arXiv subjects
Publications and source records attributed to Takashi Kato.
In this paper, we provide a new property of value at risk (VaR), which is a standard risk measure that is widely used in quantitative financial risk management. We show that the subadditivity of VaR for given loss random variables holds for any confidence level if and only if those are comonotonic. This result also gives a new equivalent condition for the comonotonicity of random vectors.
Acquiring plausible pathways on high-dimensional structural distributions is beneficial in several domains. For example, in the drug discovery field, a protein conformational pathway, i.e. a highly probable sequence of protein structural changes, is useful to analyze interactions between the protein and the ligands, helping to create new drugs. Recently, a state-of-the-art method in drug discovery was presented, which efficiently computes protein pathways using latent variables obtained from an isometric auto-encoding of the space of 3D density maps associated to protein conformations. However, our preliminary experiments show that there is room to significantly reduce the computing time. In this study, we use the Mapper algorithm, which is a Topological Data Analysis method, and present a novel variant to extract plausible conformational pathways from the isometric latent space with comparatively short running time. The extracted pathways are visualized as paths on the resulting Mapper graph. The methodological novelties are described as follows: firstly, the filter function of the Mapper algorithm is optimized so as to extract the pathways via minimization of an energy loss defined on the Mapper graph itself, while filter functions taken in the classical Mapper algorithm are fixed beforehand. The optimization is with respect to parameters of a deep neural network in the filter. Secondly, the clustering method, which defines the vertices and edges of the Mapper graph, of our algorithm, is designed by incorporating domain prior knowledge to assist the extraction. In our numerical experiments, based on an isometric latent space built on the common 50S-ribosomal dataset, the resulting Mapper graph successfully includes all the well-recognized plausible pathways. Moreover, our running time is much shorter than the above state-of-the-art counterpart.
We developed an optical phased array using an optical frequency comb and demonstrated its proof-of-principle. Optical phased arrays have been actively developed in recent years as a technology that can control the wavefront of light without any mechanical devices like phased array radar. Conventional optical phased arrays have been implemented using optical integrated circuits, but it has been difficult to achieve broadband operation with simple control. This is because control and calibration of a large number of phase modulators are required for each wavelength, and the dispersion of the waveguide makes whole bandwidth phase control of ultrashort pulses difficult. In contrast, we have developed a novel optical phased array that realizes wavefront control of ultrashort pulses generated by mode-locked laser by phase control of the comb, using high controllability of the comb and an optical array antenna with free-space optics. This is achieved by simply controlling the ratio of the two radio frequencies of the comb to realize a broadband optical phased array while suppressing environmental fluctuations. Experiments demonstrated broadband optical dot scanning at an optical frequency by forming an optical dot pattern and suppressing the environmental fluctuation by controlling the comb frequency. This innovative optical technology enables direct control of wavefronts by optical frequencies, i.e. controlling transverse modes by longitudinal modes.
We present quasi-real-time dual-comb spectroscopy (DCS) using two Yb:fiber combs with ~750 MHz repetition rates. A computational coherent averaging technique is employed to correct timing and phase fluctuations of the measured dual-comb interferogram (IGM). Quasi-real-time phase correction of 1-ms long acquisitions occurs every 1.5 seconds and is assisted by coarse radio frequency (RF) phase-locking of an isolated RF comb mode. After resampling and global offset phase correction, the RF comb linewidth is reduced from 200 kHz to ~1 kHz, while the line-to-floor ratio increases 13 dB in power in 1 ms. Using simultaneous offset frequency correction in opposite phases, we correct the aliased RF spectrum spanning three Nyquist zones, which yields an optical coverage of ~180 GHz around 1.035 $μ$m probed on a sub-microsecond timescale. The absorption profile of gaseous acetylene is observed to validate the presented technique.
