Simuloidun jäähdytyksen suppenemislause

Tämä pro gradu -tutkielma käsittelee simuloitu jäähdytys -nimisen kombinatorisen optimointimenetelmän teoriaa ja käytäntöä. Esimerkiksi kuvankäsittelyssä sovelletun algoritmin ideana on löytää annetulla joukolla määritellyn reaaliarvoisen energiafunktion globaali minimikohta sallimalla - ei pelkästään energiaa vähentäviä - vaan myös energiaa kasvattavia siirtymiä lähtöjoukon alkioiden välillä. Tilastolliseen fysiikkaan analogian omaavan, Gibbsin jakauman ominaisuuksiin pohjautuvan menetelm än matemaattisena perustana toimivat epähomogeeniset Markovin ketjut, joiden suppenemista tarkastellaan Dobrushinin kontraktiokerroinmenetelmän avulla. Simuloidun jäähdytyksen suppenemislause, joka takaa …

On first exit times and their means for Brownian bridges

For a Brownian bridge from $0$ to $y$ we prove that the mean of the first exit time from interval $(-h,h), \,\, h>0,$ behaves as $O(h^2)$ when $h \downarrow 0.$ Similar behavior is seen to hold also for the 3-dimensional Bessel bridge. For Brownian bridge and 3-dimensional Bessel bridge this mean of the first exit time has a puzzling representation in terms of the Kolmogorov distribution. The result regarding the Brownian bridge is applied to prove in detail an estimate needed by Walsh to determine the convergence of the binomial tree scheme for European options.

Time-dependent weak rate of convergence for functions of generalized bounded variation

Let $W$ denote the Brownian motion. For any exponentially bounded Borel function $g$ the function $u$ defined by $u(t,x)= \mathbb{E}[g(x{+}\sigma W_{T-t})]$ is the stochastic solution of the backward heat equation with terminal condition $g$. Let $u^n(t,x)$ denote the corresponding approximation generated by a simple symmetric random walk with time steps $2T/n$ and space steps $\pm \sigma \sqrt{T/n}$ where $\sigma > 0$. For quite irregular terminal conditions $g$ (bounded variation on compact intervals, locally H\"older continuous) the rate of convergence of $u^n(t,x)$ to $u(t,x)$ is considered, and also the behavior of the error $u^n(t,x)-u(t,x)$ as $t$ tends to $T$

Random walk approximation of BSDEs with H{\"o}lder continuous terminal condition

In this paper, we consider the random walk approximation of the solution of a Markovian BSDE whose terminal condition is a locally Hölder continuous function of the Brownian motion. We state the rate of the L2-convergence of the approximated solution to the true one. The proof relies in part on growth and smoothness properties of the solution u of the associated PDE. Here we improve existing results by showing some properties of the second derivative of u in space. peerReviewed

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Mean square rate of convergence for random walk approximation of forward-backward SDEs

AbstractLet (Y,Z) denote the solution to a forward-backward stochastic differential equation (FBSDE). If one constructs a random walk$B^n$from the underlying Brownian motionBby Skorokhod embedding, one can show$L_2$-convergence of the corresponding solutions$(Y^n,Z^n)$to$(Y, Z).$We estimate the rate of convergence based on smoothness properties, especially for a terminal condition function in$C^{2,\alpha}$. The proof relies on an approximative representation of$Z^n$and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the partial differential equation associated to the FBSDE, as well as of the finite difference equations associated to t…