ion is not needed. Inverse problems 531 Theorem 6.13. Two Gaussian measures μi = N (mi, Ci), i = 1, 2, on a Hilbert space H are either singular or equivalent. They are equivalent if and only if the following three conditions hold: (i) Im(C 1 ) = Im(C 1/2 2 ) := E, (ii) m1 −m2 ∈ E, (iii) the operator T := ( C−1/2 1 C 1/2 2 )( C−1/2 1 C 1/2 2 )∗ − I is Hilbert–Schmidt in E. In particular, choosin...

This paper is devoted to the application of B-splines to volatility modeling, specifically the calibration of the leverage function in stochastic local volatility models and the parameterization of an arbitrage-free implied volatility surface calibrated to sparse option data. We use an extension of classical B-splines obtained by including basis functions with infinite support. We first come ba...

In this paper we prove the differentiability of Lipschitz maps X → V , where X is a complete metric measure space satisfying a doubling condition and a Poincaré inequality, and V denotes a Banach space with the Radon Nikodym Property (RNP). The proof depends on a new characterization of the differentiable structure on such metric measure spaces, in terms of directional derivatives in the direct...

is to be minimized over control processes Y whose increments take values in a cone Y of R, keeping the state process X = x + B + GY in a cone X of R, k ≤ p. Here, x ∈ X, B is a Brownian motion with drift b and covariance Σ, G is a fixed matrix, and Y ◦ is the Radon–Nikodym derivative dY/d|Y |. Let L = −(1/2)trace(ΣD) − b ·D where D denotes the gradient. Solutions to the corresponding dynamic pr...

We consider the Nonlinear Schrödinger (NLS) equation with third-order dispersion and prove that Gaussian measure covariance (1??x2)?? on L2(T) is quasi-invariant for associated flow ?>1/2. This sharp improves a previous result obtained in [20] where values ?>3/4 were obtained. Also, our method completely different simpler, it based an explicit formula Radon-Nikodym derivative. obtain this latte...

We study a BSDE with random terminal time that appears in the modeling of counterparty risk in finance. We proceed by reduction of the original BSDE into a simpler BSDE posed with respect to a smaller filtration and a changed probability measure. This is done under a relaxation of the classical immersion hypothesis, stated in terms of the changed probability measure, of which we characterize th...

We study the transport property of Gaussian measures on Sobolev spaces periodic functions under dynamics one-dimensional cubic fractional nonlinear Schrödinger equation. For case second-order dispersion or greater, we establish an optimal regularity result for quasi-invariance these measures, following approach by Debussche and Tsutsumi (2021). Moreover, obtain explicit formula Radon-Nikodym de...

We derive optimality conditions and calculate approximate solutions to the problem of determining the optimal speed of mean reversion to be applied to a Gaussian state variable. The optimality criterion is the minimization of the variance of the Radon-Nikodym derivative of the measure ”with mean-reversion” with respect to the measure ”without mean-reversion” under constraints. We show that we c...

The results on the mean-variance hedging problem in Gouriéroux, Laurent and Pham (1998), Rheinländer and Schweizer (1997) and Arai (2005) are extended to discontinuous semimartingale models. When the numéraire method is used, we only assume the Radon-Nikodym derivative of the variance-optimal signed martingale measure (VSMM) is non-zero almost surely (but may be strictly negative). When discuss...