UTILITY MAXIMIZATION IN A PURE JUMP MODEL WITH PARTIAL OBSERVATION

2010 ◽  
Vol 25 (1) ◽  
pp. 29-54 ◽  
Author(s):  
Paola Tardelli
Keyword(s):  
Utility Maximization ◽  
Point Processes ◽  
Marked Point ◽  
Asset Price ◽  
Main Tool ◽  
Risky Asset ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Underlying Event ◽  
Jump Model

This article considers the asset price movements in a financial market when risky asset prices are modeled by marked point processes. Their dynamics depend on an underlying event arrivals process—a marked point process having common jump times with the risky asset price process. The problem of utility maximization of terminal wealth is dealt with when the underlying event arrivals process is assumed to be unobserved by the market agents using, as the main tool, backward stochastic differential equations. The dual problem is studied. Explicit solutions in a particular case are given.

RISK-NEUTRAL MEASURES AND PRICING FOR A PURE JUMP PRICE PROCESS

2009 ◽  
Vol 24 (1) ◽  
pp. 47-76 ◽  
Author(s):  
Anna Gerardi ◽  
Paola Tardelli
Keyword(s):  
Point Processes ◽  
Marked Point ◽  
Asset Price ◽  
Risky Asset ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Martingale Measure ◽  
Price Process ◽  
Risk Neutral Measures

This article considers the asset price movements in a financial market when risky asset prices are modeled by marked point processes. Their dynamics depend on an underlying event arrivals process, modeled again by a marked point process. Taking into account the presence of catastrophic events, the possibility of common jump times between the risky asset price process and the arrivals process is allowed. By setting and solving a suitable control problem, the characterization of the minimal entropy martingale measure is obtained. In a particular case, a pricing problem is also discussed.

scholarly journals Partially informed investors: hedging in an incomplete market with default

2015 ◽  
Vol 52 (03) ◽  
pp. 718-735 ◽  
Author(s):  
P. Tardelli
Keyword(s):  
Point Processes ◽  
Value Function ◽  
Partial Information ◽  
Marked Point ◽  
Asset Price ◽  
Risky Asset ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Control Problems ◽  
The Value Function

In a defaultable market, an investor trades having only partial information about the behavior of the market. Taking into account the intraday stock movements, the risky asset prices are modelled by marked point processes. Their dynamics depend on an unobservable process, representing the amount of news reaching the market. This is a marked point process, which may have common jump times with the risky asset price processes. The problem of hedging a defaultable claim is studied. In order to discuss all these topics, in this paper we examine stochastic control problems using backward stochastic differential equations (BSDEs) and filtering techniques. The goal of this paper is to construct a sequence of functions converging to the value function, each of these is the unique solution of a suitable BSDE.

scholarly journals Partially informed investors: hedging in an incomplete market with default

2015 ◽  
Vol 52 (3) ◽  
pp. 718-735
Author(s):  
P. Tardelli
Keyword(s):  
Point Processes ◽  
Value Function ◽  
Partial Information ◽  
Marked Point ◽  
Asset Price ◽  
Risky Asset ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Control Problems ◽  
The Value Function

In a defaultable market, an investor trades having only partial information about the behavior of the market. Taking into account the intraday stock movements, the risky asset prices are modelled by marked point processes. Their dynamics depend on an unobservable process, representing the amount of news reaching the market. This is a marked point process, which may have common jump times with the risky asset price processes. The problem of hedging a defaultable claim is studied. In order to discuss all these topics, in this paper we examine stochastic control problems using backward stochastic differential equations (BSDEs) and filtering techniques. The goal of this paper is to construct a sequence of functions converging to the value function, each of these is the unique solution of a suitable BSDE.

PRICING FOR GEOMETRIC MARKED POINT PROCESSES UNDER PARTIAL INFORMATION: ENTROPY APPROACH

2009 ◽  
Vol 12 (02) ◽  
pp. 179-207 ◽  
Author(s):  
CLAUDIA CECI ◽  
ANNA GERARDI
Keyword(s):  
Stock Price ◽  
Point Processes ◽  
Partial Information ◽  
Marked Point ◽  
Asset Price ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Martingale Measure ◽  
Minimal Entropy Martingale Measure ◽  
Risk Neutral Measures

The problem of the arbitrage-free pricing of a European contingent claim B is considered in a general model for intraday stock price movements in the case of partial information. The dynamics of the risky asset price is described through a marked point process Y, whose local characteristics depend on some unobservable jump diffusion process X. The processes Y and X may have common jump times, which means that the trading activity may affect the law of X and could be also related to the presence of catastrophic events. Risk-neutral measures are characterized and in particular, the minimal entropy martingale measure is studied. The problem of pricing under restricted information is discussed, and the arbitrage-free price of the claim B w.r.t. the minimal entropy martingale measure is computed by using filtering techniques.

Marked point processes as limits of Markovian arrival streams

1993 ◽  
Vol 30 (02) ◽  
pp. 365-372 ◽  
Author(s):  
Søren Asmussen ◽  
Ger Koole
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Marked Point ◽  
The State ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
State Transitions ◽  
Poisson Arrival ◽  
Convergence Of Moments ◽  
Environmental Process

A Markovian arrival stream is a marked point process generated by the state transitions of a given Markovian environmental process and Poisson arrival rates depending on the environment. It is shown that to a given marked point process there is a sequence of such Markovian arrival streams with the property that as m →∞. Various related corollaries (involving stationarity, convergence of moments and ergodicity) and counterexamples are discussed as well.

