M. Gaudenzi - Academia.edu (original) (raw)

Papers by M. Gaudenzi

Research paper thumbnail of The singular points binominal method for pricing American path-dependent options

The Journal of Computational Finance, 2010

Research paper thumbnail of A Seven-Positive-Solutions Theorem for a Superlinear Problem

Advanced Nonlinear Studies, 2004

We consider the superlinear boundary value problem uʺ + aμ(t)uγ+1 = 0, u(0) = 0, u(1) = 0, where ... more We consider the superlinear boundary value problem uʺ + aμ(t)uγ+1 = 0, u(0) = 0, u(1) = 0, where γ > 0 and aμ(t) is a sign indefinite weight of the form a+(t)−μa−(t). We prove, for μ positive and large, the existence of 2k − 1 positive solutions where k is the number of positive humps of aμ(t) which are separated by k − 1 negative humps. For sake of simplicity, the proof is carried on for the case k = 3 yielding to 7 positive solutions. Our main argument combines a modified shooting method in the phase plane with some properties of the blow up solutions in the intervals where the weight function is negative.

Research paper thumbnail of A new binomial algorithm for pricing American pathdependentoptions

Research paper thumbnail of Ordinary Differential Equations and Delayequations

Research paper thumbnail of Fučı́k Spectrum for a Third Order Equation

Journal of Differential Equations, 1996

Research paper thumbnail of On the Sturm Theorem for n-th order equations withweight of non constant sign

Differential Equations and Dynamical Systems, 1996

Research paper thumbnail of On the solvability of nonselfadjoint symmetricboundary value problems

Research paper thumbnail of Trasformation Theorem for Periodic Solutions Ofnondissipative Systems

Research paper thumbnail of Binomial methods for pricing american options

Research paper thumbnail of Reduction of American call barrier options to theEuropean case

Research paper thumbnail of On the existence of infinitely many solutions of n-th order boubdary value problems

Research paper thumbnail of Counting paths on a chessboard with a barrier

Nonlinear Analysis: Hybrid Systems, 2010

ABSTRACT We are dealing with the problem of counting the paths joining two points of a chessboard... more ABSTRACT We are dealing with the problem of counting the paths joining two points of a chessboard in the presence of a barrier. The formula for counting all the paths joining two distinct positions on the chessboard lying always over a barrier is well known (see for example Feller (1968) [1], Kreher and Stinson (1999) [3]). The problem is here extended to the calculation of all the possible paths of n movements which stay exactly k times, 0≤k≤n+1, over the barrier. Such a problem, motivated by the study of financial options of Parisian type, is completely solved by virtue of five different formulas depending on the initial and final positions and on the level of the barrier.

Research paper thumbnail of Comparison and Integral Comparison Theorems for Linear Differential Equations

Journal of Differential Equations, 1994

Research paper thumbnail of On an eigenvalue problem of Ahmad and Lazer for ordinary differential equations

Proceedings of the American Mathematical Society, 1987

Lazer, we show the existence of a class of nonselfadjoint eigenvalue problems related to the equa... more Lazer, we show the existence of a class of nonselfadjoint eigenvalue problems related to the equation y(n) + Xp(x)y = 0 for which the general eigenvalues comparison is not true. We use a comparison principle for the zeros of the corresponding Cauchy problem. This paper provides a contribution to the understanding of a problem raised by S. Ahmad and A. C. Lazer [1] in connection with the comparison of the eigenvalues for some multi-point boundary value problems which are not selfadjoint. One is given the equation (1) Lny + Xp(x)y = 0, where p(x) is a continuous function of constant sign on an interval /, A is a parameter, and Lny is a linear differential disconjugate operator of order n, that is, the only solution of Lny = 0 with n zeros on I (counting multiplicity) is y = 0. Let us consider the eigenvalue problem given by equation (1) and the system of boundary conditions ,, Lzy(a)=0, iG{ii,...,ik}, L]V(b)=0, JGiJu.-.Jn-k}, where o, b G I, 1 < k < n-1, Liy, i = 0,..., n-1, are the quasi-derivatives of y(x) (see [7]), and {t'i,..., ¿fc}, {ji, ■ ■ ■ ,jn-k) are two arbitrary sets of indices from the set {0,... ,n-1}. Problems of this type have been studied extensively (cf. [2, 3, 5]). In particular, Elias [5] has shown that if (-l)n_fcp(x) < 0, then the eigenvalues of problems (1) and (2) are real and nonnegative and form a divergence sequence {Am}m£N-Ahmad and Lazer [1] have considered a particular type of boundary condition (2), that is (3) y(a)=y'(a) =-= yik-1\a) = 0, y(b)=y'(b) =-=y(n-k-i\b)=0, and showed that if we set p = Pi, where p¿, i-1,2, are two continuous functions, considering the corresponding sequence of eigenvalues (A¿,m)m6N, i = 1,2, ordered by magnitude, then the condition (4)_ (-l)n-kp2(x) < (-l)"-fepi(x) < 0

