Mathematics

CHARACTERISING SMOOTHNESS OF TYPE A SCHUBERT VARIETY USING PALINDROMIC POINCARÉ POLYNOMIAL AND PLÜCKER COORDINATE METHODS.

2023-09-10T00:00:00Z

CHARACTERISING SMOOTHNESS OF TYPE A SCHUBERT VARIETY USING PALINDROMIC POINCARÉ POLYNOMIAL AND PLÜCKER COORDINATE METHODS. AFINOTAN, Patience Schubert varieties are subvarieties of the flag variety F‘n(C), a smooth complex projective variety consisting of sequences of sublinear subspaces of an n-dimensional complex vector space, ordered by inclusion. They are indexed by permutation matrices and studied in various types with important roles in algebraic geometry due to their combinatorial structures. The smoothness and singularity of Schubert variety have been characterised by various methods using the elements of the n-dimensional symmetric group. However, characterising smoothness using the exponents of the monomial of the Schubert variety and Plücker coordinate which uniquely and clearly identifies the symmetry of the Poincaré polynomial have not been established. Hence this research aims at establishing smoothness and singularity of type A Schubert varieties using the exponents of the monomials of the Schubert variety and the Jacobian criterion on the equations of the ideals of the Schubert variety obtained via the Plücker embedding. For the Schubert varieties Xσ, the cohomology of the flag varieties f : Hn−k(F‘n(C); Z) ! Hk(F‘n(C); Z) defined by f[Xσ] = [Xσ] 2 Hk(F‘n(C)) was considered, to obtain its monomials. The Poincaré polynomial was determined in order to compute the symmetry of the Schubert varieties. The flag varieties are embedded into the product of Grassmanians which is also embedded into the product of projective spaces given by the embedding map F‘n(C) = Xσ ,! Qn−1 k=1 Gr(k; n) ,! Qn−1 k=1 P 0B@ n k 1CA −1 : defined by A 7! [P12; P13; · · · ; P(n−1)n] , with Pij; 1 ≤ i < j ≤ n being the n k ! minors for Ak;n in Gr(k; n). The equations of the ideal of the Schubert varieties were obtained by taking all the minors of the matrix Schubert varieties. The rank of the Jacobian matrix and the co-dimension of the Schubert varieties were determined. The Schubert classes forms additive Z basis that generates the cohomology ring Hk(F‘4(C); Z). The basis for the cohomology ring are the geometric and algebraic basis. The algebraic basic classes xi 11xi 22 · · · ; xi mm with exponents ij = m − j forms Z basis for the cohomology ring and these basic classes are the monomials. The vPoincaré polynomial Pσ(t) = Pv≤σ tl(v) , defined with respect to the length function and via the Bruhat order, v ≤ σ =) l(v) ≤ l(σ) shows that the symmetry Pσ (t) = trPσ(t−1) Of the Poincaré polynomial is palindromic or not palindromic. The rank of the Jacobian matrix obtained using the equations of the ideal I(Xσ) derived through the embedding map is found to be equal to the co-dimension of the varieties which indicates smoothness. The exponent of the monomials xi 11xi 22 · · · xi mm of the Schubert variety Xσ have uniquely satisfied the symmetry of its Poincaré polynomial for smooth Schubert varieties and have successfully extended the underlying group from Sn to Z+n. Smoothness has successfully been generalised in terms of the differential equations using the equations defining the ideals, of the Schubert varieties through the Plücker coordinates.

FAST FOURIER TRANSFORM AND FORMULATION OF ECONOMIC RECESSION INDUCED STOCHASTIC VOLATILITY MODELS FOR AMERICAN OPTIONS COMPUTATION

