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
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.
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.
