Browsing by Subject "Malliavin calculus"

Now showing 1 - 3 of 3

Open Access
Entropy, invertibility and variational calculus of adapted shifts on Wiener space
(2009) Üstünel, A.S.
In this work we study the necessary and sufficient conditions for a positive random variable whose expectation under the Wiener measure is one, to be represented as the Radon-Nikodym derivative of the image of the Wiener measure under an adapted perturbation of identity with the help of the associated innovation process. We prove that the innovation conjecture holds if and only if the original process is almost surely invertible. We also give variational characterizations of the invertibility of the perturbations of identity and the representability of a positive random variable whose total mass is equal to unity. We prove in particular that an adapted perturbation of identity U = IW + u satisfying the Girsanov theorem, is invertible if and only if the kinetic energy of u is equal to the entropy of the measure induced with the action of U on the Wiener measure μ, in other words U is invertible ifffrac(1, 2) under(∫, W) | u |H 2 d μ = under(∫, W) frac(d U μ, d μ) log frac(d U μ, d μ) d μ . The relations with the Monge-Kantorovitch measure transportation are also studied. An application of these results to a variational problem related to large deviations is also given. © 2009 Elsevier Inc. All rights reserved.
Open Access
Hedging portfolio for a market model of degenerate diffusions
(Taylor & Francis, 2022-11-30) Çağlar, M.; Demirel, İ.; Üstünel, Süleyman
We consider a semimartingale market model when the underlying diffusion has a singular volatility matrix and compute the hedging portfolio for a given payoff function. Recently, the representation problem for such degenerate diffusions as a stochastic integral with respect to a martingale has been completely settled. This representation and Malliavin calculus established further for the functionals of a degenerate diffusion process constitute the basis of the present work. Using the Clark–Hausmann–Bismut–Ocone type representation formula derived for these functionals, we prove a version of this formula under an equivalent martingale measure. This allows us to derive the hedging portfolio as a solution of a system of linear equations. The uniqueness of the solution is achieved by a projection idea that lies at the core of the martingale representation at the first place. We demonstrate the hedging strategy as explicitly as possible with some examples of the payoff function such as those used in exotic options, whose value at maturity depends on the prices over the entire time horizon.
Open Access
Martingale representation theorem for diffusion in infinite dimensional spaces and applications
(2023-03) Aydın, Uğur
We show that square integrable martingales adapted to the filtration generated by a weak solution of a stochastic differential equation driven by a cylindrical Wiener process on a separable real Hilbert space that has the weak uniqueness property has a martingale representation driven by the martingale part of the stochastic differential equation.