IBRAHIM Dalia

This paper studies the question of filtering and maximizing terminal wealth from expected utility in a partially information stochastic volatility models. The special features is that the only information available to the investor is the one generated by the asset prices, and the unobservable processes will be modeled by a stochastic differential equations. Using the change of measure techniques, the partial observation context can be transformed into a full information context such that coefficients depend only on past history of observed prices (filter processes). Adapting the stochastic non-linear filtering, we show that under some assumptions on the model coefficients, the estimation of the filters depend on a priori models for the trend and the stochastic volatility. Moreover, these filters satisfy a stochastic partial differential equations named "Kushner-Stratonovich equations". Using the martingale duality approach in this partially observed incomplete model, we can characterize the value function and the optimal portfolio. The main result here is that the dual value function associated to the martingale approach can be expressed, via the dynamic programming approach, in terms of the solution to a semilinear partial differential equation which depends also on the filters estimate and the volatility. We illustrate our results with some examples of stochastic volatility models popular in the financial literature.

We study the filtering problem and the maximization problem of expected utility from terminal wealth in a partial information context. The special features is that the only information available to the investor is the vector of sock prices. The mean rate of return processes are not directly observed and supposed to be driven by a process $\mu_{t}$ modeled by a stochastic differential equations. The main result in this paper is to show under which assumptions on the coefficients of the model, we can estimate the unobserved market price of risks. Using the innovation approach, we show that under globally Lipschitz conditions on the coefficients of $\mu_{t}$, the filters estimate of the risks satisfy a measure-valued Kushner-Stratonovich equations. On the other hand, using the pathwise density approach, we show that under a nondegenerate assumption and some regularity assumptions on the coefficients of $\mu_{t}$, the density of the conditional distribution of $\mu_{t}$ given the observation data, can be expressed in terms of the solution to a linear parabolic partial differential equation parameterized by the observation path. Also, we can obtain an explicit formulae for the optimal wealth, the optimal portfolio and the value function for the cases of logarithmic and power utility function.

This paper studies the question of filtering and maximizing terminal wealth from expected utility in a partially information stochastic volatility models. The special features is that the only information available to the investor is the one generated by the asset prices, and the unobservable processes will be modeled by a stochastic differential equations. Using the change of measure techniques, the partial observation context can be transformed into a full information context such that coefficients depend only on past history of observed prices (filter processes). Adapting the stochastic non-linear filtering, we show that under some assumptions on the model coefficients, the estimation of the filters depend on a priori models for the trend and the stochastic volatility. Moreover, these filters satisfy a stochastic partial differential equations named "Kushner-Stratonovich equations". Using the martingale duality approach in this partially observed incomplete model, we can characterize the value function and the optimal portfolio. The main result here is that the dual value function associated to the martingale approach can be expressed, via the dynamic programming approach, in terms of the solution to a semilinear partial differential equation which depends also on the filters estimate and the volatility. We illustrate our results with some examples of stochastic volatility models popular in the financial literature.

In the context of my thesis, I was interested in the mathematical analysis of a volatility breakout indicator widely used by practitioners in the trading room. The Bollinger Bands indicator belongs to the family of so-called technical analysis methods and is therefore based exclusively on the recent history of the price considered and a principle deduced from past market observations, independently of any mathematical model. My work consists in studying the performance of this indicator in a universe that would be governed by stochastic differential equations (Black-Scholes) whose diffusion coefficient changes its value at an unknown and unobservable random time, for a practitioner wishing to maximize an objective function (for example, a certain expected utility of the portfolio value at a certain maturity).

The objective of my thesis is to study mathematically a volatility breakout indicator widely used by practitioners in the trading room. The Bollinger Bands indicator belongs to the family of so-called technical analysis methods and is therefore based exclusively on the recent history of the price considered and a principle deduced from past market observations, independently of any mathematical model. My work consists in studying the performance of this indicator in a universe governed by stochastic differential equations (Black-Scholes) whose diffusion coefficient changes its value at an unknown and unobservable random time, for a practitioner wishing to maximize an objective function (for example, a certain expected utility of the portfolio value at a certain maturity). In the framework of the model, the Bollinger indicator can be interpreted as an estimator of the time of the next break. In the case of small volatilities, we show that the behavior of the density of the indicator depends on the volatility, which makes it possible to detect, for a large enough volatility ratio, the volatility regime in which the indicator's distribution is located. Also, in the case of high volatilities, we show by an approach via the Laplace transform, that the asymptotic behavior of the indicator's distribution tails depends on the volatility. This makes it possible to detect the change in the large volatilities. Then, we are interested in a comparative study between the Bollinger indicator and the classical estimator of the quadratic variation for the detection of change in volatility. Finally, we study the optimal portfolio management which is described by a non-standard stochastic problem in the sense that the admissible controls are constrained to be functionals of the observed prices. We solve this control problem by drawing on the work of Pham and Jiao to decompose the initial portfolio allocation problem into a post-breakdown management problem and a pre-breakdown problem, and each of these problems is solved by the dynamic programming method. Thus, a verification theorem is proved for this stochastic control problem.