## Section: New Results

Keywords : jump diffusions, stochastic control.

### Stochastic control - Application in finance and assurance

Participants : B. Øksendal (Oslo University), D. Hernandez-Hernandez, M.C. Quenez, A. Sulem, P. Tankov.

B. Øksendal (Oslo University) and A.Sulem have written a second edition of their book on Stochastic control of Jump diffusions [11] . In the Second Edition there is a new chapter on optimal control of stochastic partial differential equations driven by Lévy processes. There is also a new section on optimal stopping with delayed information.

In [40] , A. Sulem and B. Øksendal consider a stochastic differential game in a financial jump diffusion market, where the agent chooses a portfolio which maximizes the utility of her terminal wealth, while the market chooses a scenario (represented by a probability measure) which minimizes this maximal utility. They show that the optimal strategy for the market is to choose an equivalent martingale measure.

A. Sulem and P. Tankov are studing pricing and hedging in markets with jumps using utility maximization and indifference pricing, and A. Sulem and B. Øksendal are studing risk-indifference pricing in these markets. D. Hernandez-Hernandez and A. Schied have solved the problem of characterization of the indifference price of derivatives for stochastic volatility models [21] .

M.C. Quenez and Daniel Hernandez-Hernandez are working on the the problem of
characterization of the variance optimal martingale measure in a
stochastic volatility model. Recall that the variance-optimal
martingale measure appears to be a key tool for characterizing the
optimal hedging strategy of the mean-variance hedging problem.
Laurent and Pham (1999) have solved the problem in terms of classical
solutions of PDEs in the particular case where the coefficients of
the model do not depend on time and price process. Quenez and Hernandez-Hernadez consider the general case. The idea is to approximate the value
function by a sequence of classical solutions of some Dirichlet
problems (which converges unifomly on each compact set of [0, T]×R^{d} ). Using this property, they derive an estimation of the gradiant of
the value function, which allows them to characterize the optimal
risk-premium (associated with ) as the gradient of the value
function (multiplied by a coefficient of the model) [42] .

M.C. Quenez and B. Jottreau are studying the problem of portfolio optimization with default. Using dynamic programming they study the aproximation of the associate HJB equation.