Reliability analysis of complex repairable systems by means of marked point processes

1980 ◽  
Vol 17 (1) ◽  
pp. 154-167 ◽  
Author(s):  
Peter Franken ◽  
Arnfried Streller
Keyword(s):  
Reliability Analysis ◽  
Point Processes ◽  
Marked Point ◽  
Regenerative Process ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Repairable Systems ◽  
Interval Reliability ◽  
Repair Facility ◽  
Markovian Systems

Starting from the theory of point processes the concept of a process with an embedded marked point process is defined. It is shown that the known formula expressing the relation between the stationary and synchronous version of a regenerative process remains valid without the assumption of independence of cycles. General formulae for stationary availability and interval reliability of complex systems with repair are also obtained. In this way generalizations of Keilson's results for Markovian systems and Ross's results for systems with separately maintained elements are presented. The formulae are applied to a two-unit parallel system with a single repair facility.

Marked point processes as limits of Markovian arrival streams

1993 ◽  
Vol 30 (2) ◽  
pp. 365-372 ◽  
Author(s):  
Søren Asmussen ◽  
Ger Koole
Keyword(s):  
Point Process ◽  
Point Processes ◽  
Marked Point ◽  
The State ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
State Transitions ◽  
Poisson Arrival ◽  
Convergence Of Moments ◽  
Environmental Process

A Markovian arrival stream is a marked point process generated by the state transitions of a given Markovian environmental process and Poisson arrival rates depending on the environment. It is shown that to a given marked point process there is a sequence of such Markovian arrival streams with the property that as m →∞. Various related corollaries (involving stationarity, convergence of moments and ergodicity) and counterexamples are discussed as well.

Reliability analysis of complex repairable systems by means of marked point processes

1980 ◽  
Vol 17 (01) ◽  
pp. 154-167 ◽  
Author(s):  
Peter Franken ◽  
Arnfried Streller
Keyword(s):  
Reliability Analysis ◽  
Point Processes ◽  
Marked Point ◽  
Regenerative Process ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Repairable Systems ◽  
Interval Reliability ◽  
Repair Facility ◽  
Markovian Systems

Starting from the theory of point processes the concept of a process with an embedded marked point process is defined. It is shown that the known formula expressing the relation between the stationary and synchronous version of a regenerative process remains valid without the assumption of independence of cycles. General formulae for stationary availability and interval reliability of complex systems with repair are also obtained. In this way generalizations of Keilson's results for Markovian systems and Ross's results for systems with separately maintained elements are presented. The formulae are applied to a two-unit parallel system with a single repair facility.

Approximations of general discrete time queues by discrete time queues with arrivals modulated by finite chains

1997 ◽  
Vol 29 (04) ◽  
pp. 1039-1059
Author(s):  
Vinod Sharma
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Continuous Time ◽  
Point Processes ◽  
Marked Point ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Discrete Time Queues ◽  
Time Queues ◽  
Finite State

Recently, Asmussen and Koole (Journal of Applied Probability 30, pp. 365–372) showed that any discrete or continuous time marked point process can be approximated by a sequence of arrival streams modulated by finite state continuous time Markov chains. If the original process is customer (time) stationary then so are the approximating processes. Also, the moments in the stationary case converge. For discrete marked point processes we construct a sequence of discrete processes modulated by discrete time finite state Markov chains. All the above features of approximating sequences of Asmussen and Koole continue to hold. For discrete arrival sequences (to a queue) which are modulated by a countable state Markov chain we form a different sequence of approximating arrival streams by which, unlike in the Asmussen and Koole case, even the stationary moments of waiting times can be approximated. Explicit constructions for the output process of a queue and the total input process of a discrete time Jackson network with these characteristics are obtained.

scholarly journals Clusterless Decoding of Position from Multiunit Activity Using a Marked Point Process Filter

2015 ◽  
Vol 27 (7) ◽  
pp. 1438-1460 ◽  
Author(s):  
Xinyi Deng ◽  
Daniel F. Liu ◽  
Kenneth Kay ◽  
Loren M. Frank ◽  
Uri T. Eden
Keyword(s):  
Real Time ◽  
Point Process ◽  
Point Processes ◽  
Marked Point ◽  
Marked Point Processes ◽  
Marked Point Process ◽  
Spike Sorting ◽  
Decoding Algorithm ◽  
Population Activity ◽  
Spiking Activity

Point process filters have been applied successfully to decode neural signals and track neural dynamics. Traditionally these methods assume that multiunit spiking activity has already been correctly spike-sorted. As a result, these methods are not appropriate for situations where sorting cannot be performed with high precision, such as real-time decoding for brain-computer interfaces. Because the unsupervised spike-sorting problem remains unsolved, we took an alternative approach that takes advantage of recent insights into clusterless decoding. Here we present a new point process decoding algorithm that does not require multiunit signals to be sorted into individual units. We use the theory of marked point processes to construct a function that characterizes the relationship between a covariate of interest (in this case, the location of a rat on a track) and features of the spike waveforms. In our example, we use tetrode recordings, and the marks represent a four-dimensional vector of the maximum amplitudes of the spike waveform on each of the four electrodes. In general, the marks may represent any features of the spike waveform. We then use Bayes’s rule to estimate spatial location from hippocampal neural activity. We validate our approach with a simulation study and experimental data recorded in the hippocampus of a rat moving through a linear environment. Our decoding algorithm accurately reconstructs the rat’s position from unsorted multiunit spiking activity. We then compare the quality of our decoding algorithm to that of a traditional spike-sorting and decoding algorithm. Our analyses show that the proposed decoding algorithm performs equivalent to or better than algorithms based on sorted single-unit activity. These results provide a path toward accurate real-time decoding of spiking patterns that could be used to carry out content-specific manipulations of population activity in hippocampus or elsewhere in the brain.

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