Modeling financial time series by stochastic processes is a challenging task and a central area of research in financial mathematics. In this paper, we break through this barrier and present Quant GANs, a data-driven model which is inspired by the recent success of generative adversarial networks (GANs). Quant GANs consist of a generator and discriminator function which utilize temporal convolutional networks (TCNs) and thereby achieve to capture longer-ranging dependencies such as the presence of volatility clusters. Furthermore, the generator function is explicitly constructed such that the induced stochastic process allows a transition to its risk-neutral distribution. Our numerical results highlight that distributional properties for small and large lags are in an excellent agreement and dependence properties such as volatility clusters, leverage effects, and serial autocorrelations can be generated by the generator function of Quant GANs, demonstrably in high fidelity.
We study two properties of semigroups of sub-Markov kernels, namely uniform conditional ergodicity and intrinsic ultracontractivity. In this paper we investigate the relationship between these two properties and we provide sufficient criteria as well as characterisations of them. In particular, our considerations show that, under suitable assumptions, the second property implies the first one. We also introduce a property called compact domination and show how this property and the parabolic boundary Harnack principle are related to the aforementioned properties. Furthermore, we apply these results in some special cases.
We consider a homogeneous fragmentation process with killing at an exponential barrier. With the help of two families of martingales we analyse the decay of the largest fragment for parameter values that allow for survival. In this respect the present paper is also concerned with the probability of extinction of the killed process.We begin our exposition by briefly reviewing what is meant by a homogeneous fragmentation process, thereby introducing some notation.
In this paper we study processes which are constructed by a convolution of a deterministic kernel with a martingale. A special emphasis is put on the case where the driving martingale is a centred Lévy process, which covers the popular class of fractional Lévy processes. As a main result we show that, under appropriate assumptions on the kernel and the martingale, the maximum process of the corresponding 'convoluted martingale' is p-integrable and we derive maximal inequalities in terms of the kernel and of the moments of the driving martingale.in terms of the kernel and the L p (P)-norm of the driving martingale. Let us illustrate our results for the case of a fractional Lévy process
In the case of neutral populations of fixed sizes in equilibrium whose
genealogies are described by the Kingman $N$-coalescent back from time $t$
consider the associated processes of total tree length as $t$ increases. We
show that the (c\`adl\`ag) process to which the sequence of compensated tree
length processes converges as $N$ tends to infinity is a process of infinite
quadratic variation; therefore this process cannot be a semimartingale. This
answers a question posed in Pfaffelhuber et al. (2011).Comment: 13 pages, 3 figure
Deep generative networks such as GANs and normalizing flows flourish in the context of high-dimensional tasks such as image generation. However, so far an exact modeling or extrapolation of distributional properties such as the tail asymptotics generated by a generative network is not available. In this paper we address this issue for the first time in the deep learning literature by making two novel contributions. First, we derive upper bounds for the tails that can be expressed by a generative network and demonstrate L p -space related properties. There we show specifically that in various situations an optimal generative network does not exist. Second, we introduce and propose copula and marginal generative flows (CM flows) which allow for an exact modeling of the tail and any prior assumption on the CDF up to an approximation of the uniform distribution. Our numerical results support the use of CM flows.
In recent times of noticeable climate change the consideration of external factors, such as weather and economic key figures, becomes even more crucial for a proper valuation of derivatives written on agricultural commodities. The occurrence of remarkable price changes as a result of severe changes in these factors motivates the introduction of different price states, each describing different dynamics of the price process. In order to include external factors we propose a two-step hybrid model based on machine learning methods for clustering and classification. First, we assign price states to historical prices using K-means clustering. These price states are also assigned to the corresponding data of external factors. Second, predictions of future price states are then obtained from short-term predictions of the external factors by means of either K-nearest neighbors or random forest classification. We apply our model to real corn futures data and generate price scenarios via a Monte Carlo simulation, which we compare to Sørensen (J Futures Mark 22(5):393–426, 2002). Thereby we obtain a better approximation of the real futures prices by the simulated futures prices regarding the error measures MAE, RMSE and MAPE. From a practical point of view, these simulations can be used to support the assessment of price risks in risk management systems or as decision support regarding trading strategies under different price states.
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