## CMAP

Grand Cascade in Petrodvorets. Photo: A.Savin, Wikimedia Commons

**Main Lecture Courses**

The summer school will be structured around two main topics:

**Numerical methods for nonlinear problems.**Minicourses will be given by Bruno Bouchard (Paris-Dauphine Univesity) and Nizar Touzi (Ecole Polytechnique)**Financial markets with arbitrage.**Minicourses will be given by Kostas Kardaras (London School of Economics) and Johannes Ruf (University College London).- In addition, a minicourse on performance measures will be given by Mikhail Zhitlukhin (Steklov mathematical institute).

** Market viability and hedging beyond NFLVR**

Kostas Kardaras (London School of Economics)

- Fundamental Theorems under a weaker market viability condition than No Free Lunch with Vanishing Risk (NFLVR).
- Hedging duality and its applications.
- Extensions to infinite-asset market models.

** References**

- The numéraire portfolio in semimartingale financial models (I. Karatzas, C. Kardaras). Finance and Stochastics, volume 11 (2007), pages 447-493.
- Finitely additive probabilities and the Fundamental Theorem of Asset Pricing (C. Kardaras). Contemporary Quantitative Finance (Platen Festschrift), Springer (2010), pages 19-34.
- No arbitrage and local martingale deflators (Yu. Kabanov, C. Kardaras, and S. Song). Preprint.

**
Arbitrage opportunities relative to the market
**

Johannes Ruf (University College London)

- Introduction to stochastic portfolio theory
- The theory of functionally generated trading strategies
- Suitable conditions guaranteeing that these trading strategies almost surely outperform the market
- The question of the existence of short-term arbitrage in market models

** References**

- Trading Strategies Generated by Lyapunov Functions (I. Karatzas, J. Ruf). Preprint.
- Volatility and Arbitrage (E. R. Fernholz, I. Karatzas, J. Ruf). Preprint.
- Stochastic portfolio theory: an overview. (R. Fernholz and I. Karatzas) Handbook of numerical analysis 15 (2009), pages 89-167.

**
Monte-Carlo approximations of backward stochastic differential equations
**

Bruno Bouchard (Paris Dauphine University)

We will study Monte-Carlo approximations of backward stochastic differential equations (BSDEs), which provide a numerical tool for the resolution of semi-linear parabolic partial differential equations (PDEs). We shall first briefly review the links between BSDEs and PDEs and then discuss the (abstract) Euler-type discrete time approximation of BSDEs (standard, reflected, with Dirichlet type boundary condition, with irregular terminal condition). The later involves the computation of conditional expectations. Two techniques will be considered. One consists in using the Malliavin calculus to rewrite condition expectations as the ratio of two unconditional ones, the second uses non-parametric regression techniques.

** References**

- Bouchard, B., I. Ekeland and N. Touzi (2004). On the Malliavin approach to Monte Carlo approximation of conditional expectations. Finance and Stochastics, 8(1), 45-71.
- Bouchard, B. and J.-F. Chassagneux (2008). Discrete-time approximation for continuously and discretely reflected BSDEs.
- Bouchard, B. and S. Menozzi (2009). Strong approximations of BSDEs in a domain. Bernoulli, 15(4), 1117-1147.
- Bouchard, B. and N. Touzi (2004). Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stochastic Processes and their applications, 111(2), 175-206.
- Bouchard, B. and X. Warin (2012). Monte-Carlo valuation of American options: facts and new algorithms to improve existing methods. In Numerical methods in finance (pp. 215-255). Springer Berlin Heidelberg.
- Chassagneux, J.-F. (2014). Linear multistep schemes for BSDEs. SIAM Journal on Numerical Analysis, 52(6), 2815-2836.
- Chassagneux, J.-F. and A. Richou (2016). Numerical simulation of quadratic BSDEs. The Annals of Applied Probability, 26(1), 262-304.
- El Karoui, N., S. Hamadène and A. Matoussi (2008). Backward stochastic differential equations and applications. Indifference pricing: theory and applications, 267-320.
- Geiss, C., S. Geiss and E. Gobet (2012). Generalized fractional smoothness and Lp-variation of BSDEs with non-Lipschitz terminal condition. Stochastic Processes and their Applications, 122(5), 2078-2116.
- Gobet, E. and A. Makhlouf (2010). Time regularity of BSDEs with irregular terminal functions. Stochastic Processes and their Applications, 120(7), 1105-1132.
- Gobet, E. and P. Turkedjiev (2015). Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions. Mathematics of Computation.

**
Branching particle representations of solutions to partial
differential equations
**

Nizar Touzi (Ecole Polytechnique)

We provide a representation of the solution of a semilinear partial differential equation by means of a branching diffusion. Unlike the backward SDE approach, such a representation induces a purely forward Monte Carlo method. We also provide applications to the unbiased simulation of stochastic differential equations.

** References**

- P. Henry-Labordère, X. Tan and N. Touzi, A numerical algorithm for a class of BSDE via branching process. Stochastic Processes and their Applications 124:1112-1140, 2014.
- P. Henry-Labordère, X. Tan and N. Touzi, Unbiased simulation of stochastic differential equations.
- P. Henry-Labordère, N. Oudjane, X. Tan, N. Touzi, and X. Warin. Branching diffusion representation of semilinear PDEs and Monte Carlo approximation

**
Performance measures: axiomatics and concrete examples
**

Mikhail Zhitlukhin (Steklov Mathematical Institute, Moscow)

I will give a brief overview of the theory of performance measures, which is an approach to evaluate investment strategies based on ideas similar to convex and coherent risk measures. We will start with abstract axiomatics, prove a representation theorem, and then look at specific examples and discuss numerical implementations.

** References**

- A. Cherny, D. Madan (2009). New measures for performance evaluation. Review of Financial Studies, 22(7), 2571-2606.
- M. Zhitlukhin (2015). A monotone modification of the Sharpe ratio. Preprint.