Research paper thumbnail of On the existence of infinitely many solutions for superlinear n-th order boundary value problems

Research paper thumbnail of Positive solutions of singular boundary value problems with indefinite weight

Bulletin of The Belgian Mathematical Society-simon Stevin, 2002

Using a generalized version of the method of lower and upper solutions, we prove existence of pos... more Using a generalized version of the method of lower and upper solutions, we prove existence of positive solutions for a class of boundary value problems for a nonlinear equation with singularities whose coefficients change sign.

Research paper thumbnail of Nonselfadjoint 4th order boundary value problems

Nonlinear Analysis: Theory, Methods & Applications, 1998

ABSTRACT Consider the following linear boundary value problem u (n) =(-1) n-k p(t)u,(1) u (i) (a)... more ABSTRACT Consider the following linear boundary value problem u (n) =(-1) n-k p(t)u,(1) u (i) (a)=0,i∈{i 1 ,⋯,i k },(2) u (j) (b)=0,j∈{j 1 ,⋯,j n-k },(3) 0≤i 1 &lt;⋯&lt;i k ≤n-1,0≤j 1 &lt;⋯&lt;j n-k ≤n-1and1≤k≤n-1· The set E⊂L ∞ is said to have the property P if for any p∈E the problem (1), (2), (3) has only the trivial solution. The authors study a connection between property P and the corresponding eigenvalue problem. The main results are basic to obtain the existence of solutions to the nonlinear problem u (n) =(-1) n-k f(t,u), (2), (3).

Research paper thumbnail of Positive solutions of superlinear boundary value problems with singular indefinite weight

Communications on Pure and Applied Analysis, 2003

Using a generalized version of the method of lower and upper solutions, we prove existence of pos... more Using a generalized version of the method of lower and upper solutions, we prove existence of positive solutions for a class of boundary value problems for a nonlinear equation with singularities whose coefficients change sign.

Research paper thumbnail of The singular points binominal method for pricing American path-dependent options

The Journal of Computational Finance, 2010

Research paper thumbnail of A Seven-Positive-Solutions Theorem for a Superlinear Problem

Advanced Nonlinear Studies, 2004

We consider the superlinear boundary value problem uʺ + aμ(t)uγ+1 = 0, u(0) = 0, u(1) = 0, where ... more We consider the superlinear boundary value problem uʺ + aμ(t)uγ+1 = 0, u(0) = 0, u(1) = 0, where γ > 0 and aμ(t) is a sign indefinite weight of the form a+(t)−μa−(t). We prove, for μ positive and large, the existence of 2k − 1 positive solutions where k is the number of positive humps of aμ(t) which are separated by k − 1 negative humps. For sake of simplicity, the proof is carried on for the case k = 3 yielding to 7 positive solutions. Our main argument combines a modified shooting method in the phase plane with some properties of the blow up solutions in the intervals where the weight function is negative.

Research paper thumbnail of A new binomial algorithm for pricing American pathdependentoptions

Research paper thumbnail of Ordinary Differential Equations and Delayequations

Research paper thumbnail of Fučı́k Spectrum for a Third Order Equation

Journal of Differential Equations, 1996

Research paper thumbnail of On the Sturm Theorem for n-th order equations withweight of non constant sign

Differential Equations and Dynamical Systems, 1996

Research paper thumbnail of On the solvability of nonselfadjoint symmetricboundary value problems

Research paper thumbnail of Trasformation Theorem for Periodic Solutions Ofnondissipative Systems

Research paper thumbnail of Binomial methods for pricing american options

Research paper thumbnail of Reduction of American call barrier options to theEuropean case

Research paper thumbnail of On the existence of infinitely many solutions of n-th order boubdary value problems

Research paper thumbnail of Counting paths on a chessboard with a barrier

Nonlinear Analysis: Hybrid Systems, 2010

ABSTRACT We are dealing with the problem of counting the paths joining two points of a chessboard... more ABSTRACT We are dealing with the problem of counting the paths joining two points of a chessboard in the presence of a barrier. The formula for counting all the paths joining two distinct positions on the chessboard lying always over a barrier is well known (see for example Feller (1968) [1], Kreher and Stinson (1999) [3]). The problem is here extended to the calculation of all the possible paths of n movements which stay exactly k times, 0≤k≤n+1, over the barrier. Such a problem, motivated by the study of financial options of Parisian type, is completely solved by virtue of five different formulas depending on the initial and final positions and on the level of the barrier.

