# A Generalized Backward Equation For One Dimensional Processes

@article{Lowther2008AGB, title={A Generalized Backward Equation For One Dimensional Processes}, author={George Lowther}, journal={arXiv: Probability}, year={2008} }

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…

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## References

SHOWING 1-10 OF 19 REFERENCES

Properties of Expectations of Functions of Martingale Diffusions

- Mathematics
- 2008

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

- Mathematics
- 1979

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

- Mathematics
- 1992

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

- Mathematics
- 2005

(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

- Mathematics, Computer Science
- 2003

It is proved that non-continuous Dirichlet processes are stable under C1 transformation.

Stochastic integration and differential equations

- Mathematics
- 1990

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

- Economics
- 1999

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

- Mathematics
- 1988

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

- Mathematics
- 2010

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