MSThD26
Functional Ito calculus and Pathdependent Partial Differential Equations
Date: August 13
Time: 13:3015:30
Room: 110
(Note: Click title to show the abstract.)
Organizer:
CONT, Rama (Imperial College London)
Abstract: The Functional Ito calculus is a nonanticipative functional calculus which extends the Ito calculus to pathdependent functionals of stochastic processes. This recently developed approach has led to new results on the representation of martingales as stochastic integrals, the derivation of FeynmanKac formulae for pathdependent functionals and a new class of functional equations known as "pathdependent PDEs", which extends the classical Kolmogorov equations to the nonMarkovian case. with interesting connections to the theory of Backward stochastic differential equations.
This MiniSymposium presents recent research on Functional Ito calculus and pathdependent PDEs and their applications to stochastic control and simulation of stochastic processes.
MSThD261
13:3014:00
Weak solutions for pathdependent Kolmogorov equations
CONT, Rama (Imperial College London)
Abstract: Pathdependent Kolmogorov equations naturally arise as the extension of the classical backward Kolmogorov equations to the case of nonMarkovian stochastic processes. We introduce a notion of Sobolevtype weak solution for linear and semilinear pathdependent PDEs and show that this notion of weak solution has a natural connection to (backward) stochastic differential equation. In particular, given a reference semimartingale X, any squareintegrable (sub)martingale in the filtration of X is characterized as a weak (sub)solution of the pathdependent Kolmogorov equation corresponding to X.
MSThD262
14:0014:30
Viscosity solutions of obstacle problems for Fully nonlinear pathdependent PDEs.
Ekren, Ibrahim (ETH Zurich)
Abstract: In this talk, we adapt the definition of viscosity solutions to the obstacle problem for fully nonlinear pathdependent PDEs with data uniformly continuous in (t,\omega), and generator Lipschitz continuous in (y,z,\gamma). We prove that our definition of viscosity solutions is consistent with the classical solutions, and satisfy a stability result. We show that the value functional defined via the second order reflected backward stochastic differential equation is the unique viscosity solution of the variational inequalities.
MSThD263
14:3015:00
Pathwise Ito Calculus for Rough Paths and Applications
Zhang, Jianfeng (Univ. of Southern California)
Abstract: The functional It\^{o} calculus has been very successful in many applications, particularly in viscosity theory for backward path dependent PDEs. In this talk we extend the theory to pathwise Ito calculus, in the rough path framework with possibly nongeometric rough paths. This is appropriate for forward problems. Some applications on (forward) stochastic PDEs will also be discussed.
MSThD264
15:0015:30
Weak approximation of martingale representations
Lu, Yi (Univ. Paris 6)
Abstract: We present a systematic method for computing explicit approximations to martingale representations for a large class of Brownian functionals. The approximations are based on a notion of pathwise functional derivative and yield a consistent estimator for the integrand in the martingale representation formula for any squareintegrable functional of the solution of an SDE with pathdependent coefficients. Explicit convergence rates are derived for functionals which are Lipschitzcontinuous in the supremum norm. The approximation and the proof of its convergence are based on the Functional Ito calculus, and require neither the Markov property, nor any differentiability conditions on the coefficients of the stochastic differential equations involved.
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Footnote: Code: TypeDateTimeRoom No.
Type : IL=Invited Lecture, SL=Special Lectures, MS=Minisymposia, IM=Industrial Minisymposia, CP=Contributed Papers, PP=Posters
Date: Mo=Monday, Tu=Tuesday, We=Wednesday, Th=Thursday, Fr=Friday
Time : A=8:309:30, B=10:0011:00, C=11:1012:10, BC=10:0012:10, D=13:3015:30, E=16:0018:00, F=19:0020:00, G=12:1013:30, H=15:3016:00
Room No.: TBA
