The strong predictable representation property of semi-martingales and the notion of enlargement ... more The strong predictable representation property of semi-martingales and the notion of enlargement of filtration meet naturally in modeling financial markets, and theoretical problems arise. Here, first, we illustrate some of them through classical examples. Then, we review recent results obtained by studying predictable martingale representations for filtrations enlarged by means of a full process, possibly with accessible components in its jump times. The emphasis is on the non-uniqueness of the martingale enjoying the strong predictable representation property with respect to the same enlarged filtration.
Bollettino della Unione Matematica Italiana B, 2003
A model of a heterogeneous population partitioned into a finite number of classes according an ex... more A model of a heterogeneous population partitioned into a finite number of classes according an exchangeable equivalence relation is studied, With this motivation the properties of exchangeable equivalence relations are investigated and, in particular, the structure of its ...
We consider a filtering problem when the state process is a reflected Brownian motion Xt and the ... more We consider a filtering problem when the state process is a reflected Brownian motion Xt and the observation process is its local time Λs, for s ≤ t. For this model we derive an approximation scheme based on a suitable interpolation of the observation process Λt. The convergence of the approximating filter to the original one combined with an explicit con-struction of the approximating filter allows us to derive the explicit form of the original filter. The last result can be obtained also by means of the Azéma martingale.
In the filtering problem considered here, the state process is a continuous time random walk and ... more In the filtering problem considered here, the state process is a continuous time random walk and the observation process is an increasing process depending deterministically on the trajectory of the state process. An explicit construction of the filter is given. This construction is then applied to a suitable approximation of a Brownian motion and to a rescaled M/M/1 queueing model. In both these cases, the sequence of the observation processes converges to a local time, and a convergence result for the respective filters is given. The case of a queueing model when the observation is the idle time is also considered.
Let M and N be an F-martingale and an H-martingale respectively on the same probability space, bo... more Let M and N be an F-martingale and an H-martingale respectively on the same probability space, both enjoying the predictable representation property. We discuss how, under the assumption of the existence of an equivalent decoupling measure for F and H, the nature of the jump times of M and N affects the representation of the FVH-martingales. More precisely we show that the multiplicity of FVH depends on the behavior of the common accessible jump times of the two martingales. Then we propose an extension of Kusuoka's representation theorem to the case when the Brownian Motion is replaced by a semi-martingale which may jump at the default time with positive probability.
We consider a filtering problem when the state process is a Brownian motion Wt and the observatio... more We consider a filtering problem when the state process is a Brownian motion Wt and the observation process is its local time Λs, for s ≤ t. For this model we derive an approximation scheme based on a suitable interpolation of the observation process Λt. The convergence of the approximating filter to the original one combined with an explicit construction of the approximating filter allows us to derive the explicit form of the original filter. Some connections with the Azéma martingale are discussed. MSC: 60J55; 60J65; 60F99; 60G35; 93E11; 60G55
Springer Proceedings in Mathematics & Statistics, 2019
Given two martingales on the same probability space, both enjoying the predictable representation... more Given two martingales on the same probability space, both enjoying the predictable representation property with respect to their own filtrations, it can happens that their quadratic covariation process enters in the martingale representation of the filtration obtained as the union of the original ones. This fact on one hand influences the multiplicity of the enlarged filtration and on the other hand it is linked to the behavior of the sharp brackets of the martingales. Here we illustrate these arguments presenting an elementary example of martingale representation in the context of progressive enlargement by an accessible random time.
A filtering problem is considered in the case when the state process is a continuous time random ... more A filtering problem is considered in the case when the state process is a continuous time random walk Xt and the observation process is its local time Lt. An explicit construction of the filter is given. This construction is then applied to a suitable approximation of a Brownian motion and to an asymptotically symmetric M/M/1 queueing model. In both these cases it has been possible to give a convergence result for the respective filters.
Let M and N be an F-martingale and an H-martingale respectively on the same probability space, bo... more Let M and N be an F-martingale and an H-martingale respectively on the same probability space, both enjoying the predictable representation property. We discuss how, under the assumption of the existence of an equivalent decoupling measure for F and H, the nature of the jump times of M and N affects the representation of the FVH-martingales.~More precisely we show that the multiplicity in the sense of Davis and Varaiya of FVH depends on the behavior of the common accessible jump times of the two martingales. Then we propose an extension of Kusuoka's representation theorem.
We show that all local martingales with respect to the initially enlarged natural filtration of a... more We show that all local martingales with respect to the initially enlarged natural filtration of a vector of multivariate point processes can be weakly represented up to the minimum among the explosion times of the components. We also prove that a strong representation holds if any multivariate point process of the vector has almost surely infinite explosion time and discrete mark’s space. Then we provide a condition under which the components of the multidimensional local martingale driving the strong representation are pairwise orthogonal.
