## CMAP

## Emmanuel GOBET - Images

### SCIENTIFIC IMAGES

#### Valuation of an American Put (optimal stopping problem), using an optimization of the exercice boundary.

Based on the boundary sensitivity formula from**Boundary sensitivities for diffusion processes in time dependent domains**(C. Costantini, E. Gobet and N. El Karoui), Applied Mathematics and Optimization, Vol.54(2), pp.159-187, 2006.Green: the optimal boundary.

Red: the iterative approximation.

Axis: x for time, y for space.

(Reload to get the boundary animation).Convergence of the value function (put price) through iterations. #### Convergence of policy value iteration in 4-d dimensional stochastic control problem.

Example from**Sensitivity analysis using Itô-Malliavin calculus and martingales. Application to stochastic optimal control**(E. Gobet and R. Munos), SIAM Journal on Control and Optimization, Vol.43(5), pp.1676-1713, 2005.The neural network to approximate the policy as a function of time and space. The evolution of the neural network parameters through the stochastic optimization algorithm. Performance of the algorithm (maximization of a probability of reaching a stochastic fly). #### Geometric convergence of the adaptive control variates algotithm.

Based on**Sequential control variates for functionals of Markov processes**(E. Gobet and S. Maire), SIAM Journal on Numerical Analysis, Vol.43(3), pp.1256-1275, 2005.**Solving BSDE with adaptive control variate**(E. Gobet et C. Labart), SIAM Journal on Numerical Analysis, Vol.48(1), pp.257-277, 2010.

2-d example, convergence of the error to 0, using only 2 (!!) simulations at each point of the grid. The solution is polynomial and polynomial interpolation is used to approximate the solutions. Each image corresponds to a new iteration (Reload to get the animation). Comparison of the geometric convergence using 2 or 100 simulations. Performance of the algorithm when the solution is not polynomial and polynomial interpolation is used (Reload to get the animation). The same type of algorithm is now designed and tested on Backward Stochastic Differential Equations (associated to semi-linear PDE). Here is the convergence for call payoff and linear driver.

Axis: x for space and y for time. The errors are larger close to the singularity (2,1) (i.e. the strike).#### Boundary shifting to compensate overestimation of exit times.

Based on**Stopped diffusion processes: overshoots and boundary correction**(E. Gobet and S. Menozzi), Stochastic Processes and their Applications, Vol.120, pp.130-162, 2010.**Advanced Monte Carlo methods for barrier and related exotic options**(E. Gobet), Handbook of Numerical Analysis, Vol. XV. Elsevier. Special Volume: Mathematical Modeling and Numerical Methods in Finance. Editor: P.G. Ciarlet. Guest Editors: Alain Bensoussan and Qiang Zhang. pp.497-528, 2009.

Convergence comparison between the standard method (discrete exit time in Red), Romberg extrapolation (pink) and the boundary correction method (blue). The advantage of the Boundary correction method over Brownian bridge techniques is its avalaibility in any dimension.