In mathematics, the theory of optimal stopping or early stopping is concerned with the problem of choosing a time to take a particular action, in order to maximise an expected reward or minimise an expected cost. Optimal stopping problems can be found in areas of statistics, economics, and mathematical finance (related to the pricing of American options). A key example of an optimal stopping problem is the secretary problem. Optimal stopping problems can often be written in the form of a Bellman equation, and are therefore often solved using dynamic programming. Definition Discrete time case Stopping rule problems are associated with two objects:

A sequence of random variables X_1, X_2, \ldots, whose joint distribution is something assumed to be known

A sequence of 'reward' functions (y_i)_{i\ge 1} which depend on the observed values of the random variables in 1:

#: y_i=y_i (x_1, \ldots ,x_i) Given those objects, the

Essays in Bank Financing

Yalda Sigrist

This thesis examines the optimal mode of financing for banks and financial institutions. The first chapter, which is a joint work with Prof. Jean-Charles Rochet, investigates how Systemically Important Financial Institutions (SIFIs) should be financed. The main specificity of such firms is that their failure imposes negative externalities on the financial system and, more broadly, on the whole economy. Since their shareholders do not internalize these externalities, they tend to provide less capital than what would be socially optimal, generating too many failures. In jurisdictions where they receive tax exemptions, CoCo bonds are a simple means of increasing the resilience of SIFIs while providing large tax benefits to banks' shareholders. However, we show that when tax revenues are properly accounted for, these CoCo bonds are detrimental to social welfare. The second chapter provides a formal model of a bail-in plan, a pre-defined contract that results in self-recapitalization of a financial institution when it is in distress and has no access to equity financing. Bail-ins have the potential to eliminate inefficient bank liquidations and ease financing constraints in bad times. Using a theoretical model of a bail-in contract where banks face time varying financing frictions, taxation and liquidation costs, I show that banks' capital structure decisions and optimal financing and pay-out policies are largely affected by the design of the contingent capital (specifically the conversion ratio). However independent of its design, for an optimal level of debt, bail-in contracts always decrease shareholders incentives to build up liquid buffers and to recapitalize in good times but also eliminate any risk-taking incentives. The third chapter presents a model of bank optimal maturity structure when banks face systemic risk through correlated investments. Banks can privately affect the probability of success of their projects. Risky short-term debt can act as a disciplinary device if it is not rolled over when an adverse interim-date signal on the quality of banks' assets is received. The optimal maturity structure is the result of the trade-off between the disciplinary benefits of short-term debt and the costs of inefficient early liquidations. I show that bank asset commonality affects this trade-off since it reduces the costs of exerting effort through positive information synergies and increases the inefficiency of early liquidations through negative fire-sale externalities. In particular, there is a more important role for the disciplinary effects of short-term debt in high correlation, high fire-sale externalities and low information synergies environments.

EPFL2017

On the optimal controllability time for linear hyperbolic systems with time-dependent coefficients

Hoài-Minh Nguyên

The optimal time for the controllability of linear hyperbolic systems in one dimensional space with one-side controls has been obtained recently for time-independent coefficients in our previous works. In this paper, we consider linear hyperbolic systems with time-varying zero-order terms. We show the possibility that the optimal time for the null-controllability becomes significantly larger than the one of the time-invariant setting even when the zero-order term is indefinitely differentiable. When the analyticity with respect to time is imposed for the zero-order term, we also establish that the optimal time is the same as in the time-independent setting.

2021

Time-independent feedbacks for the null controllability of homogeneous quasilinear hyperbolic systems in one dimensional space

Hoài-Minh Nguyên

We consider the null controllability of homogeneous quasilinear hyperbolic systems with one side controls and with nonlinear boundary condition at the other side. We present time-independent feedbacks for the null-controllability for

C^1

-solutions at any time larger than the optimal time for the linear system if the initial condition is sufficiently small. One of the key technical points is to establish the local well-posedness of quasilinear hyperbolic systems with nonlinear, non-local boundary conditions.

2020

Essays in Bank Financing

Yalda Sigrist

EPFL2017

Time-independent feedbacks for the null controllability of homogeneous quasilinear hyperbolic systems in one dimensional space

Hoài-Minh Nguyên

C^1

2020