Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userWeak error analysis via functional It^o calculus
104
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We consider autonomous stochastic ordinary differential equations (SDEs) and weak approximations of their solutions for a general class of sufficiently smooth path-dependent functionals f. Based on tools from functional It^o calculus, such as the functional It^o formula and functional Kolmogorov equation, we derive a general representation formula for the weak error $E(f(X_T)-f(tilde X_T))$, where $X_T$ and $tilde X_T$ are the paths of the solution process and its approximation up to time T. The functional $f:C([0,T],R^d)to R$ is assumed to be twice continuously Frechet differentiable with derivatives of polynomial growth. The usefulness of the formula is demonstrated in the one dimensional setting by showing that if the solution to the SDE is approximated via the linearly time-interpolated explicit Euler method, then the rate of weak convergence for sufficiently regular f is 1.
rate research
Read More
A peculiar feature of It^os calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. So, can we define a pathwise stochastic derivative of semimartingales with respect to Brownian motion that leads to a differentiation theory counterpart to It^os integral calculus? From It^os definition of his integral, such a derivative must be based on the quadratic covariation process. We give such a derivative in this note and we show that it leads to a fundamental theorem of stochastic calculus, a generalized stochastic chain rule that includes the case of convex functions acting on continuous semimartingales, and the stochastic mean value and Rolles theorems. In addition, it interacts with basic algebraic operations on semimartingales similarly to the way the deterministic derivative does on deterministic functions, making it natural for computations. Such a differentiation theory leads to many interesting applications some of which we address in an upcoming article.
In this paper we provide an It{^o}-Tanaka-Wentzell trick in a non semimartingale context. We apply this result to the study of a fractional SDE with irregular drift coefficient.
Observing that the recent developments of the recursive (product) quantization method induces a family of Markov chains which includes all standard discretization schemes of diffusions processes , we propose to compute a general error bound induced by the recursive quantization schemes using this generic markovian structure. Furthermore, we compute a marginal weak error for the recursive quantization. We also extend the recursive quantization method to the Euler scheme associated to diffusion processes with jumps, which still have this markovian structure, and we say how to compute the recursive quantization and the associated weights and transition weights.
In this note, we prove an It^o formula for the isochron map of a reaction-diffusion system. This follows the proof of a new result which states that the second derivative of the isochron map of a reaction-diffusion system is trace class. This result, in turn, is a corollary of Proposition 2.3, which guarantees that the first and second Frechet derivatives of the flow of a reaction-diffusion system with respect to initial conditions are trace class.
Using Dupires notion of vertical derivative, we provide a functional (path-dependent) extension of the It^os formula of Gozzi and Russo (2006) that applies to C^{0,1}-functions of continuous weak Dirichlet processes. It is motivated and illustrated by its applications to the hedging or superhedging problems of path-dependent options in mathematical finance, in particular in the case of model uncertainty
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
