• Corpus ID: 18564769

A Generalized Backward Equation For One Dimensional Processes

  title={A Generalized Backward Equation For One Dimensional Processes},
  author={George Lowther},
  journal={arXiv: Probability},
  • G. Lowther
  • Published 23 March 2008
  • Mathematics
  • arXiv: Probability
Suppose that a real valued process X is given as a solution to a stochastic differential equation. Then, for any twice continuously differentiable function f, the backward Kolmogorov equation gives a condition for f(t,X) to be a local martingale. We generalize the backward equation in two main ways. First, it is extended to non-differentiable functions. Second, the process X is not required to satisfy an SDE. Instead, it is only required to be a quasimartingale satisfying an integrability… 
Fitting Martingales To Given Marginals
We consider the problem of finding a real valued martingale fitting specified marginal distributions. For this to be possible, the marginals must be increasing in the convex order and have constant
From Bachelier to Dupire via optimal transport
Famously, mathematical finance was started by Bachelier in his 1900 PhD thesis where – among many other achievements – he also provided a formal derivation of the Kolmogorov forward equation. This


Properties of Expectations of Functions of Martingale Diffusions
Given a real valued and time-inhomogeneous martingale diffusion X, we investigate the properties of functions defined by the conditional expectation f(t,X_t)=E[g(X_T)|F_t]. We show that whenever g is
Multidimensional Diffusion Processes
Preliminary Material: Extension Theorems, Martingales, and Compactness.- Markov Processes, Regularity of Their Sample Paths, and the Wiener Measure.- Parabolic Partial Differential Equations.- The
Semimartingale theory and stochastic calculus
PRELIMINARIES. Monotone Class Theorems. Uniform Integrability. Essential Supremum. The Generalization of Conditional Expectation. Analytic Sets and Choquet Capacity. Lebesgue-Stieltjes Integrals.
Lectures on Lipschitz analysis
(1.1) |f(a)− f(b)| ≤ L |a− b| for every pair of points a, b ∈ A. We also say that a function is Lipschitz if it is L-Lipschitz for some L. The Lipschitz condition as given in (1.1) is a purely metric
On Non-Continuous Dirichlet Processes
It is proved that non-continuous Dirichlet processes are stable under C1 transformation.
Stochastic integration and differential equations
I Preliminaries.- II Semimartingales and Stochastic Integrals.- III Semimartingales and Decomposable Processes.- IV General Stochastic Integration and Local Times.- V Stochastic Differential
Pricing and Hedging of Derivative Securities
The theory of pricing and hedging of derivative securities is mathematically sophisticated. This book is an introduction to the use of advanced probability theory in financial economics, presenting
Variation conditionnelle des processus stochastiques
On considere (Ω, #7B-F, P (#7B-F t ) 0 ≤t≤1) un espace probabilise filtre verifiant les conditions habituelles. (X t ) 0 ≤t≤1 est un processus cadlaz adapte et integrable. On etend des resultats
Nondifferentiable functions of one-dimensional semimartingales
We consider decompositions of processes of the form Y = f(t, X t ) where X is a semimartingale. The function f is not required to be differentiable, so Ito's lemma does not apply. In the case where
Stochastic integration and differential equations, volume 21 of Applications of Mathematics
  • Stochastic Modelling and Applied Proba- bility
  • 2004