Cet article étudie les relations entre équations intégrales de Volterra forward-backward stochastiques (FBSVIE) et un système d’équations aux dérivées partielles, non locales en temps, dépendant des trajectoires (PPDE). En raison la nature du type Volterra, propriété habituelle flot, ou semigroupe, n’est pas vérifiée. Inspirés par travaux Viens–Zhang (Ann. Appl. Probab. 29 (2019) 3489–3540) Wan...

Certain Merton type consumption−investment problems under partial information are reduced to the ones of full and within framework a complete market model. Then, specializing conditionally log−Gaussian diffusion models, concrete analysis about optimal values strategies is performed by using analytical tools like Feynman−Kac formula, or HJB equations. The explicit solutions related forward-backw...

In [9], [10], B. Roynette, P. Vallois and M. Yor have studied limit theorems for Wiener processes normalized by some weight processes. In [16], K. Yano, Y. Yano and M. Yor studied the limit theorems for the one-dimensional symmetric stable process normalized by non-negative functions of the local times or by negative (killing) Feynman-Kac functionals. They call the limit theorems for Markov pro...

We present a simple derivation of a Feynman-Kac type formula to study fermionic systems. In this approach the real time or the imaginary time dynamics is expressed in terms of the evolution of a collection of Poisson processes. A computer implementation of this formula leads to a family of algorithms parametrized by the values of the jump rates of the Poisson processes. From these an optimal al...

When we studied discrete-time models we used martingale pricing to derive the Black-Scholes formula for European options. It was clear, however, that we could also have used a replicating strategy argument to derive the formula. In this part of the course, we will use the replicating strategy argument in continuous time to derive the Black-Scholes partial differential equation. We will use this...

We propose a nonequilibrium sampling method for computing free energy profiles along a given reaction coordinate. The method consists of two parts: a controlled Langevin sampler that generates nonequilibrium bridge paths conditioned by the reaction coordinate, and Jarzynski’s formula for reweighting the paths. Our derivation of the equations of motion of the sampler is based on stochastic pertu...

We present a simple derivation of a Feynman-Kac type formula to study fermionic systems. In this approach the real time or the imaginary time dynamics is expressed in terms of the evolution of a collection of Poisson processes. A computer implementation of this formula leads to a family of algorithms parametrized by the values of the jump rates of the Poisson processes. From these an optimal al...

Fermionic Brownian paths are defined as paths in a space para-metrised by anticommuting variables. Stochastic calculus for these paths, in conjunction with classical Brownian paths, is described; Brownian paths on supermanifolds are developed and applied to establish a Feynman-Kac formula for the twisted Laplace-Beltrami operator on differential forms taking values in a vector bundle. This form...

Feynman-Kac transforms driven by discontinuous additive functionals are studied in this paper for a large class of Markov processes. General gauge and conditional gauge theorems are established for such transforms. Furthermore, the L2-infinitesimal generator of the Schrödinger semigroup given by a non-local Feynman-Kac transform is determined in terms of its associated bilinear form.