# SMS 2016 Summer School: Dynamics of Biological Systems

### Organizers :

# Backward stochastic equations and equilibrium pricing

We propose an equilibrium framework within which to price financial securities written on non-tradable underlyings such as temperature indices. We analyze a financial market with a finite set of agents whose preferences are described by a convex dynamic risk measure generated by the solution of a backward stochastic differential equation. The agents are exposed to financial and non-financial risk factors. They can hedge their financial risk in the stock market and trade a security whose payoff depends on both financial and external risk factors. We prove an existence and uniqueness of equilibrium result for security price and characterize the equilibrium market price of risk in terms of a solution to a non-linear BSDE.

# Dual space multigrid strategies for variational data assimilation

4D variational data assimilation problems in geophysical fluids consist in solving nonlinear least square problems.The incremental 4D-Var method is a popular technique to tackle high dimensional problems as it is equivalent to applying a truncated Gauss-Newton iteration. The development of efficient numerical techniques, like the restricted preconditioned conjugate gradient (RPCG) method, allow to solve the embedded quadratic optimization subproblem (also called inner loop) in observation space. This approach becomes computationally attractive when the number of observations is much smaller than the dimension of the control space (for instance the dimension of the initial state vector). However, the amount of observations to assimilate can still be large even in this favorable case. In order to reduce the computing costs of the Incremental 4D-Var in observation space, we proposed an observation-thinning strategy that exploits an adaptive structure of the observations in the spirit of multigrid techniques. The thinned observation set is defined using a hierarchy of observations, from coarsest to finest level. Starting from the coarsest set of observations, observations from the next level will be included in the observation set according to the influence they have on the solution, as measured by an estimate on the solution variation between two consecutive levels. Numerical experiments performed in toy models highlighted the benefits of this approach considering both the reductions in the computing costs/time and the amount of assimilated observations. Based on this multilevel approach, we investigate the introduction of multigrid techniques in the resolution of the optimization problem in order to speed up the convergence of the dual space iterative solver. Such techniques aim at exploiting the smoothing properties of iterative solvers by introducing coarse grid correction steps that can efficiently remove the large scale components of the error.

# 2016 Statistical Society of Canada Student Conference

### Organizer :

Thuva Vanniyasingam (McMaster University)

### Web site : http://www.ssc.ca/en/meetings/2016/student_conference

# Rare DNA variants, Analysis of family studies

### Organizers

### Aurélie Labbe (McGill University)

Alexandre Bureau (Université Laval)

### Web site : http://www.crm.umontreal.ca/2016/Family16/index_e.php

# Part B Examination - Michele Fornea

**PRINCIPAL **

# Part B Examination - Nicholas Simard

**PRINCIPAL **

# TBA

# On the number of planar Eulerian orientations

### Web site : http://www.lacim.uqam.ca

# Seventh Montreal Industrial Problem Solving Workshop