Randomised Markov Bridges and Applications in Finance

Abstract

We consider the filtering problem of estimating a hidden random variable X by noisy observations. The noisy observation process is given by a randomised Markov bridge (RMB) of which terminal value is set to X. That is, at the terminal time T, the noise of the bridge process vanishes and the hidden random variable X is revealed. We give the explicit filtering formula, also known as the Bayesian posterior probability formula, for a general RMB. It turns out that the posterior probability is given by a function of time, the initial and the current observations of the RMB, and the prior distribution of X. With these results at hand, we then go on to applying RMBs to the modelling of (multi-curve) interest rates and further to the pricing of financial assets and insurance liabilities. References: A. Macrina and J. Sekine (2014) Filtering with Randomised Markov Bridges. http://arxiv.org/abs/1411.1214 S. Crepey, A. Macrina, T. M. Nguyen and D. Skovmand (2015) Rational Multi-Curve Models with Counterparty-Risk. http://arxiv.org/abs/1502.07397 A. Macrina (2014) Heat Kernel Models for Asset Pricing. International Journal of Theoretical and Applied Finance, 17 (7), 1-34.