arXiv subjects
Publications and source records attributed to Blandine Berard Bergery.
We propose a mathematical model for one pattern of charts studied in technical analysis: in a phase of consolidation, the price of a risky asset goes down $ξ$ times after hitting a resistance level. We construct a mathematical strategy and we calculate the expectation of the wealth for the logaritmic utility function. Via simulations, we compare the strategy with the standard one.
Through a regularization procedure, few approximation schemes of the local time of a large class of one dimensional processes are given. We mainly consider the local time of continuous semimartingales and reversible diffusions, and the convergence holds in ucp sense. In the case of standard Brownian motion, we have been able to determine a rate of convergence in $L^2$, and a.s. convergence of some of our schemes.
We give some approximations of the local time process $(L_t^x)_{t\geqslant 0}$ at level $x$ of the real Brownian motion $(X_t)$. We prove that $ \frac{2}ε\int_0^{t} X_{(u+ε)\wedge t}^+ \indi_{\{X_u \leqslant 0\}} du + \frac{2}ε\int_0^{t} X_{(u+ε) \wedge t}^- \indi_{\{X_u>0\}} du$ and $\frac{4}ε\int_0^{t} X_u^- \indi_{\{X_{(u+ε) \wedge t} > 0\}} du$ converge in the ucp sense to $L_t^0$, as $ε\to 0$. We show that $ \frac{1}ε\int_0^t (\indi_{\{x<X_{s+ε}\}} - \indi_{\{x<X_{s}\}}) (X_{s+ε}-X_{s})ds$ goes to $L_t^x$ in $L^2(Ω)$ as $ε\to 0$, and that the rate of convergence is of order $ε^α$, for any $α< {1/4}$.