Stochastic Analysis with Financial Applications

Stochastic analysis has a variety of applications to biological systems as well as physical and engineering problems, and its applications to finance and insurance have bloomed exponentially in recent times. The goal of this book is to present a broad overview of the range of applications of stochastic analysis and some of its recent theoretical developments. This includes numerical simulation, error analysis, parameter estimation, as well as control and robustness properties for stochastic equations. The book also covers the areas of backward stochastic differential equations via the (non-linear) G-Brownian motion and the case of jump processes. Concerning the applications to finance, many of the articles deal with the valuation and hedging of credit risk in various forms, and include recent results on markets with transaction costs.
Contributors:
T.R. Bielecki
N. Bouleau
S. Chakraborty
T.S. Chiang
S.N. Cohen
J.M. Corcuera
S. Crépey
A.B. Cruzeiro
L. Denis
J. Duan
R.J. Elliott
S. Fang
M. Fukasawa
F.Q. Gao
B. Goldys
S. Han
Y. Ishikawa
M. Jeanblanc
H. Jiang
B. Jourdain
A. Kohatsu-Higa
E.T. Kolkovska
H. Lee
L. Li
J.A. López-Mimbela
J. Luo
B. Øksendahl
J. Ren
M. Rutkowski
E. Shamarova
S.J. Sheu
A. Sulem
A. Takeuchi
N. Vaytis
R. Wang
J. Wei
J. Wu
J. Yang
H. Yang
K. Yasuda
X. Zhang

Stochastic analysis has a variety of applications to biological systems as well as physical and engineering problems, and its applications to finance and insurance have bloomed exponentially in recent times. The goal of this book is to present a broad overview of the range of applications of stochastic analysis and some of its recent theoretical developments. This includes numerical simulation, error analysis, parameter estimation, as well as control and robustness properties for stochastic equations. The book also covers the areas of backward stochastic differential equations via the (non-linear) G-Brownian motion and the case of jump processes. Concerning the applications to finance, many of the articles deal with the valuation and hedging of credit risk in various forms, and include recent results on markets with transaction costs.

Part I: Stochastic Analysis.- Dirichlet forms for Poisson measures and Lévy processes: the lent particle method.- Backward stochastic difference equations with finite states.- On a forward-backward stochastic system associated to the Burgers equation.- Quantifying model uncertainties in complex systems.- On the estimate for commutators in DiPerna-Lions theory.- Approximation theorem for stochastic differential equations driven by G-Brownian motion.- Stochastic flows for nonlinear SPDEs driven by linear multiplicative space-time white noises.- Optimal stopping problem associated with jump-diffusion processes.- A review of recent results on approximation of solutions of stochastic differential equations.- Strong consistency of Bayesian estimator under discrete observations and unknown transition density.- Stability of a nonlinear equation related to a spatially-inhomogeneous branching process.- Exponentially stable stationary solutions for delay stochastic evolution equations.- Robust stochastic control and equivalent martingale measures.- Multivalued stochastic differential questions driven by point processes.- Logarithmic derivatives of densities for jump processes.- Part II: Financial Applications.- Convertible bonds in a defaultable diffusion model.- A geometric approach to option pricing with transaction costs in discrete models.- Completeness and hedging in a Lévy bond market.- Asymptotically efficient discrete hedging.- Estimating joint default probability by efficient importance sampling with applications from bottom up.- Market models of forward CDS spreads.- Optimal threshold dividend strategies under the compound Poisson model with regime switching.