We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

The free-space Green’s function is used to evaluate electromagnetic wave scattering and radiation from currents in free space. This section investigates two examples using the free-space Green’s function, namely, radiation from a sheet current and radiation from a shell current.

Green’s Functions: Applications

We propose a time-space discretization scheme for quasi-linear PDEs. The algorithm relies on the theory of fully coupled Forward-Backward SDEs, which provides an efficient probabilistic representation of this type of equations. The derivated algorithm holds for strong solutions defined on any interval of arbitrary length. As a bypass product, we obtain a discretization procedure for the underlying FBSDE.

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A Forward-Backward Stochastic Algorithm For Quasi-Linear PDEs

Adaptive multidimensional integration: VEGAS enhanced

Since its migration from the Asian honey bee (Apis cerana) to the European honey bee (Apis mellifera), the ectoparasitic mite Varroa destructor has emerged as a major issue for beekeeping worldwide. Due to a short history of coevolution, the host-parasite relationship between A. mellifera and V. destructor is unbalanced, with honey bees suffering infestation effects at the individual, colony and population levels. Several control solutions have been developed to tackle the colony and production losses due to Varroa, but the burden caused by the mite in combination with other biotic and abiotic factors continues to increase, weakening the beekeeping industry. In this synthetic review, we highlight the main advances made between 2015 and 2020 on V. destructor biology and its impact on the health of the honey bee, A. mellifera. We also describe the main control solutions that are currently available to fight the mite and place a special focus on new methodological developments, which point to integrated pest management strategies for the control of Varroa in honey bee colonies.

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Varroa destructor: how does it harm Apis mellifera honey bees and what can be done about it?

We consider the exact path sampling of the squared Bessel process and some other continuous-time Markov processes, such as the CIR model, constant elasticity of variance diffusion model, and hypergeometric diffusions, which can all be obtained from a squared Bessel process by using a change of variable, time and scale transformation, and/or change of measure. All these diffusions are broadly used in mathematical finance for modelling asset prices, market indices, and interest rates. We show how the probability distributions of a squared Bessel bridge and a squared Bessel process with or without absorption at zero are reduced to randomized gamma distributions. Moreover, for absorbing stochastic processes, we develop a new bridge sampling technique based on conditioning on the first hitting time at zero. Such an approach allows us to simplify simulation schemes. New methods are illustrated with pricing path-dependent options.

Exact Simulation of Bessel Diffusions

This paper develops bridge sampling path integral algorithms for pricing path-dependent options under a new class of nonlinear state dependent volatility models. Path-dependent option pricing is considered within a new (dual) Bessel family of semimartingale diffusion models, as well as the constant elasticity of variance (CEV) diffusion model, arising as a particular case of these models. The transition p.d.f.s or pricing kernels are mapped onto an underlying simpler squared Bessel process and are expressed analytically in terms of modified Bessel functions. We establish precise links between pricing kernels of such models and the randomized gamma distributions, and thereby demonstrate how a squared Bessel bridge process can be used for exact sampling of the Bessel family of paths. A Bessel bridge algorithm is presented which is based on explicit conditional distributions for the Bessel family of volatility models and is similar in spirit to the Brownian bridge algorithm. A special rearrangement and splitting of the path integral variables allows us to combine the Bessel bridge sampling algorithm with either adaptive Monte Carlo algorithms, or quasi-Monte Carlo techniques for significant numerical efficiency improvement. The algorithms are illustrated by pricing Asian-style and lookback options under the Bessel family of volatility models as well as the CEV diffusion model.

Pricing path-dependent options on state dependent volatility models with a bessel bridge

We present some further developments in the construction and classification of new solvable one-dimensional diffusion models having transition densities, and other quantities that are fundamental to derivatives pricing, representable in analytically closed form Our approach is based on so-called diffusion canonical transformations that produce a large class of multiparameter nonlinear local volatility diffusion models that are mapped onto various simpler diffusions Using an asymptotic analysis, we arrive at a rigorous boundary classification as well as a characterization with respect to probability conservation and the martingale property of the newly constructed diffusions Specifically, we analyze and classify in detail four main families of driftless regular diffusion models that arise from the underlying squared Bessel process (the Bessel family), Cox–Ingersoll–Ross process (the confluent hypergeometric family), the Ornstein-Uhlenbeck diffusion (the OU family), and the Jacobi diffusion (the hypergeometric family) We show that the Bessel family is a superset of the constant elasticity of variance model without drift The Bessel family, in turn, is nested by the confluent hypergeometric family For these two families we find further subfamilies of conservative strict supermartingales and nonconservative martingales with an exit boundary For the new classes of nonconservative regular diffusions we also derive analytically exact first exit time densities that are given in terms of generalized inverse Gaussians and extensions As for the two other new models, we show that the OU family of processes are conservative strict martingales, whereas the Jacobi family are nonconservative nonmartingales Considered as asset price diffusion models, we also show that these models demonstrate a wide range of local volatility shapes and option implied volatility surfaces that include various pronounced skew and smile patterns

On properties of analytically solvable families of local volatility diffusion models

Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science

We consider the exact path sampling of the squared Bessel process and some
other continuous-time Markov processes, such as the CIR model, constant
elasticity of variance diffusion model, and hypergeometric diffusions, which
can all be obtained from a squared Bessel process by using a change of
variable, time and scale transformation, and/or change of measure. All these
diffusions are broadly used in mathematical finance for modelling asset prices,
market indices, and interest rates. We show how the probability distributions
of a squared Bessel bridge and a squared Bessel process with or without
absorption at zero are reduced to randomized gamma distributions. Moreover, for
absorbing stochastic processes, we develop a new bridge sampling technique
based on conditioning on the first hitting time at zero. Such an approach
allows us to simplify simulation schemes. New methods are illustrated with
pricing path-dependent options.

Roman N. Makarov

Papers

Exact Simulation of Bessel Diffusions

Pricing path-dependent options on state dependent volatility models with a bessel bridge

On properties of analytically solvable families of local volatility diffusion models

Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science

Exact Simulation of Bessel Diffusions