Stochastic Processes and Applications to Mathematical Finance

PREFACE.

Program.

CONTENTS.

We review a general mathematical link between utility and information theory appearing in a simple financial market model with two kinds of small investors: insiders, whose extra information is stored in an enlargement of the less informed agents' filtration. The insider's expected logarithmic utility increment is described in terms of the information drift, i.e. the drift one has to eliminate in order to perceive the price dynamics as a martingale from his perspective. We describe the information drift in a very general setting by natural quantities expressing the conditional laws of the better informed view of the world. This on the other hand allows to identify the additional utility by entropy related quantities known from information theory.

No abstract received.

We formulate an operational definition for absence of model-free arbitrage in a financial market, in terms of a set of minimal requirements for the pricing rule prevailing in the market and without making reference to any ‘objective’ probability measure. We show that any pricing rule verifying these properties can be represented as a conditional expectation operator with respect to a probability measure under which prices of traded assets follow martingales. Our result does not require any notion of “reference” probability measure and is consistent with the formulation of model calibration problems in option pricing.

We consider a multidimensional financial model with mild conditions on the underlying asset price process. The trading is only allowed at some fixed discrete times and the strategy is constrained to lie in a closed convex cone. We show how the minimal cost of a super hedging strategy can be easily computed by a backward recursive scheme. As an application, when the underlying asset follows a stochastic differential equation including stochastic volatility or Poisson jumps, we compute those super-replication prices for a range of European and American style options, including Asian, Lookback or Barrier Options. We also perform some multidimensional computations.

The model developed is a structural valuation model of risky debt which includes a dynamic of debt. The value of the firm follows either a Brownian dynamic or a mixed Brownian-Poisson dynamic. The default threshold is endogenous; thus the optimal default threshold changes as the firm's value changes. Another optimal control problem is setting with respect to a couple (coupon or dividend policy, stopping time). Hence a local time is introduced in the firm value dynamic, being the optimal coupon or dividend policy. Once again an optimal threshold is given.

We consider the problem of computing some basic quantities such as defaultable bond prices and survival probabilities in a credit risk model according to the intensity based approach. We let the default intensities depend on an external factor process that we assume is not observable. We use stochastic filtering to successively update its distribution on the basis of the observed default history. On one hand this allows us to capture aspects of default contagion (information-induced contagion). On the other hand it allows us to evaluate the above quantities also in our incomplete information context. We consider in particular affine credit risk models and show that in such models the nonlinear filter can be computed via a recursive procedure. This then leads to an explicit expression for the filter that depends on a finite number of sufficient statistics of the observed interarrival times for the defaults provided one chooses an initial distribution for the factor process that is of the Gamma type.

No abstract received.

No abstract received.

In this paper we develop a valuation model when there exist two competitive firms that face irreversible investment decisions under the demand uncertainty. We propose a numerical procedure to derive both project values and equilibrium strategies in the duopolistic environment. In numerical examples we consider two different economic conditions, which are labelled first mover advantage and second mover advantage, and examine the effects of these conditions on the equilibrium strategies as well as the project values. We show that these conditions cause significant changes in the equilibrium strategies of both firms.

We study Asian strike put options of Europian, American and game type. After changing the variable, we show that the study of these Asian strike options is reduced to the study of Europian, American and game type options for a one dimensional nonstationary Brownian motion (diffusion process) y_t with a suitable pay-off function. We get the exercise region of the holder for the American type option and the exercise regions of the holder and the writer for the game type option. Then we study the early exercise premium for the American type option and the early exercise premium of the holder and the early cancellation fee of the writer for the game type option.

For the study of the early cancellation fee of the writer, we need an extension of Itô's formula, which involves the local time of the price process y_t. We establish the generalized Itô's formula in Appendix.

This is a short report on a talk given at the “short communications” session. An article which contains the full proofs and several extensions of the results in this report is now in the preparation, and it will appear in the near future.

Higher order particle methods can provide an accurate description of an evolving family of measures, but frequently the number of particles used in the description explodes as we iterate. In this paper we present a general method to simplify the support of the intermediate measures used in the iteration without increasing the error in the particle approximation by more than a constant factor. We describe two algorithms that can be used to simplify the support of a discrete measure and give an application to the cubature on Wiener space method developed by Lyons, Victoir [13].

A verification theorem for an impulse control problem related to optimal investment with transaction costs is studied. A risk-sensitive variational inequality (QVI) of ergodic type concerns the problem. The pair (u⊂l) of a function u and a constant l is considered to be a solution of the QVI. The constant determines the value of the impulse control problem and an optimal strategy is constructed from the function u. In proving the verification theorem we have uniqueness of the constant l of the solution of the QVI.

In this paper we introduce and study Bessel processes which take values in a d-dimensional nonnegative cone of and are constructed via the multi-dimensional Kingman convolution. We prove that every d-variate Bessel process is a stationary independent increments-type process. Moreover, a stochastic integral with respect to with the convergence in distribution is defined.

In the present paper we study multiply selfdecomposable probability measures (SDPM) and processes and prove their integral representations. Similarly, the multiple s-selfdecomposability case is treated. Our results extend some of known results due to Urbanik, K., Jurek, Z., Rosinski, J. and Rajput, B.S. As an application, following Cartea and Howinson ([1]) we introduce the Damped-Lévy-mixed - stable process which leads to a mathematical model for option pricing.

No abstract received.

We present some recent developments on optimal quantization methods for numerically feasible solutions to discrete-time optimization problems under partial information. The main problem in effective implementation is in growing dimension of the the approximating filters. We overcome this difficulty by performing a quantization of the pair process filter-observation. Dynamic programming is then applied to solve the approximated optimization problem. Several numerical applications in finance are presented for the pricing of American option or for hedging problems in the context of partially observed stochastic volatility models.