In our previous paper [1], we proposed a non-standardization of the concept of convolution in order to construct an extended Wiener measure using nonstandard analysis by E. Nelson [2]. In this paper, we consider Ito’...In our previous paper [1], we proposed a non-standardization of the concept of convolution in order to construct an extended Wiener measure using nonstandard analysis by E. Nelson [2]. In this paper, we consider Ito’s integral with respect to the extended Wiener measure and extend Ito’s formula for Ito’s process. Because of doing the extension of Ito’s formula, we could treat stochastic differential equations in the sense of nonstandard analysis. In this framework, we need the nonstandardization of convolution again. It was not yet proved in the last paper, therefore we shall provide the proof.展开更多
Following Konno [1], it is natural to ask: What is the Ito’s formula for the discrete time quantum walk on a graph different than Z, the set of integers? In this paper we answer the question for the discrete time qua...Following Konno [1], it is natural to ask: What is the Ito’s formula for the discrete time quantum walk on a graph different than Z, the set of integers? In this paper we answer the question for the discrete time quantum walk on Z2, the square lattice.展开更多
Standing on a different view point from Anderson, we prove that the extended Wiener process defined by Anderson satisfies the definition of the Wiener process in standard analysis, for example the Wiener process at ti...Standing on a different view point from Anderson, we prove that the extended Wiener process defined by Anderson satisfies the definition of the Wiener process in standard analysis, for example the Wiener process at time t obeys the normal distribution N(0,t) by showing the central limit theorem. The essential theory used in the proof is the extended convolution property in nonstandard analysis which is shown by Kanagawa, Nishiyama and Tchizawa (2018). When processing the extension by non-standardization, we have already pointed out that it is needed to proceed the second extension for the convolution, not only to do the first extension for the delta function. In Section 2, we shall introduce again the extended convolution as preliminaries described in our previous paper. In Section 3, we shall provide the extended stochastic process using a hyper number N, and it satisfies the conditions being Wiener process. In Section 4, we shall give a new proof for the non-differentiability in the Wiener process.展开更多
In this paper, the one-dimensional time-homogenuous lto’s stochastic differential equations, which have degenerate and discontinuous diffusion coefficients, are considered. The non-confluent property of solutions is ...In this paper, the one-dimensional time-homogenuous lto’s stochastic differential equations, which have degenerate and discontinuous diffusion coefficients, are considered. The non-confluent property of solutions is showed under some local integrability condition on the diffusion and drift coefficients. The strong comparison theorem for solutions is also established.展开更多
The study analyses some problems arising in stochastic volatility models by using Ito’s lemma and its applications to boundary Cauchy problem by giving the solution of vanilla option pricing models satisfying the par...The study analyses some problems arising in stochastic volatility models by using Ito’s lemma and its applications to boundary Cauchy problem by giving the solution of vanilla option pricing models satisfying the partial differential equation obtained by assuming stochastic volatility in replication problems and risk neutral probability.展开更多
The proportion of the favorable among voters to a nominee might change over times and depend on different factors for example: talent, reputation, party and even name order on election. The unobservable factors which ...The proportion of the favorable among voters to a nominee might change over times and depend on different factors for example: talent, reputation, party and even name order on election. The unobservable factors which might have minor impacts on the approval rate are modelized by random elements. The approval rate is initially described by the differential equation and then by the random differential equation including the above unobservable factors. We figure out the formula of the solution for the stochastic differential equation and simulate these solutions to identify the changes of the approval rate over time.展开更多
In the present paper we first obtain the comparison principle for the nonlinear stochastic differentialdelay equations with Markovian switching. Later, using this comparison principle, we obtain some stabilitycriteria...In the present paper we first obtain the comparison principle for the nonlinear stochastic differentialdelay equations with Markovian switching. Later, using this comparison principle, we obtain some stabilitycriteria, including stability in probability, asymptotic stability in probability, stability in the pth mean, asymptoticstability in the pth mean and the pth moment exponential stability of such equations. Finally, an example isgiven to illustrate the effectiveness of our results.展开更多
Let M = {M<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>} be a continuous square integrable martingale and A = {A<sub>z</sub>, z∈ R<sub>+</sub><sup>2</...Let M = {M<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>} be a continuous square integrable martingale and A = {A<sub>z</sub>, z∈ R<sub>+</sub><sup>2</sup>} be a continuous adapted increasing process. Consider the following stochastic partial differential equations in the plane: dX<sub>z</sub>=α（z, X<sub>z</sub>）dM<sub>2</sub>+β（z,X<sub>z</sub>）dA<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>, X<sub>z</sub>=Z<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>, where R<sub>+</sub><sup>2</sup>=[0,+∞)×[0,+∞) and R<sub>+</sub><sup>2</sup> is its boundary, Z is a continuous stochastic process on R<sub>+</sub><sup>2</sup>. We establish a new theorem on the pathwise uniqueness of solutions for the equation under a weaker condition than the Lipschitz one. The result concerning the one-parameter analogue of the problem we consider here is immediate （see [1, Theorem 3.2]）. Unfortunately, the situation is much more complicated for two-parameter process and we believe that our result is the first one of its kind and is interesting in itself. We have proved the existence theorem for the equation in.展开更多
文摘In our previous paper [1], we proposed a non-standardization of the concept of convolution in order to construct an extended Wiener measure using nonstandard analysis by E. Nelson [2]. In this paper, we consider Ito’s integral with respect to the extended Wiener measure and extend Ito’s formula for Ito’s process. Because of doing the extension of Ito’s formula, we could treat stochastic differential equations in the sense of nonstandard analysis. In this framework, we need the nonstandardization of convolution again. It was not yet proved in the last paper, therefore we shall provide the proof.
