Quadratic optimal functional quantization of stochastic processes and numerical applications

In this paper, we present an overview of the recent developments of functional quantization of stochastic processes, with an emphasis on the quadratic case. Functional quantization is a way to approximate a process, viewed as a Hilbert-valued random variable, using a nearest neighbour projection on a finite codebook. A special emphasis is made on the computational aspects and the numerical applications, in particular the pricing of some path-dependent European options.Comment: 41 page

Strong low-frequency radio flaring from Cygnus X-3 observed with LOFAR

We present Low-Frequency Array (LOFAR) 143.5-MHz radio observations of flaring activity during 2019 May from the X-ray binary Cygnus X-3. Similar to radio observations of previous outbursts from Cygnus X-3, we find that this source was significantly variable at low frequencies, reaching a maximum flux density of about 5.8 Jy. We compare our LOFAR light curve with contemporaneous observations taken at 1.25 and 2.3 GHz with the RATAN-600 telescope, and at 15 GHz with the Arcminute Microkelvin Imager (AMI) Large Array. The initial 143.5-MHz flux density level, $\sim$2 Jy, is suggested to be the delayed and possibly blended emission from at least some of the flaring activity that had been detected at higher frequencies before our LOFAR observations had begun. There is also evidence of a delay of more than four days between a bright flare that initially peaked on May 6 at 2.3 and 15 GHz, and the corresponding peak ($\gtrsim$ 5.8 Jy) at 143.5 MHz. From the multi-frequency light curves, we estimate the minimum energy and magnetic field required to produce this flare to be roughly 10$^{44}$ erg and 40 mG, respectively, corresponding to a minimum mean power of $\sim$10$^{38}$ erg s$^{-1}$. Additionally, we show that the broadband radio spectrum evolved over the course of our observing campaign; in particular, the two-point spectral index between 143.5 MHz and 1.25 GHz transitioned from being optically thick to optically thin as the flare simultaneously brightened at 143.5 MHz and faded at GHz frequencies