Rough path analysis provides a fresh perspective on Ito's important theory of stochastic differential equations. Key theorems of modern stochastic analysis (existence and limit theorems for stochastic flows, Freidlin-Wentzell theory, the Stroock-Varadhan support description) can be obtained with dramatic simplifications. Classical approximation results and their limitations (Wong-Zakai, McShane's counterexample) receive 'obvious' rough path explanations. Evidence is building that rough paths will play an important role in the future analysis of stochastic partial differential equations and the authors include some first results in this direction. They also emphasize interactions with other parts of mathematics, including Caratheodory geometry, Dirichlet forms and Malliavin calculus. Based on successful courses at the graduate level, this up-to-date introduction presents the theory of rough paths and its applications to stochastic analysis. Examples, explanations and exercises make the book accessible to graduate students and researchers from a variety of fields.
It is well known that there is a mathematical equivalence between 'solving' parabolic partial differential equations (PDEs) and 'the integration' of certain functionals on Wiener space. Monte Carlo simulation of stochastic differential equations (SDEs) is a naive approach based on this underlying principle. In finite dimensions, it is well known that cubature can be a very effective approach to integration. We discuss the appropriate extension of this idea to Wiener space. In the process we develop high-order numerical schemes valid for high-dimensional SDEs and semielliptic PDEs.
For a d-dimensional diffusion of the form d X t = µ(X t )dt + σ (X t )dW t and continuous functions f and g, we study the existence and uniqueness of adapted processes Y , Z , , and A solving the second-order backward stochastic differential equation (2BSDE)If the associated PDEhas a sufficiently regular solution, then it follows directly from Itô's formula that the processessolve the 2BSDE, where L is the Dynkin operator of X without the drift term. The main result of the paper shows that if f is Lipschitz in Y as well as decreasing in and the PDE satisfies a comparison principle as in the theory of viscosity solutions, then the existence of a solution (Y, Z , , A) to the 2BSDE implies that the associated PDE has a unique continuous viscosity solution v and the process Y is of the form Y t = v(t, X t ), t ∈ [0, T ]. In particular, the 2BSDE has at most one solution. This provides a stochastic representation for solutions of fully nonlinear parabolic PDEs. As a consequence, the numerical treatment of such PDEs can now be approached by Monte Carlo methods.
Large classes of multi-dimensional Gaussian processes can be enhanced with stochastic Lévy area(s). In a previous paper, we gave sufficient and essentially necessary conditions, only involving variational properties of the covariance. Following T. Lyons, the resulting lift to a "Gaussian rough path" gives a robust theory of (stochastic) differential equations driven by Gaussian signals with sample path regularity worse than Brownian motion.The purpose of this sequel paper is to establish convergence of Karhunen-Loeve approximations in rough path metrics. Particular care is necessary since martingale arguments are not enough to deal with third iterated integrals. An abstract support criterion for approximately continuous Wiener functionals then gives a description of the support of Gaussian rough paths as the closure of the (canonically lifted) Cameron-Martin space.
The authors present a new simple algorithm to approximate weakly stochastic differential equations in the spirit of [10][16]. They apply it to the problem of pricing Asian options under the Heston stochastic volatility model, and compare it with other known methods. It is shown that the combination of the suggested algorithm and quasi-Monte Carlo methods makes computations extremely fast.
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