We demonstrate balanced-detection dual-comb spectroscopy (DCS) using two free-running mechanical sharing dual-comb fiber lasers assisted by an all-computational digital phase correction algorithm. The mutual coherence between the combs allows us perform mode-resolved spectroscopy of gaseous hydrogen cyanide by digitally compensating residual timing and offset frequency fluctuations of the dual-comb signal. Setting the repetition rate difference between the combs to 500 Hz (1.5 kHz) yields more than 2000 resolved radio frequency comb lines after phase correction in a 3-dB bandwidth centered at 1560 nm of wavelength. Through coadding the corrected interferograms (IGMs), we obtain a single time-domain trace with a SNR of 6378 (13960) and 12.64 (13.77) bits of dynamic range in 1 second of averaging. The spectral SNR of the coadded trace reaches 529 (585), corresponding to a figure of merit of SNR of 1.3$\times$10$^6$ (1.4$\times$10$^6$). The measured absorption spectrum of hydrogen cyanide agrees well with the HITRAN database.
Dual-comb microscopy (DCM), an interesting imaging modality based on the optical-frequency-comb (OFC) mode and image pixel one-to-one correspondence, benefits from scan-less full-field imaging and simultaneous confocal amplitude and phase imaging. However, the two fully frequency-stabilized OFC sources requirement hampers DCM practicality due to the complexity and costs. Here, a bidirectional single-cavity dual-comb fiber laser (SCDCFL) is adopted as a DCM low-complexity OFC source. Computational image correction reduces the image blur caused by the SCDCFL residual timing jitter. Nanometer-order step surface profilometry with a 14.0 nm uncertainty highlights the image-corrected DCM effectiveness. The proposed method enhances the DCM versality and practicality.
This paper demonstrates that the precise phase controllability of an optical frequency comb enables all-optical signal processing for the first time. A novel all-optical Hilbert transform is presented with precise control of relative carrier-phase and envelope of optical pulse train based on frequency control utilizing an optical frequency comb. With the proposed all-optical signal processing method, fully non-scanning one-shot three-dimensional (3D) imaging can be realized with high image resolution. The technique can be applied to coherent phase imaging simultaneously. A precise pair of 90$^\circ$-phase-shifted optical pulses over the entire spectral bandwidth can be generated based on the precise optical phase controllability of an optical frequency comb, thereby facilitating a real-time and precise all-optical Hilbert transform to obtain amplitude and phase of optical signal in a single shot of ultrashort pulses. In our experiments, we realized single-shot 3D imaging with an uncertainty of 5 $\rmμ$m and obtained a surface profile with a resolution of 200 $\times$ 200 pixels.
We study the asymptotic behavior of the difference $Δρ^{X, Y}_α:= ρ_α(X + Y) - ρ_α(X)$ as $α\rightarrow 1$, where $ρ_α$ is a risk measure equipped with a confidence level parameter $0 < α< 1$, and where $X$ and $Y$ are non-negative random variables whose tail probability functions are regularly varying. The case where $ρ_α$ is the value-at-risk (VaR) at $α$, is treated in Kato (2017). This paper investigates the case where $ρ_α$ is a spectral risk measure that converges to the worst-case risk measure as $α\rightarrow 1$. We give the asymptotic behavior of the difference between the marginal risk contribution and the Euler contribution of $Y$ to the portfolio $X + Y$. Similarly to Kato (2017), our results depend primarily on the relative magnitudes of the thicknesses of the tails of $X$ and $Y$. We also conducted a numerical experiment, finding that when the tail of $X$ is sufficiently thicker than that of $Y$, $Δρ^{X, Y}_α$ does not increase monotonically with $α$ and takes a maximum at a confidence level strictly less than $1$.
In this study, we extend the optimal execution problem with convex market impact function studied in Kato (2014) to the case where the market impact function is S-shaped, that is, concave on $[0, \bar {x}_0]$ and convex on $[\bar {x}_0, \infty )$ for some $\bar {x}_0 \geq 0$. We study the corresponding Hamilton-Jacobi-Bellman equation and show that the optimal execution speed under the S-shaped market impact is equal to zero or larger than $\bar {x}_0$. Moreover, we provide some examples of the Black-Scholes model. We show that the optimal strategy for a risk-neutral trader with small shares is the time-weighted average price strategy whenever the market impact function is S-shaped.
In this study, we introduce an explicit trading-volume process into the Almgren-Chriss model, which is a standard model for optimal execution. We propose a penalization method for deriving a verification theorem for an adaptive optimization problem. We also discuss the optimality of the volume-weighted average-price strategy of a risk-neutral trader. Moreover, we derive a second-order asymptotic expansion of the optimal strategy and verify its accuracy numerically.
