## List of invited speakers:

• Aurelien Alfonsi, Ecole des Ponts ParisTech.
• Etienne Chevalier, Université d'Evry.
• Romuald Elie, Université Paris Est.
• Zorana Grbac, Université Paris 7.
• Shao Hui, NUS.
• Steve Kou, NUS.
• Nabil Kazi-Tani, Université de Lyon.
• Junbeom Lee, NUS.
• Gilles Pagès, Université Paris 6.
• Seyoung Park, NUS.
• Agnès Sulem, INRIA.
• Adrien Richou, Université de Bordeaux.
• Chao Zhou, NUS.

Organisers: JF Chassagneux (P7), Z. Chao (NUS), M. Dai (NUS), N. Frikha (P7) & S. Kou (NUS).
Local organisers: Jean-François Chassagneux & Noufel Frikha.

## Timetable & Registration:

• The workshop will start at 1.30pm on Thursday October, 6 2016 and finish at 6pm on Friday October, 7 2016.
• Registration is free but mandatory, please send an email to the organisers.
• The programme with titles and abstract is available here.

## Information:

• The workshop will take place in the Sophie Germain Building of Université Paris Diderot campus. On Thursday October, 6, it will be located in Room 1009 (first floor) and on Friday October, 7 in Room 2015.

This workshop is funded by an USPC-NUS joint research projects grant.

## Schedule:

Thursday 6 October
Room 1009
Friday 7 October
Room 2015
9.00-9.40
9.45-10.25 C. Zhou
10.30-11.00 Coffee break
11.00-11.40 A. Sulem
11.45-12.25 J. Lee
12.30-13.45 welcome at 13.30 lunch break
13.45-14.25 R. Elie Lunch break
14.30-15.10 A. Richou N. Kazi-Tani
15.15-15.55 S. Park G. Pagès
16.00-16.30 Coffee break Coffee break
16.30-17.10 E. Chevalier A. Alfonsi
17.15-17.55 S. Hui S. Kou

## Abstracts:

Aurelien Alfonsi, Ecole des Ponts ParisTech: Maximum Likelihood Estimation for Wishart processes

In the last decade, there has been a growing interest to use Wishart processes for modelling, especially for financial applications. However, there are still few studies on the estimation of its parameters. Here, we study the Maximum Likelihood Estimator (MLE) in order to estimate the drift parameters of a Wishart process. We obtain precise convergence rates and limits for this estimator in the ergodic case and in some nonergodic cases. We check that the MLE achieves the optimal convergence rate in each case. Motivated by this study, we also present new results on the Laplace transform that extend the recent findings of Gnoatto and Grasselli and are of independent interest.
Joint work with A. Kebaier and C. Rey.

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Etienne Chevalier, Université d'Evry: Optimal investment and capital structure strategies

We study the problem of determining optimal controls on the dividend, investment and capital structure strategies of a firm operating under uncertain environment and risk constraints. First, we allow the company to make investment decisions by acquiring or selling producing assets whose value is governed by a stochastic process. The firm may face liquidity costs when it decides to buy or sell assets. We formulate this problem as a multi-dimensional mixed singular and multi-switching control problem and use a viscosity solution approach. We then focus on the modeling of a control problem on the capital structure, dividend and investment policy of a bank operating under solvability constraints. We assume that the bank's assets consist of both clients' deposits and shareholders' equity. The managers of the bank may invest in either risky assets or in risk-free assets. The objective of the manager is to optimize the bank shareholders' value, ie. the cumulative dividend distributed over the life time of the bank while controlling its solvability. We allow the bank to seek recapitalization or to issue new capital should they fall under financial difficulties. This problem is formulated as a combined impulse control, regular and singular control problem and show that a quasi-explicit solution may be obtained. We further enrich our studies with some numerical illustrations.

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Romuald Elie, Université Paris Est: Optimal incentives for mean field agents in interaction

We consider a model where a Principal requires to design separate contracts with a large number of Agents in interaction. We focus on the optimal design of these contracts, and study in particular the mean field limit of this problem. Considering an infinite number of agents in Nash equilibrium, this leads to the consideration of mean field FBSDE, and allows to rewrite the corresponding problem as a control one on McKean-Vlasov SDEs. We solve completely and discuss the asymptotic convergence for special cases, which go beyond the usual linear-quadratic framework.
This is a joint work with Dylan Possamai (Univ. Paris-Dauphine) and Thibaut Mastrolia (Ecole Polytechnique).

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Zorana Grbac, Université Paris 7: Lévy forward price approach for multiple yield curves and low/negative interest rates

In this talk we present a framework for discretely compounding interest rates which is based on the forward price process approach. This approach has a number of advantages in particular in the current market environment. Compared to the classical Libor market models, it allows in a natural way for negative interest rates and has superb calibration properties even in the presence of extremely low rates. Moreover, the measure changes along the tenor structure are simplified significantly. These properties make it an excellent base for a post-crisis multiple curve setup. Three variants for multiple curve constructions are discussed. As driving processes we use time-inhomogeneous Lévy processes, which allow to derive semi-explicit formulas for the valuation of various interest rate products using Fourier transform techniques.
This is joint work with E. Eberlein and Ch. Gerhart.

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Shao Hui, NUS: Gini curve and top incomes

This paper discusses why the top incomes are so critical for inequality measures from both theoretical and empirical viewpoints. By using the weighted expected utility theory, we propose an axiomatic framework to characterize a truncated relative inequality measure that does not take the top incomes of the income distribution into account. Under suitable conditions, the truncated relative inequality measure reduces to the truncated Gini coefficient, which allows us to define the Gini curves, on which every point is a truncated Gini coefficient excluding the top incomes. The properties of the Gini curve are examined. In terms of empirical study, we investigate the evolutions of income inequalities for the overall population and the population without the top incomes, in particular, we find although the income inequality for the overall population has not changed much, the Gini coefficient excluding the top incomes has been significantly decreased from 1980 to 2010. In contrast to the fundamental Lorenz curve, our Gini curves are able to capture the changes of the top incomes and are appropriate as an inequality measure.

