Mathematics

In this work, we address the convergence of a finite element approximation of the minimizer of the Freidlin-Wentzell (F-W) action functional for non-gradient dynamical systems perturbed by small noise. The F-W theory of large deviations is a rigorous mathematical tool to study small-noise-induced transitions in a dynamical system. The central task in the application of F-W theory of large deviations is to seek the minimizer and minimum of the F-W action functional. We discretize the F-W action functional using linear finite elements, and establish the convergence of the approximation through Γ-convergence.

A general stochastic integration theory for adapted and instantly independent stochastic processes arises when we consider anticipative stochastic differential equations. In Part I of this thesis, we conduct a deeper research on the general stochastic integral introduced by W. Ayed and H.-H. Kuo in 2008. We provide a rigorous mathematical framework for the integral in Chapter 2, and prove that the integral is well-defined. Then a general Itô formula is given. In Chapter 3, we present an intrinsic property, near-martingale property, of the general stochastic integral, and Doob–Meyer’s decomposition for near-submartigales. We apply the new stochastic integration theory to several kinds of anticipative stochastic differential equations and discuss their properties in Chapter 4. In Chapter 5, we apply our results to a general Black–Scholes model with anticipative initial conditions to obtain the properties of the market with inside information and the pricing strategy, which can help us...

In this paper, we study the solutions of a stochastic differential equation with various anticipating initial conditions. We show that the conditional expectation of the solution of such a stochastic differential equation is not simply the solution of the corresponding stochastic differential equation with initial condition taken as the conditional expectation of the anticipating initial condition. We derive the conditional expectation of the solution in general, and apply it to the special case of anticipating initial condition given by Hermite polynomials. We also extend the class of initial conditions to functions of Wiener integrals.

A stochastic process Xt is called a near-martingale with respect to a filtration {Ft} if E[Xt|Fs] = E[Xs|Fs] for all s ≤ t. It is called a nearsubmartingale with respect to {Ft} if E[Xt|Fs] ≥ E[Xs|Fs] for all s ≤ t. Near-martingale property is the analogue of martingale property when the Itô integral is extended to non-adapted integrands. We prove that Xt is a near-martingale (near-submartingale) if and only if E[Xt|Ft] is a martingale (near-submartingale, respectively). Doob-Meyer decomposition theorem is extended to near-submartingale. We study stochastic differential equations with anticipating initial conditions and obtain a relationship between such equations and the associated stochastic differential equations of the Itô type.

We review a new stochastic integral for adapted and instantly independent stochastic processes and show that it is well-defined. Then we prove a unified Itô formula for the new stochastic integral. This general formula is used to produce several interesting special cases of the Itô formula. Then we apply these formulas to study exponential processes and stochastic differential equations involving the new stochastic integral.

In this work, we construct a minimum action method for dynamical systems with constant time delays. The minimum action method (MAM) plays an important role in seeking the most probable transition pathway induced by small noise. There exist two formulations of the minimum action method: one is the geometric formulation based on the Maupertuis principle, and the other one is the temporal formulation. The geometric formulation relies on the conservation of Hamiltonian corresponding to the Freidlin–Wentzell action functional. For systems with time delays, the Hamiltonian does not conserve due to the explicit dependence on the time delay, which implies that the geometric MAM is not applicable. We work with the temporal formulation of MAM for problems with time delays. By defining an auxiliary path, we remove the optimization with respect to time through the optimal linear time scaling. The pointwise correspondence between the auxiliary path and the delayed transition path is dealt with b...

The time evolution of the probability distribution of a stochastic differential equation follows the Fokker-Planck equation, which usually has an unbounded, high-dimensional domain. Inspired by our early study in [9], we propose a mesh-free Fokker-Planck solver, in which the solution to the Fokker-Planck equation is now represented by a neural network. The presence of the differential operator in the loss function improves the accuracy of the neural network representation and reduces the the demand of data in the training process. Several high dimensional numerical examples are demonstrated.

A probabilistic approach for estimating sample qualities for stochastic differential equations is introduced in this paper. The aim is to provide a quantitative upper bound of the distance between the invariant probability measure of a stochastic differential equation and that of its numerical approximation. In order to extend estimates of finite time truncation error to infinite time, it is crucial to know the rate of contraction of the transition kernel of the SDE. We find that suitable numerical coupling methods can effectively estimate such rate of contraction, which gives the distance between two invariant probability measures. Our algorithms are tested with several low and high dimensional numerical examples.

Computing the invariant probability measure of a randomly perturbed dynamical system usually means solving the stationary Fokker-Planck equation. This paper studies several key properties of a novel data-driven solver for low-dimensional Fokker-Planck equations proposed in [15]. Based on these results, we propose a new `block solver' for the stationary Fokker-Planck equation, which significantly improves the performance of the original algorithm. Some possible ways of reducing numerical artifacts caused by the block solver are discussed and tested with examples.