Research paper thumbnail of Comparison and Integral Comparison Theorems for Linear Differential Equations

Journal of Differential Equations, 1994

Research paper thumbnail of On an eigenvalue problem of Ahmad and Lazer for ordinary differential equations

Proceedings of the American Mathematical Society, 1987

Lazer, we show the existence of a class of nonselfadjoint eigenvalue problems related to the equa... more Lazer, we show the existence of a class of nonselfadjoint eigenvalue problems related to the equation y(n) + Xp(x)y = 0 for which the general eigenvalues comparison is not true. We use a comparison principle for the zeros of the corresponding Cauchy problem. This paper provides a contribution to the understanding of a problem raised by S. Ahmad and A. C. Lazer [1] in connection with the comparison of the eigenvalues for some multi-point boundary value problems which are not selfadjoint. One is given the equation (1) Lny + Xp(x)y = 0, where p(x) is a continuous function of constant sign on an interval /, A is a parameter, and Lny is a linear differential disconjugate operator of order n, that is, the only solution of Lny = 0 with n zeros on I (counting multiplicity) is y = 0. Let us consider the eigenvalue problem given by equation (1) and the system of boundary conditions ,, Lzy(a)=0, iG{ii,...,ik}, L]V(b)=0, JGiJu.-.Jn-k}, where o, b G I, 1 < k < n-1, Liy, i = 0,..., n-1, are the quasi-derivatives of y(x) (see [7]), and {t'i,..., ¿fc}, {ji, ■ ■ ■ ,jn-k) are two arbitrary sets of indices from the set {0,... ,n-1}. Problems of this type have been studied extensively (cf. [2, 3, 5]). In particular, Elias [5] has shown that if (-l)n_fcp(x) < 0, then the eigenvalues of problems (1) and (2) are real and nonnegative and form a divergence sequence {Am}m£N-Ahmad and Lazer [1] have considered a particular type of boundary condition (2), that is (3) y(a)=y'(a) =-= yik-1\a) = 0, y(b)=y'(b) =-=y(n-k-i\b)=0, and showed that if we set p = Pi, where p¿, i-1,2, are two continuous functions, considering the corresponding sequence of eigenvalues (A¿,m)m6N, i = 1,2, ordered by magnitude, then the condition (4)_ (-l)n-kp2(x) < (-l)"-fepi(x) < 0

Research paper thumbnail of On the existence of infinitely many solutions for superlinear n-th order boundary value problems

Research paper thumbnail of Positive solutions of singular boundary value problems with indefinite weight

Bulletin of The Belgian Mathematical Society-simon Stevin, 2002

Using a generalized version of the method of lower and upper solutions, we prove existence of pos... more Using a generalized version of the method of lower and upper solutions, we prove existence of positive solutions for a class of boundary value problems for a nonlinear equation with singularities whose coefficients change sign.

Research paper thumbnail of Nonselfadjoint 4th order boundary value problems

Nonlinear Analysis: Theory, Methods & Applications, 1998

ABSTRACT Consider the following linear boundary value problem u (n) =(-1) n-k p(t)u,(1) u (i) (a)... more ABSTRACT Consider the following linear boundary value problem u (n) =(-1) n-k p(t)u,(1) u (i) (a)=0,i∈{i 1 ,⋯,i k },(2) u (j) (b)=0,j∈{j 1 ,⋯,j n-k },(3) 0≤i 1 &lt;⋯&lt;i k ≤n-1,0≤j 1 &lt;⋯&lt;j n-k ≤n-1and1≤k≤n-1· The set E⊂L ∞ is said to have the property P if for any p∈E the problem (1), (2), (3) has only the trivial solution. The authors study a connection between property P and the corresponding eigenvalue problem. The main results are basic to obtain the existence of solutions to the nonlinear problem u (n) =(-1) n-k f(t,u), (2), (3).

Research paper thumbnail of Positive solutions of superlinear boundary value problems with singular indefinite weight

Communications on Pure and Applied Analysis, 2003

Using a generalized version of the method of lower and upper solutions, we prove existence of pos... more Using a generalized version of the method of lower and upper solutions, we prove existence of positive solutions for a class of boundary value problems for a nonlinear equation with singularities whose coefficients change sign.