Katia Colaneri

Nonlinear filtering for jump diffusion observations

We deal with the filtering problem of a general jump diffusion process, X, when the observation process, Y, is a correlated jump diffusion process having common jump times with X. In this setting, at any time t the σ-algebra provides all the available information about X t , and the central goal is to characterize the filter, π t , which is the conditional distribution of X t given observations . To this end, we prove that π t solves the Kushner-Stratonovich equation and, by applying the filtered martingale problem approach (see Kurtz and Ocone (1988)), that it is the unique weak solution to this equation. Under an additional hypothesis, we also provide a pathwise uniqueness result.

A benchmark approach to risk-minimization under partial information

The goal of this paper is to investigate (locally) risk-minimizing hedging strategies under the benchmark approach in a financial semimartingale market model where there are restrictions on the available information. More precisely, we characterize the optimal strategy as the integrand appearing in the Galtchouk–Kunita–Watanabe decomposition of the benchmarked contingent claim under partial information and provide its description in terms of the integrand in the classical Galtchouk–Kunita–Watanabe decomposition under full information via dual predictable projections. Finally we show how these results can be applied to unit-linked life insurance contracts.

Hedging of unit-linked life insurance contracts with unobservable mortality hazard rate via local risk-minimization

In this paper we investigate the local risk-minimization approach for a combined financial-insurance model where there are restrictions on the information available to the insurance company. In particular we assume that, at any time, the insurance company may observe the number of deaths from a specific portfolio of insured individuals but not the mortality hazard rate. We consider a financial market driven by a general semimartingale and we aim to hedge unit-linked life insurance contracts via the local risk-minimization approach under partial information. The Follmer–Schweizer decomposition of the insurance claim and explicit formulas for the optimal strategy for pure endowment and term insurance contracts are provided in terms of the projection of the survival process on the information flow. Moreover, in a Markovian framework, this leads to a filtering problem with point process observations.

Shall I Sell or Shall I Wait? Optimal Liquidation Under Partial Information with Price Impact

We study the problem of a trader who wants to maximize the expected revenue from liquidating a given stock position. We model the stock price dynamics as a geometric pure jump process with local characteristics driven by an unobservable finite-state Markov chain and by the liquidation rate. This reflects uncertainty about activity of other traders and feedback effects from trading. We use stochastic filtering to reduce the optimization problem under partial information to an equivalent one under complete information. This leads to a control problem for piecewise deterministic Markov processes (PDMPs). We apply control theory for PDMPs to our problem. In particular, we derive the optimality equation for the value function, we characterize the value function as viscosity solution of the associated dynamic programming equation, and we prove a novel comparison result. The paper concludes with a detailed analysis of a specific example. We present numerical results illustrating the impact of partial information and feedback effects on the value function and on the optimal liquidation rate.

The F\"ollmer-Schweizer decomposition under incomplete information

In this paper we study the Föllmer–Schweizer decomposition of a square integrable random variable with respect to a given semimartingale S under restricted information. Thanks to the relationship between this decomposition and that of the projection of with respect to the given information flow, we characterize the integrand appearing in the Föllmer–Schweizer decomposition under partial information in the general case where is not necessarily adapted to the available information level. For partially observable Markovian models where the dynamics of S depends on an unobservable stochastic factor X, we show how to compute the decomposition by means of filtering problems involving functions defined on an infinite-dimensional space. Moreover, in the case of a partially observed jump-diffusion model where X is described by a pure jump process taking values in a finite dimensional space, we compute explicitly the integrand in the Föllmer–Schweizer decomposition by working with finite dimensional filters. Finally, we use our achievements in a financial application where we compute the optimal hedging strategy under restricted information for a European put option and provide a comparison with that under complete information.

The Value of Information for Optimal Portfolio Management

We study the value of information for a manager who invests in a stock market to optimize the utility of her future wealth. We consider an incomplete financial market model with a mean reverting market price of risk that cannot be directly observed by the manager. The available information is represented by the filtration generated by the stock price process. We solve the classical Merton problem for an incomplete market under partial information by means of filtering techniques and the martingale approach.

Pairs Trading under Drift Uncertainty and Risk Penalization

In this work, we study the dynamic portfolio optimization problem related to the pairs trading, which is an investment strategy that matches a long position in one security with a short position in an another security with similar characteristics. The relation between pairs, called spread, is modeled by a Gaussian mean-reverting process whose drift rate is modulated by an unobservable continuous-time finite state Markov chain. Using the classical stochastic filtering theory, we reduce this problem with partial information to the one with complete information and solve it for the logarithmic utility function, where the terminal wealth is penalized by the riskiness of the portfolio according to the realized volatility of the wealth process. We characterize optimal dollar-neutral strategies as well as optimal value functions under both full and partial information and show that the certainty principle holds for the optimal portfolio strategy. Finally, we provide a numerical analysis for a simple example with a two-state Markov chain.

Portfolio optimization for a large investor controlling market sentiment under partial information

We consider an investor faced with the utility maximization problem in which the risky asset price process has pure-jump dynamics affected by an unobservable continuous-time finite-state Markov chain, the intensity of which can also be controlled by actions of the investor. Using the classical filtering theory, we reduce this problem with partial information to one with full information and solve it for logarithmic and power utility functions. In particular, we apply control theory for piece-wise deterministic Markov processes (PDMP) to our problem and derive the optimality equation for the value function and characterize the value function as the unique viscosity solution of the associated dynamic programming equation. Finally, we provide a toy example, where the unobservable state process is driven by a two-state Markov chain, and discuss how investors ability to control the intensity of the state process affects the optimal portfolio strategies as well as the optimal wealth under both partial and full information cases.

Unit-linked life insurance policies: optimal hedging in partially observable market models

In this paper we investigate the hedging problem of a unit-linked life insurance contract via the local risk-minimization approach, when the insurer has a restricted information on the market. In particular, we consider an endowment insurance contract, that is a combination of a term insurance policy and a pure endowment, whose final value depends on the trend of a stock market where the premia the policyholder pays are invested. To allow for mutual dependence between the financial and the insurance markets, we use the progressive enlargement of filtration approach. We assume that the stock price process dynamics depends on an exogenous unobservable stochastic factor that also influences the mortality rate of the policyholder. We characterize the optimal hedging strategy in terms of the integrand in the Galtchouk–Kunita–Watanabe decomposition of the insurance claim with respect to the minimal martingale measure and the available information flow. We provide an explicit formula by means of predictable projection of the corresponding hedging strategy under full information with respect to the natural filtration of the risky asset price and the minimal martingale measure. Finally, we discuss applications in a Markovian setting via filtering.