Vol 29, No 147 (2024)

The article proposes a universal approach to constructing Monte Carlo methods for pricing options with payoffs depending on the joint distribution of the final position of the Lévy process $X_T$ and its infimum ${\mathcal{I}}_T$ (or supremum ${\mathcal{S}}_T$). We derive approximate formulas for the conditional cumulative distribution functions of the Lévy process $P(X_T<x|{\mathcal{S}}_T=y)$ ($P(X_T<x|{\mathcal{I}}_T=y)$), which are expressed through the partial derivative with respect to $y$ of the joint cumulative distribution function $P(X_T\text{}<x,{\mathcal{S}}_T\text{}<y)$ ($P(X_T\text{}<x,{\mathcal{I}}_T\text{}<y)\text{}$) and the density of the infimum (or supremum) at the final moment of time. By applying the Laplace transform to the joint cumulative distribution function of the Lévy process and its extremum, we use the approximate Wiener–Hopf factorization to represent the image of its partial derivative. By inverting the Laplace transform using the Gaver–Stehfest algorithm, we find the desired conditional cumulative distribution function. The developed algorithm for simulating the joint position of the Lévy process and its extremum at a given point in time consists of two key stages. At the first stage, we simulate the extremum value of the Lévy process based on the approximation of its cumulative distribution function $P({\mathcal{S}}_T<x)$ (or $P({\mathcal{I}}_T<x)$). In the second step, we simulate the final value of the Lévy process based on the approximation of the conditional cumulative distribution function of the final position of the Lévy process relative to its extremum. The universality of the Monte Carlo method we developed lies in the implementation of a uniform approach for a wide class of Lévy processes, in contrast to classical approaches, when simulations are essentially based on the features of the probability distribution associated with the simulated random process or its extrema. In our approach, it is enough to know the characteristic exponent of the Lévy process. The most time-consuming computational unit for simulating a random variable based on a known cumulative distribution function can be effectively implemented using neural networks and accelerated through parallel computing. Thus, on the one hand, the approach we propose is suitable for a wide class of Lévy models, on the other hand, it can be combined with machine learning methods.

The article considers a partial differential equation of the form

$\frac{\partial u}{\partial t}ft\mathrm{,}x\mathrm{,}y\mathrm{,}u\mathrm{,}\frac{\partial u}{\partial x}\mathrm{,}\frac{\partial u}{\partial y}\mathrm{,}\frac{{\partial }^{2}u}{\partial x^2}\mathrm{,}\frac{{\partial }^{2}u}{\partial y^2}\mathrm{,}\frac{{\partial }^{2}u}{\partial x\partial y}x\mathrm{,}y\in D\subset {ℝ}^2\mathrm{,}t\ge \mathrm{0,}$

with respect to an unknown function $u\mathrm{,}$ defined in a domain $D$ of spatial variables $x\mathrm{,}y$ and for $t\ge 0.$ A method for finding an approximate solution is proposed. The equation under consideration is replaced by an approximate one by introducing the shift operator $S:D\to D,$ which allows replacing at each step of the calculations the unknown values of the function $u(x,y,t)$ on the right side with the values $u\left(S\right(x,y),t),$ obtained at the previous step. The idea of the proposed method goes back to the idea of the Tonelli method, known for differential equations with respect to functions of one variable (with ordinary, not partial derivatives). The advantages of the proposed method are the simplicity of the obtained iteration relation and the possibility of application to a wide class of equations and boundary conditions. In the article, iteration formulas are obtained for solving a boundary value problem with the Dirichlet condition for spatial variables and with an initial or boundary condition for the variable $t\mathrm{.}$ Based on the proposed method, an approximate solution is obtained for a specific initial-boundary value problem for the heat conductivity equation in a square domain.

The classical Suslov problem of the motion of a rigid body with a fixed point is well known and has been studied in detail. In this paper, an omniwheel implementation of the Suslov problem is proposed. The controlled motion of a rigid body with a fixed point in the presence of scleronomic nonholonomic constraints and rheonomic artificial kinematic constraint is considered. The rigid body rotates around a fixed point, rolls around a spherical shell from the inside and contacts it by means of omniwheels with a differential actuator. We believe that the omniwheels are in contact with the spherical shell only at one point. In order to subordinate the motion of the rigid body to an artificial rheonomic constraint, a differential actuator creates control torques on omniwheels. Based on the d’Alembert–Lagrange principle, equations of motion of the mechanical system with indeterminate multipliers specifying constraint reactions are constructed. The problem is reduced to the study of a non-autonomous two-dimensional dynamical system. Using the generalized Poincaré transformation, the study of a two-dimensional dynamical system is reduced to the study of the stability of a one-parameter family of fixed points for a system of differential equations with a degenerate linear part. We determine numerical parameters for which phase trajectories of the system are bounded and for which phase trajectories of the system are unbounded. The results of the study are illustrated graphically. Based on numerical integration, maps for the period (Poincaré sections) and a map of dynamic regimes are constructed to confirm the Feigenbaum scenario of transition to chaotic dynamics.

We consider a family of complex operator functions whose domain and range of values are included in the real Banach algebra of bounded linear complex operators acting in the Banach space of complex vectors over the field of real numbers. It is shown that the study of a function from this family can be reduced to the study of a pair of real operator functions of two real operator variables. The main elementary functions of this family are considered: power function; exponent; trigonometric functions of sine, cosine, tangent, cotangent, secant, cosecant; hyperbolic sine, cosine, tangent, cotangent, secant, cosecant; the main property of the exponent is proved. A complex Euler operator formula is obtained. Relations that express sine and cosine in terms of the exponent are found. For the trigonometric functions of sine and cosine, addition formulas are justified. The periodicity of the exponent and trigonometric functions of sine, cosine, tangent, cotangent is proved; reduction formulas for these functions are provided. The main complex operator trigonometric identity is obtained. Equalities connecting trigonometric and hyperbolic functions are found. The main complex operator hyperbolic identity is established. For the hyperbolic functions of sine and cosine, addition formulas are indicated. As an example of an elementary function from the family of complex operator functions under consideration, a rational function is considered, a special case of which is the characteristic operator polynomial of a linear homogeneous differential equation of n-th order with constant bounded operator coefficients in a real Banach space.