Talks in financial and insurance mathematics

This talk presents two recent papers on backward stochastic differential equations and the foundations of continuous-time recursive utility.

In the first part, we establish existence, uniqueness, monotonicity, concavity, and a utility gradient inequality for continuous-time recursive utility in the Epstein-Zin parametrization with relative risk aversion and elasticity of intertemporal substitution greater than unity. This is done in a fully general semimartingale setting, extending existing results for Brownian filtrations as in Schroder and Skiadas (1999) and Xing (2015).

In the second part, building on an abstract framework for dynamic nonlinear expectations that comprises g-, G- and random G-expectations, we develop a theory of backward nonlinear expectation equations (BNEEs). Such equations can be thought of as backward stochastic differential equations under nonlinear expectations. We provide existence, uniqueness, and stability results and establish convergence of the associated discrete-time nonlinear aggregations. In the context of recursive preferences, BNEEs emerge naturally when ambiguity is taken into account; we apply our results to show that discrete-time recursive utility with ambiguity converges to the nonlinear stochastic differential utility introduced formally by Chen and Epstein (2002) and Epstein and Ji (2014).

How would the introduction of a small trading friction such as a transaction tax affect financial markets? To answer questions of this kind, one needs to consider equilibrium models, where prices are determined endogenously. Indeed, taxes change agents' individual decision making, which in turn affects the market prices determined by their interactions. The new market environment then again alters the agents' behavior, leading to a notoriously intractable fixed point problem.

In this talk we report on recent progress using asymptotic techniques for small trading frictions. In this practically relevant limiting regime, explicit solutions become available for many of the arising singular control problems, bringing analytical results for the equilibrium problem within reach.

(The talk is based on joint works with Martin Herdegen and Jan Kallsen)

In statistical inference it is of fundamental importance to obtain the sampling distribution of statistics. However, we often encounter situations where the exact distribution cannot be obtained in closed form, or even if it is obtained, it might be of little use because of its complexity. One practical way of getting around the problem is to provide reasonable approximations of the distribution function and its quantiles, along with extra information on their possible errors. It can be made with help of Edgeworth and Cornish–Fisher expansions (EE and CFE resp.) (see, e.g., Ulyanov V.V., Cornish-Fisher Expansions, International Encyclopedia of Statistical Science (Ed. M.Lovric), Springer-Verlag, Berlin–Heidelberg, 2011, p.312–315). Starting from 90-s the interest for CFE stirred up because of intensive study of risk measures (Value at Risk, Expected Shortfall etc.)

In the talk we discuss new approaches to CFE appeared in last years, in particular generalized CFE and rearrangements to monotonize CFE. Generalized CFE (non-normal limit distributions) appear naturally for statistics based on samples of random size. We consider as well the computable error bounds for CFE in the case when there are error bounds for EE of the distributions of statistics. The different applications for risk measurement and construction of the confidence intervals will be also discussed.