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On some Brownian functionals and their applications to moments in the lognormal stochastic volatility model

100%
Jakubowski J., Wiśniewolski M.
Studia Mathematica
|
2013
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tom 219
|
nr 3
201-224
EN
We find a probabilistic representation of the Laplace transform of some special functional of geometric Brownian motion using squared Bessel and radial Ornstein-Uhlenbeck processes. Knowing the transition density functions of these processes, we obtain closed formulas for certain expectations of the relevant functional. Among other things we compute the Laplace transform of the exponent of the T transforms of Brownian motion with drift used by Donati-Martin, Matsumoto, and Yor in a variety of identities of duality type between functionals of Brownian motion. We also present links between geometric Brownian motion and Markov processes studied by Matsumoto and Yor. These results have wide applications. As an example of their use in financial mathematics we find the moments of processes representing the asset price in the lognormal volatility model.
2
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Pricing bonds and CDS in the model with rating migration induced by a Cox process

100%
Jakubowski J., Niewęgłowski M.
Banach Center Publications
|
2008
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tom 83
|
nr 1
159-182
EN
We investigate the properties of a rating migration process assuming that it is given by subordination of a discrete time Markov chain and a Cox process. The problem of pricing of defaultable bonds with fractional recovery of par value with rating migration and credit default swaps is considered. As an example of applications of our results, we give an explicit solution to the pricing problem in a model with short rate and intensity processes given by the solution of a two-dimensional Ornstein-Uhlenbeck equation with a Lévy noise.
3
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Defaultable bonds with an infinite number of Lévy factors

100%
Jakubowski J., Niewęgłowski M.
Applicationes Mathematicae
|
2010
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tom 37
|
nr 3
275-307
EN
A market with defaultable bonds where the bond dynamics is in a Heath-Jarrow-Morton setting and the forward rates are driven by an infinite number of Lévy factors is considered. The setting includes rating migrations driven by a Markov chain. All basic types of recovery are investigated. We formulate necessary and sufficient conditions (generalized HJM conditions) under which the market is arbitrage-free. Connections with consistency conditions are discussed.
4
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Conditional Markov chains - construction and properties

81%
Bielecki T., Jakubowski J., Niewęgłowski M.
Banach Center Publications
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2015
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tom 105
|
nr 1
33-42
EN
In this paper we study finite state conditional Markov chains (CMCs). We give two examples of CMCs, one which admits intensity, and another one, which does not admit an intensity. We also give a sufficient condition under which a doubly stochastic Markov chain is a CMC. In addition we provide a method for construction of conditional Markov chains via change of measure.
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