2022-12-08T00:00:00Z

FAST FOURIER TRANSFORM AND FORMULATION OF ECONOMIC RECESSION INDUCED STOCHASTIC VOLATILITY MODELS FOR AMERICAN OPTIONS COMPUTATION BANKOLE, Philip Ajibola Economic recession has become a global and reccurring phenomenon which poses worrisome uncertainties on assets’ returns in financial markets. Various stochastic models have been formulated in response to price instability in financial markets. However, the existing stochastic volatility models did not incorporate the concept of economic recession and induced volatility-uncertainty for options price valuation in a recessed economic setting. Therefore, this study was geared towards the formulation of economic recession-induced stochastic models for price computation. Stochastic modelling methods with probabilistic uncertainty measure were used to formulate two new volatility models incorporating economy recession volatility uncertainties. The Feynman-Kac formula was applied to derive the characteristic functions for the two novel models. The derived characteristic functions were used to obtain an inverse-Fourier analytic formula for European and American-style options. A modified Carr and Madan Fast-Fourier Transform (FFT)-algorithm was used to obtain an approximate solution for the American-call option, and a class of Multi-Assets option in multi-dimensions. Itˆo Calculus was used to obtain an explicit formula for a Factorial function Black-Scholes Partial Differential Equations (BS-PDE) for American options subject to moving boundary conditions. The FFT call-prices accuracy test at varied fineness grid points N was investigated using an FFT-algorithm via Maple, taking BS-prices as benchmark. Sample paths and numerical simulations were generated via software codes for the control regimeswitching Triple Stochastic Volatility Heston-like (TSVH) model. The derived Uncertain Affine Exponential-Jump Model (UAEM) with recession, induced stochastic-volatility and stochastic-intensity, and a control regime-switching Triple Stochastic Volatility Heston-like (TSVH) formulated with respect to economy recession volatility uncertainties are:  d ln S(t) = r − q − λ(t)m dt + pσ(t)dWs(t) + (eν − 1)dN(t), S(0) = S0 > 0 dσ(t) = κσ β∗ + βrec − σ(t) dt + ξσpσ(t)dWσ(t), σ(0) = σ0 > 0 dλ(t) = κλθ − λ(t) dt + ξλpλ(t)dWλ(t), λ(0) = λ0 > 0. and  dyt = r − q dt + pv1(t)dW1(t) + pv2(t)dW2(t) + αpv3(t)dW3(t) , S(0) = S0 > 0 dv1(t) = κ1θ1 − v1(t) dt + σ1pv1(t)dWc1(t), v1(0) = v10 > 0. dv2(t) = κ2θ2 − v2(t) dt + σ2pv2(t)dWc2(t), v2(0) = v20 > 0. dv3(t) = α κ3θ3 − v3(t) dt + σ3pv3(t)dWc3(t) recession, v3(0) = v30 > 0 respectively, where α was a binary control parameter defined as: α := 0, if the economy is not in recession; 1, if the economy is in recession. The inverse-Fourier analytic formulae for European-style and American-style calloptions obtained for the UAEM-process are: Ecall T (k) = exp(−αk) π Z ∞ 0 e−(rT +iuk)φτ u − (α + 1)i × α2 + α − u2 − i(2α + 1)u α4 + 2α3 + α2 + 2(α2 + α) + 1 u2 + u4 du ivand At(k) = exp(−αk) π Z ∞ 0 e−(rT +iuk) × φτu − (α + 1)i α2 + α − u2 − i(2α + 1)u α4 + 2α3 + α2 + 2(α2 + α) + 1 u2 + u4 du + Pt, respectively where Pt is premium price. The approximate solution obtained for American-call option via FFT-algorithm for the UAEM-process was: A τ (ku) ≈ exp(−αk) π NX j =1 e−iuj ζη(j−1)(u−1) eiϖujψT (uj)η + Pt, where 1 ≤ u ≤ N and ζη = 2π N . The derived multi-Assets options prices formula in n-dimension was: VT (k1,p1, k2,p2 · · · , kn,pn) ≈ e−(α1k1,p1 +α2k2,p2 +...+αnkn,pn ) (2π)n Ω(k1,p1, k2,p2, · · · , kn,pn) nY j =1 hj, where 0 ≤ p1, p2, · · · , pn ≤ N − 1 and Ω(k1,p1, k2,p2, · · · , kn,pn) = N1−1 X m1 =1 N1−1 X m2 =1 · · · N1−1 X mn=1 e − 2π N (m1− N2 )(p1− N2 )+(m2− N2 )(p2− N2 )+···+(mn− N2 )(pn− N2 ) × ψT (u1, u2, · · · , un). The derived explicit formula for the Factorial function BS-PDE was: S(T) = S(t0) exp hn! r + 1 2(n − 1)σ2 T − t0 + n!σ W(t) − W(t0) i, S(t0) ̸= 0), and the TSVH call pricing formula derived was: C(K) = Ste−qτP1 − Ke−rτP2 such that P1 = 1 2 + 1 π Z ∞ 0 ℜ exp(−iω ln K) fω − i; yt, v1(t), v2(t), v3(t) iωSte(r−q)τ dω P2 = 1 2 + 1 π Z ∞ 0 ℜ exp(−iω ln K) fω; yt, v1(t), v2(t), v3(t) iω dω, and fω − i; yt, v1(t), v2(t), v3(t) = exp A(τ, ω) + B0(τ, ω)yt + B1(τ, ω)v1(t) + B2(τ, ω)v2(t) + B3(τ, ω)v3(t) where A, B0, B1, B2, B3 are coefficient terms of the stochastic processes yt, v1(t), v2(t), v3(t). The options prices obtained from An Uncertain Affine Exponential-Jump Model with Recession, induced stochastic-volatility and stochastic-intensity and a control regimeswitching Triple Stochastic Volatility Heston-like model, were efficient in terms of probable future payoffs and applicable in financial markets, for options valuation in recessed and recession-free economy states.

