Anna Gerardi

A MODEL FOR HIGH FREQUENCY DATA UNDER PARTIAL INFORMATION: A FILTERING APPROACH

A general model for intraday stock price movements is studied. The asset price dynamics is described by a marked point process Y, whose local characteristics (in particular the jump-intensity) depend on some unobservable hidden state variable X. The dynamics of Y and X may be strongly dependent. In particular the two processes may have common jump times, which means that the actual trading activity may affect the law of X and could be also related to the possibility of catastrophic events. The agents, in this model, are restricted to observing past asset prices. This leads to a filtering problem with marked point process observations. The conditional law of X given the past asset prices (the filter) is characterized as the unique weak solution of the Kushner–Stratonovich equation. An explicit representation of the filter is obtained by the Feyman–Kac formula using a linearization method. This representation allows us to provide a recursive algorithm for the filter computation.

Pricing For Geometric Marked Point Processes Under Partial Information: Entropy Approach

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.

Nonlinear Filtering Equation of a Jump Process with Counting Observations

We study the filtering problem of an Rd-valued pure jump process when the observations is a counting process. We assume that the dynamic of the state and the observations may be strongly dependent and that the two processes may jump together. Weak and pathwise uniqueness of solution of the Kushner–Stratonovich equation are discussed.

Existence of Optimal Controls for Partially Observed Jump Processes

A partially observable control problem for an Rd-valued jump process with counting observations is studied. The state and the observations may be strongly dependent and, in particular, the two processes may jump together. An equivalent separated problem is introduced and the existence of an optimal control for the separated problem is obtained in the class of relaxed and generalized controls. Equivalence between the initial problem and the relaxed generalized separated control problem is discussed.

Optimal Control and Filtering of the Reproduction Law of a Branching Process

A finite horizon control problem for the reproduction law of a branching process is studied. Some examples with complete information are tackled via the Hamilton–Jacobi–Bellman equation. A partially observable control of the cardinality of the population using the information given by the splitting process is formulated. Though there is correlation between the state and the observations and the observation process has unbounded intensity, a Girsanov-type change of probability measure can be set and the filtering equation for the unnormalized conditional distribution (the Zakai equation) can be derived. Strong uniqueness for the Zakai equation and, as a consequence, also for the Kushner–Stratonovich equation is obtained. A separated control problem is introduced, in which the dynamics are represented by the splitting process and the unnormalized conditional distribution. By the strong uniqueness for the Zakai equation, equivalence between the partially observable control problem and the separated one is proved.

Wealth optimization and dual problems for jump stock dynamics with stochastic factor

An incomplete financial market is considered with a risky asset and a bond. The risky asset price is a pure jump process whose dynamics depends on a jump-diffusion stochastic factor describing the activity of other markets, macroeconomics factors or microstructure rules that drive the market. With a stochastic control approach, maximization of the expected utility of terminal wealth is discussed for utility functions of constant relative risk aversion type. Under suitable assumptions, closed form solutions for the value functions and for the optimal strategy are provided and verification results are discussed. Moreover, the solution to the dual problems associated with the utility maximization problems is derived.

Risk-neutral measures and pricing for a pure jump price process: A stochastic control approach

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.

An approximation method for controlled discrete jump processes under partial observations

The paper is concerned with control and filtering of a discrete jump Markov process when only the total number of jumps is observed. A partially observable control problem is considered and equivalence between this problem and the separated one is proved. A technique to approximate the value function, under some suitable assumptions, is presented.