Lecturer: |
M.C. Quenez |

Period: |
Term 2 |

ECTS: |
3 |

Schedule: |
3 hours per week |

**Programme : **

In this course, we present the theory of backwards as well as their applications to finance.

Backwards (also called backward stochastic differential equations) are stochastic differential equations for which the terminal condition is given. The mathematical tools which are introduced in

this course are very useful to study market models with imperfections, including in particular the case of some funding constraints, the case when the seller is a ``large" investor (whose strategy

may impact the prices), the case of a market model with uncertainty ... In this type of model, the dynamics of the wealth are non-linear, contrary to the classical case. The price of an option

with payoff and maturity T then corresponds to the solution of a backward (of a non-linear type) with terminal condition, which induces on the market a non-linear pricing system with

respect to the payoff. The techniques of backwards are also very useful in stochastic control, in particular to study some portfolio optimization problems, as well as dynamic risk-measures.

- Introduction. Some useful properties (martingale inequalities ...).
- Existence and uniqueness of the solution of a backward equation.
- A representation property of the solution of a linear BSDE, and application to the pricing of European options in a perfect market model.
- Comparison theorem for backwards.
- Backwards et optimization: a verification theorem, and a necessary and suffcient optimality condition.
- Application to the non-linear pricing of options in an imperfect market. Examples.
- Markovian case: links between non-linear backwards and non-linear PDEs.
- Dynamic risk-measures induced by non-linear backwards.
- Backwards with default jump. Application to the non-linear pricing in a market with default and funding constraints and/or repurchase agreement (case of CVA, for example).

**Bibliographie :**

Crépey, S (2015) Bilateral Counterparty Risk under Funding Constraints- Part II CVA. Mathematical Finance, 25, 23-50.

Dumitrescu, R., Grigorova, M., Quenez M.C., Sulem A. (2018), BSDEs with default jump, in Computation and Combinatorics in Dynamics, Stochastics and Control - Abel Symposium, August 2016}, E. Celledoni et al. (Eds) vol 13. Springer, p. 233--263, https://doi.org/10.1007/978-3-030-01593-0\_9, available at https://hal.inria.fr/hal-01799335.

El Karoui N., Kapoudjian C., Pardoux E., Peng S. et M.C.Quenez (1997), Reflected solutions of Backward SDEs and related obstacle problems

for PDEs, The Annals of Probability, 25,2, 702-737.

El Karoui N. et M.C. Quenez (1996), Non-linear Pricing Theory and Backward Stochastic Differential

Equations, Financial Mathematics, Lectures Notes in Mathematics 1656, Bressanone, 1996,

Editor:W.J.Runggaldier, collection Springer,1997.

Pham H. (2007), Optimisation et contrôle stochastique appliqués à la finance, Springer Chap.5.

Quenez M.C. (2010), Backward Stochastic Differential Equations (BSDEs) and Reflected BSDEs, Encyclopedia of Quantitative Finance.