In this work, we address the convergence of the finite element approximation of the minimizer of the Freidlin-Wentzell (F-W) action functional for a non-gradient dynamical system perturbed by small noise. The F-W theory of large deviations is a rigorous mathematical tool to study small-noise-induced transitions in a dynamical system. The central task in the application of F-W theory of large deviations is to seek the minimizer and minimum of the F-W action functional. We discretize the F-W action functional using linear finite element, and establish the convergence of the approximate minimizer through $\Gamma$-convergence.

Computing the invariant probability measure of a randomly perturbed dynamical system usually means solving the stationary Fokker-Planck equation. This paper studies several key properties of a novel data-driven solver for low-dimensional Fokker-Planck equations proposed in . Based on these results, we propose a new "block solver" for the stationary Fokker-Planck equation, which significantly improves the performance of the original algorithm. Some possible ways of reducing numerical artifacts caused by the block solver are discussed and tested with examples.

A $n\times n$ matrix $A$, which has a certain sign-symmetric structure ($J$--sign-symmetric), is studied in this paper. It is shown that such a matrix is similar to a nonnegative matrix. The existence of the second in modulus positive eigenvalue $\lambda_2$ of a $J$--sign-symmetric matrix $A$, or an odd number $k$ of simple eigenvalues, which coincide with the $k$-th roots of $\rho(A)^k$, is proved under the additional condition that its second compound matrix is also $J$--sign-symmetric. The conditions when a $J$--sign-symmetric matrix with a $J$--sign-symmetric second compound matrix has complex eigenvalues, which are equal in modulus to $\rho(A)$, are given. Comment: 14 pages

The existence of the second (according to the module) eigenvalueλ2of a completely continuous nonnegative operatorAis proved under the conditions thatAacts in the spaceLp(Ω)orC(Ω)and its exterior squareA∧Ais also nonnegative. For the case when the operatorsAandA∧Aare indecomposable, the simplicity of the first and second eigenvalues is proved, and the interrelation between the indices of imprimitivity ofAandA∧Ais examined. For the case whenAandA∧Aare primitive, the difference (according to the module) ofλ1andλ2from each other and from other eigenvalues is proved.

This paper is devoted to the generalization of the theory of total positivity. We say that a linear operator A : R n → R n is generalized totally positive (GTP), if its jth exterior power ∧ j A preserves a proper cone K j ⊂ ∧ j R n for every j = 1,. .. , n. We also define generalized strictly totally positive (GSTP) operators. We prove that the spectrum of a GSTP operator is positive and simple, moreover, its eigenvectors are localized in special sets. The existence of invariant cones of finite ranks is shown under some additional conditions. Some new insights and alternative proofs of the well-known results of Gantmacher and Krein describing the properties of TP and STP matrices are presented.

We establish the eigenvalue interlacing property (i.e. the smallest real eigenvalue of a matrix is less than the smallest real eigenvalue of any its principal submatrix) for the class of matrices, introduced by Kotelyansky (all principal and all almost principal minors of these matrices are positive). We show that certain generalizations of Kotelyansky and totally positive matrices also possess this property. We prove some interlacing inequalities for the other eigenvalues of Kotelyansky matrices.

A generalization of the definition of an oscillatory matrix based on the theory of cones is given in this paper. The positivity and simplicity of all the eigenvalues of a generalized oscillatory matrix are proved. The classes of generalized even and odd oscillatory matrices are introduced. Spectral properties of the obtained matrices are studied. Criteria of generalized even and odd oscillation are given. Examples of generalized even and odd oscillatory matrices are presented.

A new class of sign-symmetric matrices is introduced in this paper. Such matrices are called J-sign-symmetric. The spectrum of a J-sign-symmetric irreducible matrix is studied under the assumption that its second compound matrix is also J-sign-symmetric. The conditions for such matrices to have complex eigenvalues on the spectral circle are given. The existence of two positive simple eigenvalues λ 1 > λ 2 > 0 of a J-sign-symmetric irreducible matrix A is proved under some additional conditions. The question when the approximation of a J-sign-symmetric matrix with a J-sign-symmetric second compound matrix by strictly Jsign-symmetric matrices with strictly J-sign-symmetric second compound matrices is possible is also answered in this paper.

In this paper, we study the continued fraction y(s, r) which satisfies the equation y(s, r)y(s + 2r, r) = (s + 1)(s + 2r − 1) for r > 1 2. This continued fraction is a generalization of the Brouncker's continued fraction b(s). We extend the formulas for the first and the second logarithmic derivatives of b(s) to the case of y(s, r). The asymptotic series for y(s, r) at ∞ are also studied. The generalizations of some Ramanujan's formulas are presented.

The tensor and exterior squares of a completely continuous non-negative linear operator A acting in the ideal space X(Ω) are studied. The theorem representing the point spectrum (except, probably, zero) of the tensor square A ⊗ A in the terms of the spectrum of the initial operator A is proved. The existence of the second (according to the module) positive eigenvalue λ 2 , or a pair of complex adjoint eigenvalues of a completely continuous non-negative operator A is proved under the additional condition, that its exterior square A ∧ A is also nonnegative.