Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Stochastic Equations in Infinite Dimensions

Optimal consumption and portfolio policies when asset prices follow a diffusion process

This book is a comprehensive treatment of inference for hidden Markov models, including both algorithms and statistical theory. Topics range from filtering and smoothing of the hidden Markov chain to parameter estimation, Bayesian methods and estimation of the number of states. In a unified way the book covers both models with finite state spaces and models with continuous state spaces (also called state-space models) requiring approximate simulation-based algorithms that are also described in detail. Many examples illustrate the algorithms and theory. This book builds on recent developments to present a self-contained view.

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Inference in Hidden Markov Models

Using only the Boson canonical commutation relations and the Riemann-Lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes. This leads to a noncommutative Ito product formula, a realisation of the classical Poisson process in Fock space which gives a noncommutative central limit theorem, the construction of solutions of certain noncommutative stochastic differential equations, and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps. The classical Ito product formula for stochastic differentials with respect to Brownian motion and the Poisson process is a special case.

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Quantum Ito's formula and stochastic evolutions

Spot and Futures Markets.- An Introduction to Financial Derivatives.- Discrete-time Security Markets.- Benchmark Models in Continuous Time.- Foreign Market Derivatives.- American Options.- Exotic Options.- Volatility Risk.- Continuous-time Security Markets.- Fixed-income Markets.- Interest Rates and Related Contracts.- Short-Term Rate Models.- Models of Instantaneous Forward Rates.- Market LIBOR Models.- Alternative Market Models.- Cross-currency Derivatives.

Martingale Methods in Financial Modelling

Statistics of Random Processes I: General Theory

Statistics of Random Processes II. Applications

The stability problem of the Wonham filter with respect to initial conditions is addressed. The case of ergodic signals is revisited in view of a gap in the classic work of H. Kunita (1971). We give new bounds for the exponential stability rates, which do not depend on the observations. In the nonergodic case, the stability is implied by identifiability conditions, formulated explicitly in terms of the transition intensities matrix and the observation structure.

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Asymptotic stability of the wonham filter: ergodic and nonergodic signals

Let ${\frak z}$ be a stochastic exponential, i.e., ${\frak z}_t=1+\int_0^t{\frak z}_{s-}\,dM_s,$ of a local martingale $M$ with jumps $\triangle M_t>-1$. Then ${\frak z}$ is a nonnegative local martingale with ${\bf E}\,{\frak z}_t\le 1$. If ${\bf E}\,{\frak z}_{_T}= 1$, then ${\frak z}$ is a martingale on the time interval $[0,T]$. The martingale property plays an important role in many applications. It is therefore of interest to give natural and easily verifiable conditions for the martingale property. In this paper, the property ${\bf E}\,{\frak z}_{_T}=1$ is verified with the so-called linear growth conditions involved in the definition of parameters of $M$, proposed by Girsanov [Theory Probab. Appl., 5 (1960), pp. 285--301]. These conditions generalize the Benes idea [SIAM J. Control, 9 (1971), pp. 446--475] and avoid the technology of piecewise approximation. These conditions are applicable even if the Novikov [Theory Probab. Appl., 24 (1979), pp. 820--824] and Kazamaki [Tohoku Math. J., 29 (1977),...

When a Stochastic Exponential Is a True Martingale. Extension of the Beneš Method

Stability problem of the Wonham filter with respect to initial conditions is addressed. The case of ergodic signals is revisited in view of a gap in the classic work of H. Kunita (1971). We give new bounds for the exponential stability rates, which do not depend on the observations. In the non-ergodic case, the stability is implied by identifiability conditions, formulated explicitly in terms of the transition intensities matrix and the observation structure.

Robert Liptser

Papers

Statistics of Random Processes I: General Theory

Statistics of Random Processes II. Applications

Asymptotic stability of the wonham filter: ergodic and nonergodic signals

When a Stochastic Exponential Is a True Martingale. Extension of the Beneš Method

Asymptotic stability of the Wonham filter for ergodic and nonergodic signals