A filtering problem is considered in the case when the state process is a continuous time random ... more A filtering problem is considered in the case when the state process is a continuous time random walk and the observation process is its local time. An explicit construction of the filter is given. This construction is then applied to a suitable approximation of a Brownian motion and to a rescaled M/M/1 queueing model. In both these cases a convergence result for the respective filters is given. The case of a queueing model when the observation is the idle time is also considered. MSC: 60K20; 60J55; 60F99; 60G35; 93E11; 60G55
The strong predictable representation property of semi-martingales and the notion of enlargement ... more The strong predictable representation property of semi-martingales and the notion of enlargement of filtration meet naturally in modeling financial markets, and theoretical problems arise. Here, first, we illustrate some of them through classical examples. Then, we review recent results obtained by studying predictable martingale representations for filtrations enlarged by means of a full process, possibly with accessible components in its jump times. The emphasis is on the non-uniqueness of the martingale enjoying the strong predictable representation property with respect to the same enlarged filtration.
Bollettino della Unione Matematica Italiana B, 2003
A model of a heterogeneous population partitioned into a finite number of classes according an ex... more A model of a heterogeneous population partitioned into a finite number of classes according an exchangeable equivalence relation is studied, With this motivation the properties of exchangeable equivalence relations are investigated and, in particular, the structure of its ...
We consider a filtering problem when the state process is a reflected Brownian motion Xt and the ... more We consider a filtering problem when the state process is a reflected Brownian motion Xt and the observation process is its local time Λs, for s ≤ t. For this model we derive an approximation scheme based on a suitable interpolation of the observation process Λt. The convergence of the approximating filter to the original one combined with an explicit con-struction of the approximating filter allows us to derive the explicit form of the original filter. The last result can be obtained also by means of the Azéma martingale.
In the filtering problem considered here, the state process is a continuous time random walk and ... more In the filtering problem considered here, the state process is a continuous time random walk and the observation process is an increasing process depending deterministically on the trajectory of the state process. An explicit construction of the filter is given. This construction is then applied to a suitable approximation of a Brownian motion and to a rescaled M/M/1 queueing model. In both these cases, the sequence of the observation processes converges to a local time, and a convergence result for the respective filters is given. The case of a queueing model when the observation is the idle time is also considered.
Let M and N be an F-martingale and an H-martingale respectively on the same probability space, bo... more Let M and N be an F-martingale and an H-martingale respectively on the same probability space, both enjoying the predictable representation property. We discuss how, under the assumption of the existence of an equivalent decoupling measure for F and H, the nature of the jump times of M and N affects the representation of the FVH-martingales. More precisely we show that the multiplicity of FVH depends on the behavior of the common accessible jump times of the two martingales. Then we propose an extension of Kusuoka's representation theorem to the case when the Brownian Motion is replaced by a semi-martingale which may jump at the default time with positive probability.
We consider a filtering problem when the state process is a Brownian motion Wt and the observatio... more We consider a filtering problem when the state process is a Brownian motion Wt and the observation process is its local time Λs, for s ≤ t. For this model we derive an approximation scheme based on a suitable interpolation of the observation process Λt. The convergence of the approximating filter to the original one combined with an explicit construction of the approximating filter allows us to derive the explicit form of the original filter. Some connections with the Azéma martingale are discussed. MSC: 60J55; 60J65; 60F99; 60G35; 93E11; 60G55
Springer Proceedings in Mathematics & Statistics, 2019
Given two martingales on the same probability space, both enjoying the predictable representation... more Given two martingales on the same probability space, both enjoying the predictable representation property with respect to their own filtrations, it can happens that their quadratic covariation process enters in the martingale representation of the filtration obtained as the union of the original ones. This fact on one hand influences the multiplicity of the enlarged filtration and on the other hand it is linked to the behavior of the sharp brackets of the martingales. Here we illustrate these arguments presenting an elementary example of martingale representation in the context of progressive enlargement by an accessible random time.
A filtering problem is considered in the case when the state process is a continuous time random ... more A filtering problem is considered in the case when the state process is a continuous time random walk Xt and the observation process is its local time Lt. An explicit construction of the filter is given. This construction is then applied to a suitable approximation of a Brownian motion and to an asymptotically symmetric M/M/1 queueing model. In both these cases it has been possible to give a convergence result for the respective filters.
Let M and N be an F-martingale and an H-martingale respectively on the same probability space, bo... more Let M and N be an F-martingale and an H-martingale respectively on the same probability space, both enjoying the predictable representation property. We discuss how, under the assumption of the existence of an equivalent decoupling measure for F and H, the nature of the jump times of M and N affects the representation of the FVH-martingales.~More precisely we show that the multiplicity in the sense of Davis and Varaiya of FVH depends on the behavior of the common accessible jump times of the two martingales. Then we propose an extension of Kusuoka's representation theorem.
We show that all local martingales with respect to the initially enlarged natural filtration of a... more We show that all local martingales with respect to the initially enlarged natural filtration of a vector of multivariate point processes can be weakly represented up to the minimum among the explosion times of the components. We also prove that a strong representation holds if any multivariate point process of the vector has almost surely infinite explosion time and discrete mark’s space. Then we provide a condition under which the components of the multidimensional local martingale driving the strong representation are pairwise orthogonal.
A filtering problem is considered in the case when the state process is a continuous time random ... more A filtering problem is considered in the case when the state process is a continuous time random walk and the observation process is its local time. An explicit construction of the filter is given. This construction is then applied to a suitable approximation of a Brownian motion and to a rescaled M/M/1 queueing model. In both these cases a convergence result for the respective filters is given. The case of a queueing model when the observation is the idle time is also considered. MSC: 60K20; 60J55; 60F99; 60G35; 93E11; 60G55
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