Asymptotic expansion for the characteristic function of a multiscale stochastic volatility model

Abstract: We give the first order asymptotic correction for the characteristicfunction of the log-return of an asset price process whose volatility is driven bytwo diffusion processes on two different time scales. In particular we considera fast mean reverting process with reverting scale1\u1eband a slow mean revertingprocess with scale \u3b4, and we perform the expansion for the associated charac-teristic function, at maturity time T > 0, in powers of 1a \u1eb and 1a \u3b4. The latterresult, according, e.g., to [2, 3, 8, 11], can be exploited to compute the fair pricefor an option written on the asset of interest

Delayed Forward-Backward stochastic PDE's driven by non Gaussian Lévy noise with application in finance

From the very first results, the mathematical theory of financial markets has undergone several changes, mostly due to financial crises who forced the mathematical-economical community to change the basic assumptions on which the whole theory is founded. Consequently a new mathematical foundation were needed. In particular, the 2007/2008 credit crunch showed the word that a new financial theoretical framework was necessary, since several empirical evidences emerged that aspects that were neglected prior to these years were in fact fundamental if one has to deal with financial markets. The goal of the present thesis goes in this direction; we aim at developing rigorous mathematical instruments that allow to treat fundamental problems in modern financial mathematics. In order to do so, the talk is thus divided into three main parts, which focus on three different topics of modern financial mathematics. The first part is concerned with delay equations. In particular, we will prove Feynman-Kac type result for BSDE's with time-delayed generator, as well as an ad hoc Ito formula for delay equations with jumps. The second part deal with infinite dimensional analysis and network models, focusing in particular on existence and uniqueness results for infinite dimensional SPDE's on networks with general non-local boundary conditions. The last part treats the topic of rigorous asymptotic expansions, providing a small noise asymptotic expansion for SDE with Lévy noise with several concrete application to financial models

FIRST ORDER CORRECTION FOR THE CHARACTERISTIC FUNCTION OF A MULTIDIMENSIONAL AND MULTISCALE STOCHASTIC VOLATILITY MODEL

Abstract: The present work generalizes the results obtained in [3] to a d > 1dimensional setting. In particular we give the first order asymptotic correctionfor the characteristic function of the log-return of a multidimensional asset priceprocess whose volatility is driven by two diffusion processes on two different timescales. We consider a fast mean reverting process with reverting scale 1\u1eb anda slow mean reverting process with scale \u3b4, and we perform the expansion forthe associated characteristic function, at maturity time T > 0, in powers of 1a\u1eb and 1a\u3b4. Latter result, according, e.g., to [2, 4, 9, 12], can be exploitedto numerically analyze the fair price of a structured option written on d > 1assets

LIE SYMMETRY APPROACH TO THE CEV MODEL

Abstract: Using a Lie algebraic approach we explicitly provide both the probabilitydensity function of the constant elasticity of variance (CEV) process andthe fundamental solution for the associated pricing equation. In particular wereduce the CEV stochastic differential equation (SDE) to the SDE characterizingthe Cox, Ingersoll and Ross (CIR) model, being the latter easier to treat.The fundamental solution for the CEV pricing equation is then obtained followingtwo methods. We first recover a fundamental solution via the invariantsolution method, while in the second approach we exploit Lie classical result onclassification of linear partial differential equations (PDEs). In particular wefind a map which transforms the pricing equation for the CIR model into anequation of the form v\u3c4 = vyy 12 Ay2 v whose fundamental solution is known. Then,by inversion, we obtain a fundamental solution for the CEV pricing equation

A lending scheme for a system of interconnected banks with probabilistic constraints of failure

We derive a closed form solution for an optimal control problem related to an interbank lending schemes subject to terminal probability constraints on the failure of banks which are interconnectedthrough a financial network. The derived solution applies to a real banks network by obtaining ageneral solution when the aforementioned probability constraints are assumed for all the banks. We also present a direct method to compute the systemic relevance parameter for each bank within the networ

Optimal control for the stochastic fitzhugh-nagumo model with recovery variable

In the present paper we derive the existence and uniqueness of the solution for the optimal control problem governed by the stochastic FitzHugh-Nagumo equation with recovery variable. Since the drift coefficient is characterized by a cubic non-linearity, standard techniques cannot be applied, instead we exploit the Ekeland\u2019s variational principle

SMALL NOISE EXPANSION FOR THE L\uc9VY PERTURBED VASICEK MODEL

We present rigorous small noise expansion results for a L\ue9vy perturbed Vasicek model. Estimates for the remainders as well as an application to ZCB pricing are also provided

Stochastic reaction-diffusion equations on networks with dynamic time-delayed boundary conditions

We consider a reaction-diffusion equation on a network subjected to dynamic boundary conditions, with time delayed behavior, also allowing for multiplicative Gaussian noise perturbations. Exploiting semigroup theory, we rewrite the aforementioned stochastic problem as an abstract stochastic partial differential equation taking values in a suitable product Hilbert space, for which we prove the existence and uniqueness of a mild solution. Eventually, a stochastic optimal control application is studied

Optimal control of stochastic FitzHugh-Nagumo equation

This paper is concerned with existence and uniqueness of solution for the the optimal control problem governed by the stochastic FitzHugh-Nagumo equation driven by a Gaussian noise. First order conditions of optimality are also obtained

Stochastic systems with memory and jumps

Stochastic systems with memory naturally appear in life science, economy, and finance. We take the modelling point of view of stochastic functional delay equations and we study these structures when the driving noises admit jumps. Our results concern existence and uniqueness of strong solutions, estimates for the moments and the fundamental tools of calculus, such as the It\uf4 formula. We study the robustness of the solution to the change of noises. Specifically, we consider the noises with infinite activity jumps versus an adequately corrected Gaussian noise. The study is presented in two different frameworks: we work with random variables in infinite dimensions, where the values are considered either in an appropriate -type space or in the space of c\ue0dl\ue0g paths. The choice of the value space is crucial from the modelling point of view, as the different settings allow for the treatment of different models of memory or delay. Our techniques involve tools of infinite dimensional calculus and the stochastic calculus via regularisation