SUITABLE MEASURES OF JUMPS IN STOCHASTIC MODELS FOR STOCK MARKET INDICES

2022-02-01T00:00:00Z

SUITABLE MEASURES OF JUMPS IN STOCHASTIC MODELS FOR STOCK MARKET INDICES ADEOSUN, Mabel Eruore The dynamics of the stock market indices log-returns (∆(lnS~t)) have been characterised by non-normal features such as upward and downward jumps of different measures, asymmetric and leptokurtic features. The Bi-Power Variation (BPV) process has been used to develop jump-estimators to detect jumps in (∆(lnS~t). However, the existing jump-estimators are restricted to the BPV case of the Realised Multi-Power Variation (RMPV) processes, and existing models do not accommodate these features. Therefore, this study was designed to construct unrestricted Particular Higher-Order Cases (PHOC) jump-estimators, and build suitable models that can accommodate the jumps and non-normal features found in (∆(lnS~t). The limits in probability and distribution were used to derive the Jump Test Models (JTM) in the PHOC of the RMPV processes. The JTM were used to test for jumps under the null hypothesis (H0) of no jump in (∆(lnS~t) at a 5% level of significance in three stock markets, namely: Nigerian, UK, and Japan. These were used to build the dynamics of (∆(lnS~t). The convolution of densities and the L´ evy-It^ o decomposition methods were used to derive the Probability Density Functions (PDF) and the L´ evy-Khintchine (LK) formulae of two novel skewed models. The maximum likelihood estimation method was used to estimate the optimal values of the parameters in the models. The Kolmogorov-Smirnov, Anderson-Darling statistics, and the basic moments were used to test the suitability of these models to the empirical stock market data and compared with three existing models viz: Black-Scholes (BS), normal and double-exponential jump-diffusion models. The JTM derived for the PHOC of the RMPV processes were: ^ Z m = ∆−0:5 µ −m 2=mfXg[r1;:::;rm] ∆;t fXg[2] ∆;t − 1! ’RMP V rmax 1; p^q^2 −1, for m = 2 · · · 10, where, fXg[r1;:::;rm] ∆;t ; fXg[2] ∆;t; ’RMP V ; p^and ^ q are the RMPV, realised variance, asympvtotic variance, estimators of bi-power and quad-power variation, respectively. Jumps in ∆(lnS~t) were observed and H0 was rejected. The dynamics of ∆(lnS~t) was derived as: ∆(lnS~t) = (µ− 1 2σ2)∆+σ∆Wt +J(Qu j )∆Ntu +J(Qd j)∆Ntd, where,µ; σ; Wt, J(Qu j ); J(Qd j); Ntu and Ntd are respectively drift and volatility parameters, standard Brownian motion, upward and downward jump measures with intensities λu j and λd j, respectively. The PDF of the Asymmetric-Laplace (AL) and the Modified DoubleRayleigh (MDR) were models derived were: f∆(ln S~t)(x) = (1−λ∆t) σp∆t ’ x−(µ− 1 2 σ2)∆t σp∆t + ∆t pκα2λu j exp 2α1µj+α1σ2∆t 2 exp − x − µ − 12σ2 ∆t α1 Φ aµj + qkα2λd jexp 2α2µj+α2σ2∆t 2 exp − x − µ − 12σ2 ∆t α2 Φb − µj and f∆(ln S~t)x = (1−λ∆t) σp∆t ’ x−(µ− 1 2 σ2)∆t σp∆t + pηexp θ2−ρ # 2θexp −(µj − θ)2 # +θpπ#Φa(µj)−µjpπ#Φa(µj) −qηexp ^ θ^2−ρ^ #^ #2^exp −(µj − θ^)2 #^ !+ θpπ#^Φb(µj) − µjpπ#^Φb(−µj) ∆t, where, θ = σju(µd∆t)+µjσ2∆t (σju+σ2∆t) , ρ = µ2 j σ2∆t (σju+σ2∆t), # = 2σ2∆tσju (σju+σ2∆t), θ^ = σjd(µd∆t)+µjσ2∆t (σjd+σ2∆t) , ^ ρ = µ2 j σ2∆t (σjd+σ2∆t), #^ = 2σ2∆tσjd (σjd+σ2∆t), η = λu j (σju)’ x−(µ− 1 2 σ2)∆t σp∆t and ^ η = λd j (σjd)’ x−(µ− 1 2 σ2)∆t σp∆t . The derived LK formulae of the novel models were: (u) = iuµ − 1 2σ2u2 − λu j pkα1 α1−iu + λd j qkα2 α2+iu eiuµj + λd jqk + λu j pk and (u) = iuµ − 1 2σ2µ2 − pλu j σu j + qλd j σd j + pλu j σu j + qλd j σd j eiuµj, respectively.The optimal values of the parameters: (µd; σ; α1; α2; pk; qk; λu j ; λd j; µj) and (µd; σ; σju; σjd; p; qλu j ; λd j; µj) in the models were obtained. The AL and MDR were models fit the empirical distributions better than the existing models, having the BS model in the worst-case scenario. The jump test models of the particular higher-order cases were found to be better jump-estimators. The asymmetric-Laplace and modified double-Rayleigh jumpdiffusion models proved more suitable for capturing jumps and non-normal features in the stock market indices log-returns.

