We consider three classes of linear differential equations on distribution functions, with a fractional order α ∈ [0, 1]. The integer case α = 1 corresponds to the three classical extreme families. In general, we show that there is a unique distribution function solving these equations, whose underlying random variable is expressed in terms of an exponential random variable and an integral transform of an independent α-stable subordinator. From the analytical viewpoint, this distribution is in one-to-one correspondence with a Kilbas-Saigo function for the Weibull and Fréchet cases, and with a Le Roy function for the Gumbel case.

We introduce a statistical simulation method for dynamic portfolio valuation and risk management building on machine learning with kernels. We learn the dynamic value process of a portfolio from a finite sample of its terminal cumulative cash flow. The learned value process is given in closed form thanks to a suitable choice of the kernel. We develop an asymptotic theory and prove convergence and a central limit theorem. We derive dimension-free sample error bounds and concentration inequalities. Numerical examples show good results for a relatively small training sample size. Keywords: dynamic portfolio valuation, kernel ridge regression, learning theory, reproducing kernel Hilbert space, portfolio risk management MSC (2010) Classification: 68T05, 91G60 JEL Classification: C15, G32 1 Introduction Valuation, risk measurement, and hedging form an integral task in portfolio risk management for banks, insurance companies, and other financial institutions. Portfolio risk arises because the values of constituent assets and liabilities of the portfolio change over time in response to changes in the underlying risk factors, e.g., interest rates, equity prices, real-estate prices, foreign exchange rates, credit spreads, etc. The quantification and management of this risk requires a stochastic model of the dynamic portfolio value process.In the absence of a liquid market for price discovery of such a portfolio, its value has to be derived from the cash flow that it generates. Examples of such portfolios include path-dependent options, structured products, such as barrier reverse convertibles, mortgage-backed securities, and life insurance products featuring mortality and longevity risks. More formally, we henceforth assume that all financial values and cash flows are discounted by some numeraire, e.g., the money market account that earns the short-term interest rates.

We characterize the complete monotonicity of the Kilbas-Saigo function on the negative half-line. We also provide the exact asymptotics at −∞, and uniform hyperbolic bounds are derived. The same questions are addressed for the classical Le Roy function. The main ingredient for the proof is a probabilistic representation of these functions in terms of the stable subordinator.

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We consider three classes of linear differential equations on distribution functions, with a fractional order α ∈ [0, 1]. The integer case α = 1 corresponds to the three classical extreme families. In general, we show that there is a unique distribution function solving these equations, whose underlying random variable is expressed in terms of an exponential random variable and an integral transform of an independent α−stable subordinator. From the analytical viewpoint, this law is in one-to-one correspondence with a Kilbas-Saigo function for the Weibull and Fréchet cases, and with a Le Roy function for the Gumbel case. By the stochastic representation, we can derive several analytical properties for the latter special functions, extending known features of the classical Mittag-Leffler function, and dealing with monotonicity, complete monotonicity, infinite divisibility, asymptotic behaviour at infinity, uniform hyperbolic bounds.