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Junbeom Lee, NUS: Explicit representation of XVA

In this talk, we will investigate some conditions to represent derivative price under XVA explicitly. As long as we consider different borrowing/lending rate, XVA problem becomes a semi-linear equation and this makes finding explicit solution of XVA difficult. However, It will be shown that the associated valuation problem is actually linear under some proper conditions so we can find explicit representation of the present value of derivatives under XVA. Moreover, the conditions mentioned above is mild in the sense that it can be obtained by choosing adequate covenants between the dealer and counterparty.

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Steve Kou, NUS: Simulating Risk Measures

Risk measures, such as value-at-risk and expected shortfall, are widely used in risk management, as exemplified in the Basel Accords proposed by Bank for International Settlements. We propose a simple general framework, allowing dependent samples, to compute these risk measures via simulation. The framework consists of two steps: In the C-step, we control the relative error in the simulation by computing the necessary sample size needed for simulation, using a newly derived asymptotic expansion of the relative errors for dependent samples; in the S-step, the risk measures are computed by using sorting algorithms. Numerical experiments indicate that the algorithm is easy to implement and fast, compared to existing methods, even at the 0.001 quantile level. We also give a comparison of the relative errors of value-at-risk and expected shortfall.
This is a joint work with Wei Jiang.

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Nabil Kazi-Tani, Université de Lyon: Three points suffice

We consider the problem of optimally stopping a continuous-time Markov process with a stopping time satisfying a given expectation constraint. We first reformulate the problem as a linear optimization problem, over a set of probability measures satisfying some moment constraints. To do so, we extend the balayage approach of Chacon and Walsh to the Skorokhod embedding problem for general Markov processes. This also allows us to reduce the optimization over a set of atomic measures. Our main result is the following: it is sufficient to consider stopping times such that the stopped process has a law that is a weighted sum of 3 Dirac measures. In other words: stopping at three points is enough. Several examples will illustrate that result. ​
This is a joint work with Stefan Ankirchner (University of Jena), Maike Klein (University of Jena) and Thomas Kruse (University of Duisburg-Essen).

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Gilles Pagès, Université Paris 6: Non-asymptotic Gaussian Estimates for the Recursive Approximation of the Invariant Measure of a Brownian Diffusion

We obtain non-asymptotic Gaussian concentration bounds for the difference between the invariant measure $\nu$ of an ergodic Brownian diffusion process and the empirical distribution of an approximating scheme with decreasing time step along a suitable class of (smooth enough) test functions $f$ such that $f-\nu(f)$ is a coboundary of the infinitesimal generator. We show that these bounds can still be improved when the (squared) Fröbenius norm of the diffusion coefficient lies in this class. We apply these bounds to design computable non-asymptotic confidence intervals for the approximating scheme. As a theoretical application, we finally derive non-asymptotic deviation bounds for the almost sure Central Limit Theorem.
This is a joint work with I. Honoré et S. Menozzi (Evry University).

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Seyoung Park, NUS: Optimal Consumption and Investment with Uninsurable Income Risks

We present an optimal consumption and investment model for a borrowing and short sale constrained investor with uninsurable labor income risks. The model is able to generate empirically plausible values for marginal propensities to consume. We also find that there exists a target wealth-to-income ratio below which the investor never participates in stock market when labor income is highly correlated with stock market.
This is a joint work with Min Dai and Shan Huang.

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Agnès Sulem, INRIA: Game options in an imperfect financial market with default

We study pricing and superhedging strategies for game options in an imperfect financial market with default. We extend the results obtained by Kifer in a perfect market model to the case of imperfections in the market taken into account via the nonlinearity of the wealth dynamics. In this framework, the pricing system is expressed as a nonlinear ${\cal E}^g$-expectation/evaluation induced by a nonlinear Backward Stochastic Differential Equation (BSDE) with default jump with driver $g$. We prove that the superhedging price of a game option coincides with the value function of a corresponding generalized Dynkin game expressed in terms of the ${\cal E}^g$-evaluation. The proofs of these results are based on links between generalized Dynkin games and doubly reflected BSDEs. We then address the case of ambiguity on the model, - for example ambiguity on the default probability - and characterize the superhedging price of a game option as the value function of a mixed generalized Dynkin game.
Joint work with Roxana Dumitrescu (Humboldt-Universität zu Berlin) and Marie-Claire Quenez (Université Paris-Diderot).

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Adrien Richou, Université de Bordeaux: Numerical approximation of switching problems

We use the representation of Switching Problems as obliquely reflected BSDEs to obtain a discrete time approximation scheme of the solution. We thus focus on the disretization of the obliquely reflected BSDEs. By proving a stability result for the Euler scheme associated to the BSDE, we are able to obtain a rate of convergence in the Lipschitz setting and under the same structural conditions on the generator as the one required for the existence and uniqueness of a solution to the obliquely reflected BSDE.
This is a joint work with Jean-François Chassagneux.

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Zhou Chao, NUS: Dynamic Programming for Stochastic Control Problems with Expectation Constraints

We prove a dynamic programming principle for stochastic optimal control problems with expectation constraints by measurable selection approach. Since target problems, state constraints problems, quantile hedging and efficient hedging can all be reformulated into expectation constraints, we apply our results to prove the corresponding dynamic programming principle for these classes of stochastic control problems in a continuous but non-Markovian setting.
This is a joint work with Yulong ZHOU.

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