GRONWALL-BELLMAN-BIHARI TYPE INEQUALITY AND HYERS-ULAM AND HYERS-ULAM-RASSIAS STABILITIES OF CERTAIN CLASSES OF NONLINEAR SECOND AND THIRD ORDER DIFFERENTIAL EQUATIONS

2022-12-01T00:00:00Z

GRONWALL-BELLMAN-BIHARI TYPE INEQUALITY AND HYERS-ULAM AND HYERS-ULAM-RASSIAS STABILITIES OF CERTAIN CLASSES OF NONLINEAR SECOND AND THIRD ORDER DIFFERENTIAL EQUATIONS FAKUNLE, Ilesanmi The concept of stability in differential equations is of immense importance particularly for determination of properties of solutions of nonlinear equations which cannot be readily solved to obtain closed analytic solutions. Stability in the sense of Hyers-Ulam(H-U) and Hyers-Ulam-Rassias(H-U-R) has been considered for linear Ordinary Differential Equations(ODE) due to their solutions which are easily determined. However, cases of nonlinear ODE of second and third orders have received little attention. This research was therefore designed to establish the stability of nonlinear second and third order in the sense of H-U and H-U-R. Variants of the perturbed second order ODE of the form u00(t) + f(t; u(t); u0(t)) = P(t; u(t); u0(t)) and third order ODE of the form u000(t) + f(t; u(t); u0(t); u00(t)) = P(t; u(t); u0(t)) were reduced to their equivalent integral equations, where t is an independent real variable; f; P; and u are continuous functions of their argument. Extension of Gronwall-Bellman-Bihari(G-B-B) type inequalities having the same number of integrals as the equivalent integral equations were developed. These integral inequalities were used to prove the existence of H-U and H-U-R stability. They were also used to estimate the H-U and H-U-R constants for each of the variants of the equations. The newly developed G-B-B type inequalities of nonlinear integrals obtained were: u(t) ≤ u0 + L Rtt0 f(s)u(s)ds + Rtt0 f(s)r(s)(Rts0 ρ(τ)$(u(τ))dτ)ds and u(t) ≤ u0 + T Rtt0 r(s)β(s)ds + L Rtt0 h(s)$u(s))ds where f; r; ρ; $; β and h are continuous functions and T; L and u0 are positive constants. The nonlinear second order ODE were found to possess H-U stability and H-U constants K21 = L(1 + 1 2λ2 + jq(ξ; u(ξ); u0(ξ); u00(ξ))j + jp(u(ρ); u0(ρ))jΩ−1(Ω(1) + neM) vand K22 = (αL(ξ) + λ2 2α(ξ))Ω−1(Ω(1) + λ2 α(ξ)n$(F−1(F(1) + λ2 α(ξ)m))F−1(F(1) + m λ2 α(ξ)); respectively, where q; p; Ω; Ω−1; F; F−1; $; h are functions of their argument and λ; ξ; ρ; M; n and m are constants. The newly developed G-B-B type inequality for two nonlinear integrals u(t) ≤ ρ(t) + T Rtt0 r(s)β(s)ds + L Rtt0 h(s)$(u(s))ds: where ρ(t) a monotonic, nonnegative, continuous function, with this nonlinear second order ODE were found to possess H-U-R stability and the H-U-R constants C’ 21 = Ω(Ω(1) + m(η + η2)$(F−1(F(1) + l))F−1(F(1) + l) and C’ 22 = Γ−1(Γ(1) + mηnγ(F−1(F(1) + l)))F−1(F(1) + l) where Γ; Γ−1 and γ are functions of their argument and η; l are constants. The new G-B-B type inequality for three nonlinear integrals was: u(t) ≤ D + T Rtt0 r(s)β(u(s))ds + B Rtt0 h(s)$(u(s))ds + L Rtt0 g(s)γ(u(s))ds; where B and D are positive constants. The nonlinear third order ODE were found to possess H-U stability with H-U constants given by K31 = L+δL F−1(F(1) + d(λ)λδ φ(SX))SX; where S = Ω−1(Ω(1) + nλδ γ(X))X and X = F−1(F(1) + mλδ2) and K32 = L+jr(u(κ))jL 2φju0(ξ)j Γ−1(Γ(1) + C4qr(Ω−1(Ω(1) + C3mg(H))Ω−1(Ω(1) + C2n))) Ω−1(Ω(1) + C3mg)H; where H = F−1(F(1) + C2n); C2 = 2ju0(η)jλ 2φju0(ξ)j; C3 = λ2 2φju0(ξ)j and C4 = λn+1 2φju0(ξ)j; δ; constants and g; r; φ are functions of their argument. The newly developed G-B-B type inequality with the three nonlinear integrals : u(t) ≤ ρ(t)+T Rtt0 r(s)β(u(s))ds+B Rtt0 h(s)$(u(s))ds+L Rtt0 g(s)γ(u(s))ds; where ρ(t) a monotonic, nonnegative, nondecreasing and continuous function. The nonlinear third order ODE were also found to possess H-U-R stability with H-U-R constants C’ 31 = 1δ Υ−1(Υ(1) + ηδnρ1γ(Ω−1(Ω(1) + 1δρ3α(Y ))y)Ω−1(Ω(1) + 1δρ3α(Y ))Y; where Y = F−1(F(1)+ηδρ1) and C’32 = 1δΩ−1(Ω(1)+d1λn η !(F−1(F(1)+d2h(λ) η )))F−1(F(1)+ d2h(λ) η ) with ρ1; ρ2 and ρ3 are constants. A generalisation of the existing results on Hyers-Ulam and Hyers-Ulam-Rassias stability to nonlinear ordinary differential equations was achieved. This can also be used to achieve the stability of the other differential equations.