Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe It{^o}-Tanaka Trick: a non-semimartingale approach
107
0
0.0
(
0
)
Authors
Laure Coutin
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
The objective of this paper is to study the local time and Tanaka formula of symmetric $G$-martingales. We introduce the local time of $G$-martingales and show that they belong to $G$-expectation space $L_{G}^{2}(Omega _{T})$. The bicontinuous modification of local time is obtained. We finally give the Tanaka formula for convex functions of $G$-martingales.
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 the balayage formula, we prove an inequality between the measures associated to local times of semimartingales. Our result extends the comparison theorem of local times of Ouknine $(1988)$, which is useful in the study of stochastic differential equations. The inequality presented in this paper covers the discontinuous case. Moreover, we study the pathwise uniqueness of some stochastic differential equations involving local time of unknown process.
We prove existence and uniqueness of strong solutions for a class of semilinear stochastic evolution equations driven by general Hilbert space-valued semimartingales, with drift equal to the sum of a linear maximal monotone operator in variational form and of the superposition operator associated to a random time-dependent monotone function defined on the whole real line. Such a function is only assumed to satisfy a very mild symmetry-like condition, but its rate of growth towards infinity can be arbitrary. Moreover, the noise is of multiplicative type and can be path-dependent. The solution is obtained via a priori estimates on solutions to regularized equations, interpreted both as stochastic equations as well as deterministic equations with random coefficients, and ensuing compactness properties. A key role is played by an infinite-dimensional Doob-type inequality due to Metivier and Pellaumail.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