We study the asymptotic behavior of the difference between the values at risk VaR(L) and VaR(L+S) for heavy tailed random variables L and S for application in sensitivity analysis of quantitative operational risk management within the framework of the advanced measurement approach of Basel II (and III). Here L describes the loss amount of the present risk profile and S describes the loss amount caused by an additional loss factor. We obtain different types of results according to the relative magnitudes of the thicknesses of the tails of L and S. In particular, if the tail of S is sufficiently thinner than the tail of L, then the difference between prior and posterior risk amounts VaR(L+S) - VaR(L) is asymptotically equivalent to the expectation (expected loss) of S.
The volume weighted average price (VWAP) execution strategy is well known and widely used in practice. In this study, we explicitly introduce a trading volume process into the Almgren-Chriss model, which is a standard model for optimal execution. We then show that the VWAP strategy is the optimal execution strategy for a risk-neutral trader. Moreover, we examine the case of a risk-averse trader and derive the first-order asymptotic expansion of the optimal strategy for a mean-variance optimization problem.
In this short note, we study an optimization problem of expected implementation shortfall (IS) cost under general shaped market impact functions. In particular, we find that an optimal strategy is a VWAP (volume weighted average price) execution strategy when the market model is a Black-Scholes type with stochastic clock and market trading volume is large.
This paper is a continuation of Ishitani and Kato (2015), in which we derived a continuous-time value function corresponding to an optimal execution problem with uncertain market impact as the limit of a discrete-time value function. Here, we investigate some properties of the derived value function. In particular, we show that the function is continuous and has the semigroup property, which is strongly related to the Hamilton-Jacobi-Bellman quasi-variational inequality. Moreover, we show that noise in market impact causes risk-neutral assessment to underestimate the impact cost. We also study typical examples under a log-linear/quadratic market impact function with Gamma-distributed noise.
We study an optimal execution problem with uncertain market impact to derive a more realistic market model. We construct a discrete-time model as a value function for optimal execution. Market impact is formulated as the product of a deterministic part increasing with execution volume and a positive stochastic noise part. Then, we derive a continuous-time model as a limit of a discrete-time value function. We find that the continuous-time value function is characterized by a stochastic control problem with a Levy process.
We study an optimal execution problem in a continuous-time market model that considers market impact. We formulate the problem as a stochastic control problem and investigate properties of the corresponding value function. We find that right-continuity at the time origin is associated with the strength of market impact for large sales, otherwise the value function is continuous. Moreover, we show the semi-group property (Bellman principle) and characterise the value function as a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. We introduce some examples where the forms of the optimal strategies change completely, depending on the amount of the trader's security holdings and where optimal strategies in the Black-Scholes type market with nonlinear market impact are not block liquidation but gradual liquidation, even when the trader is risk-neutral.
We study an optimal execution problem in the presence of market impact where the security price follows a geometric Ornstein-Uhlenbeck process, which implies the mean-reverting property, and show that the optimal strategy is a mixture of initial/terminal block liquidation and gradual intermediate liquidation. The mean-reverting property describes a price recovery effect that is strongly related to the resilience of market impact, as described in several papers that have studied optimal execution in a limit order book (LOB) model. It is interesting that despite the fact that the model in this paper is different from the LOB model, the form of our optimal strategy is quite similar to those obtained for an LOB model. Moreover, we discuss what properties cause gradual liquidation as an optimal strategy by studying various cases and find out that not only "convexity of market impact function" but also "price recovery effect" (or, in other words, transience of market impact) are essential to make a trader execute the security gradually to mitigate the effect of market impact.
A one-factor asset pricing model with an Ornstein--Uhlenbeck process as its state variable is studied under partial information: the mean-reverting level and the mean-reverting speed parameters are modeled as hidden/unobservable stochastic variables. No-arbitrage pricing formulas for derivative securities written on a liquid asset and exponential utility indifference pricing formulas for derivative securities written on an illiquid asset are presented. Moreover, a conditionally linear filtering result is introduced to compute the pricing/hedging formulas and the Bayesian estimators of the hidden variables.