other traders we use a shadow price approach, similar to Kallsen and Muhle-Karbe (Ann Appl Probab, forthcoming) for models with proportional transaction costs, to show that the optimal strategy consists of using market orders to keep the proportion of wealth invested in the risky asset within certain boundaries, similar to the result for proportional transaction costs, while within these boundaries limit orders are used to profit from the bid–ask spread. Although the given best-bid and best-ask price processes are geometric Brownian motions the resulting shadow price process possesses jumps.

/pdf/optimal-portfolios-of-a-small-investor-in-a-limit-order-19o6rl5xb1.pdf

Christoph Kühn

Papers

Pricing derivatives of American and game type in incomplete markets

Fractional Brownian motion as a weak limit of Poisson shot noise processes - with applications to finance.

Callable puts as composite exotic options

Convertible Bonds: Financial Derivatives of Game Type

Optimal portfolios of a small investor in a limit order market: a shadow price approach