London School of Economics

Given a deterministically time-changed Brownian motion Z starting from 1, whose time-change V (t) satisfies $V (t) > t$ for all $t>=0$, we perform an explicit construction of a process X which is Brownian motion in its own filtration and that hits zero for the first time at V (s), where $s:= inf {t > 0 : Z_t = 0}$. We also provide the semimartingale decomposition of $X >$ under

the filtration jointly generated by X and Z. Our construction relies on a combination of enlargement of filtration and filtering techniques. The resulting process X may be viewed as the analogue of a 3-dimensional Bessel bridge starting from 1 at time 0 and ending at 0 at the random time $V (s)$.

We call this a dynamic Bessel bridge since V(s) is not known in advance. Our study is motivated by insider trading models with default risk.(this is a joint work with Luciano Campi and Umut Cetin)

In this talk, we focus on the analytical properties of a decentralized auction, namely the PAUSE Auction Procedure. We prove that the revenue of the auctioneer from PAUSE is greater than or equal to the profit from the well-known VCG auction when there are only two bidders and provide lower bounds on the profit for arbitrary number of bidders. Based on these bounds and observations from auctions with few items, we propose a modification of the procedure that increases the profit. We believe that this study, which is still in progress, will be a milestone in designing better decentralized auctions since it is the first analytical study on such auctions with promising results.

We develop a methodology to optimally design a financial issue to hedge

non-tradable risk on financial markets.The modeling involves a minimization

of the risk borne by issuer given the constraint imposed by a buyer who

enters the transaction if and only if her risk level remains below a given

threshold. Both agents have also the opportunity to invest all their residual

wealth on financial markets but they do not have the same access to financial

investments. The problem may be reduced to a unique inf-convolution problem

involving some transformation of the initial risk measures.