arXiv subjects

Oleksii Patsiuk

Publications and source records attributed to Oleksii Patsiuk.

3 recordsLinked to original sources

arXiv2018

Symmetry reduction and exact solutions of the non-linear Black--Scholes equation

In this paper, we investigate the non-linear Black--Scholes equation: $$u_t+ax^2u_{xx}+bx^3u_{xx}^2+c(xu_x-u)=0,\quad a,b>0,\ c\geq0.$$ and show that the one can be reduced to the equation $$u_t+(u_{xx}+u_x)^2=0$$ by an appropriate point transformation of variables. For the resulting equation, we study the group-theoretic properties, namely, we find the maximal algebra of invariance of its in Lie sense, carry out the symmetry reduction and seek for a number of exact group-invariant solutions of the equation. Using the results obtained, we get a number of exact solutions of the Black--Scholes equation under study and apply the ones to resolving several boundary value problems with appropriate from the economic point of view terminal and boundary conditions.

q-fin.MF↗

arXiv2014

Lie group classification and invariant exact solutions of the generalized Kompaneets equations

In this paper, from the group-theoretic point of view it is investigated such class of the generalized Kompaneets equations (GKEs): $$u_t=\frac1{x^2}\cdot\left[x^4(u_x+f(u))\right]_x, \ (t,x) \in \mathbb{R}_{+} \times \mathbb{R}_{+},$$ where $u=u(t,x)$, $u_t=\frac{\partial u}{\partial t}$, $u_x=\frac{\partial u}{\partial x}$, $u_{xx}=\frac{\partial^2 u}{\partial x^2}$; $f(u)$ is an arbitrary smooth function of the variable $u$. Using the Lie--Ovsiannikov algorithm, the group classification of the class under study is carried out. It is shown that the kernel algebra of the full groups of the GKEs is the one-dimensional Lie algebra $\mathfrak{g}^\cap=\langle \partial_t \rangle$. Using the direct method, the equivalence group $G^\sim$ of the class is found. It is obtained six non-equivalent (up to the equivalence transformations from the group $G^\sim$) GKEs that allow wider invariance algebras than $\mathfrak{g}^\cap$. It is shown that, among the non-linear equations from the class, the GKE with the function $f(u)=u^{\frac43}$ has the maximal symmetry properties, namely, it admits a three-dimensional maximal Lie invariance algebra. Using the obtained operators, it is found all possible non-equivalent group-invariant exact solutions of the GKE under consider.

math.AP↗

arXiv2011

On self-adjoint operators in Krein spaces constructed by Clifford algebra Cl_2

Let $J$ and $R$ be anti-commuting fundamental symmetries in a Hilbert space $\mathfrak{H}$. The operators $J$ and $R$ can be interpreted as basis (generating) elements of the complex Clifford algebra ${\mathcal C}l_2(J,R):={span}\{I, J, R, iJR\}$. An arbitrary non-trivial fundamental symmetry from ${\mathcal C}l_2(J,R)$ is determined by the formula $J_{\vecα}=α_{1}J+α_{2}R+α_{3}iJR$, where ${\vecα}\in\mathbb{S}^2$. Let $S$ be a symmetric operator that commutes with ${\mathcal C}l_2(J,R)$. The purpose of this paper is to study the sets $Σ_{J_{\vecα}}$ ($\forall{\vecα}\in\mathbb{S}^2$) of self-adjoint extensions of $S$ in Krein spaces generated by fundamental symmetries ${J_{\vecα}}$ (${J_{\vecα}}$-self-adjoint extensions). We show that the sets $Σ_{J_{\vecα}}$ and $Σ_{J_{\vecβ}}$ are unitarily equivalent for different ${\vecα}, {\vecβ}\in\mathbb{S}^2$ and describe in detail the structure of operators $A\inΣ_{J_{\vecα}}$ with empty resolvent set.

math.FA↗