The talk concerns the use of backward stochastic differential equations (BSDEs) in mathematical finance In particular we aim at dealing with concrete challenging questions arising in contexts which are not covered by the standard approach à la Black-Scholes (BS).
Empirical evidence has pointed out that assuming the Geometric Brownian motion as a model for the stock prices' behaviors it is not fully satisfactory lacking of an accurate description of financial data. Discrepancies between the BS forecasted prices and real data, for instance, have their roots in properties of trajectories of the Brownian motion, implied volatility in option market, kurtosis and skewness of asset returns. Great improvements has been achieved taking into account jump perturbations in the asset price Motion using Lévy processes, generalizing the classical BS setup by allowing the stock prices to jump while preserving the independence and stationarity of returns. We start analyzing Poisson and Compound Poisson process, then we move to fundamental properties of Lévy processes, giving a meaning to the integral with respect a jump processes. The latter allows us to manage general Itô-Lévy processes, extending the notion of Lévy Processes while still preserving their mathematical structure. We provide basic results and properties of Itô-Lévy processes which are fundamental to explain results for BSDEs.
We use BSDEs theory to deal with mathematical financial problem, in particular for pricing and hedging purposes. Moreover we interpret a general BSDE as the wealth process equation of a portfolio subject to a riskless asset and a risky security including a jump component in addition to a Brownian Motion to model the underlying asset’s behavior. The structure of such results are similar to those obtained in the BS framework but with a fundamental difference; market with jumps is incomplete and hence the equivalent martingales measure is not unique.
We also sketch applications of comparison and optimization results for dynamic risk measures induced by a BSDE with jumps.
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