文摘Following Konno [1], it is natural to ask: What is the Ito’s formula for the discrete time quantum walk on a graph different than Z, the set of integers? In this paper we answer the question for the discrete time quantum walk on Z2, the square lattice.
文摘Standing on a different view point from Anderson, we prove that the extended Wiener process defined by Anderson satisfies the definition of the Wiener process in standard analysis, for example the Wiener process at time t obeys the normal distribution N(0,t) by showing the central limit theorem. The essential theory used in the proof is the extended convolution property in nonstandard analysis which is shown by Kanagawa, Nishiyama and Tchizawa (2018). When processing the extension by non-standardization, we have already pointed out that it is needed to proceed the second extension for the convolution, not only to do the first extension for the delta function. In Section 2, we shall introduce again the extended convolution as preliminaries described in our previous paper. In Section 3, we shall provide the extended stochastic process using a hyper number N, and it satisfies the conditions being Wiener process. In Section 4, we shall give a new proof for the non-differentiability in the Wiener process.
文摘In this paper, the one-dimensional time-homogenuous lto’s stochastic differential equations, which have degenerate and discontinuous diffusion coefficients, are considered. The non-confluent property of solutions is showed under some local integrability condition on the diffusion and drift coefficients. The strong comparison theorem for solutions is also established.
文摘The study analyses some problems arising in stochastic volatility models by using Ito’s lemma and its applications to boundary Cauchy problem by giving the solution of vanilla option pricing models satisfying the partial differential equation obtained by assuming stochastic volatility in replication problems and risk neutral probability.
文摘The proportion of the favorable among voters to a nominee might change over times and depend on different factors for example: talent, reputation, party and even name order on election. The unobservable factors which might have minor impacts on the approval rate are modelized by random elements. The approval rate is initially described by the differential equation and then by the random differential equation including the above unobservable factors. We figure out the formula of the solution for the stochastic differential equation and simulate these solutions to identify the changes of the approval rate over time.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 10171009) Tianyuan Young Fund of China (Grant No. 10226009).
文摘In the present paper we first obtain the comparison principle for the nonlinear stochastic differentialdelay equations with Markovian switching. Later, using this comparison principle, we obtain some stabilitycriteria, including stability in probability, asymptotic stability in probability, stability in the pth mean, asymptoticstability in the pth mean and the pth moment exponential stability of such equations. Finally, an example isgiven to illustrate the effectiveness of our results.
基金Supported by the National Science Foundationthe Postdoctoral Science Foundation of China
文摘Let M = {M<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>} be a continuous square integrable martingale and A = {A<sub>z</sub>, z∈ R<sub>+</sub><sup>2</sup>} be a continuous adapted increasing process. Consider the following stochastic partial differential equations in the plane: dX<sub>z</sub>=α（z, X<sub>z</sub>）dM<sub>2</sub>+β（z,X<sub>z</sub>）dA<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>, X<sub>z</sub>=Z<sub>z</sub>, z∈R<sub>+</sub><sup>2</sup>, where R<sub>+</sub><sup>2</sup>=[0,+∞)×[0,+∞) and R<sub>+</sub><sup>2</sup> is its boundary, Z is a continuous stochastic process on R<sub>+</sub><sup>2</sup>. We establish a new theorem on the pathwise uniqueness of solutions for the equation under a weaker condition than the Lipschitz one. The result concerning the one-parameter analogue of the problem we consider here is immediate （see [1, Theorem 3.2]）. Unfortunately, the situation is much more complicated for two-parameter process and we believe that our result is the first one of its kind and is interesting in itself. We have proved the existence theorem for the equation in.
