White noise analysis for Lévy processes

We construct a white noise theory for Lévy processes. The starting point of this theory is a chaos expansion for square integrable random variables. We use this approach to Malliavin calculus to prove the following white noise generalization of the Clark-Haussmann-Ocone formula for Lévy processes

     

Default swap games driven by spectrally negative Lévy processes

This paper studies game-type credit default swaps that allow the protection buyer and seller to raise or reduce their respective positions once prior to default. This leads to the study of an optimal stopping game subject to early default termination. Under a structural credit risk model based on spectrally negative Lévy processes, we apply the principles of smooth and continuous fit to identify the equilibrium exercise strategies for the buyer and the seller. We then rigorously prove the existence of the Nash equilibrium and compute the contract value at equilibrium. Numerical examples are provided to illustrate the impacts of default risk and other contractual features on the players’ exercise timing at equilibrium.     

Bilateral gamma distributions and processes in financial mathematics

We present a class of Lévy processes for modelling financial market fluctuations: bilateral Gamma processes. Our starting point is to explore the properties of bilateral Gamma distributions, and then we turn to their associated Lévy processes. We treat exponential Lévy stock models with an underlying bilateral Gamma process as well as term structure models driven by bilateral Gamma processes, and apply our results to a set of real financial data (DAX 1996–1998).     

Optimal consumption, investment and insurance with insurable risk for an investor in a Lévy market

Numerous researchers have applied the martingale approach for models driven by Lévy processes to study optimal investment problems. The aim of this paper is to apply the martingale approach to obtain a closed form solution for the optimal investment, consumption and insurance strategies of an individual in the presence of an insurable risk when the insurable risk and risky asset returns are described by Lévy processes and the utility is a constant absolute risk aversion (CARA). The model developed in this paper can potentially be applied to absorb large insurable losses in the absence of insurance protection and to examine the level of diminishing current utility and consumption.     

Continuous time trading of a small investor in a limit order market

We provide a mathematical framework to model continuous time trading of a small investor in limit order markets. We show how elementary strategies can be extended in a suitable way to general continuous time strategies containing orders with infinitely many different limit prices. The general limit buy order strategies are predictable processes with values in the set of nonincreasing demand functions. It turns out that our strategy set of limit and market orders is closed, but limit orders can turn into market orders when passing to the limit, and any element can be approximated by a sequence of elementary strategies.     

A new existence result for quadratic BSDEs with jumps with application to the utility maximization problem

In this study, we consider the exponential utility maximization problem in the context of a jump–diffusion model. To solve this problem, we rely on the dynamic programming principle to express the value process of this problem in terms of the solution of a quadratic BSDE with jumps. Since the quadratic BSDE¹ under study is driven by both a Wiener process and a Poisson random measure having a Lévy measure with infinite mass, our main task is therefore to establish a new existence result for the specific BSDE introduced.     

Information,no-arbitrage and completeness for asset price models with a change point

We consider a general class of continuous asset price models where the drift and the volatility functions, as well as the driving Brownian motions, change at a random time τ$\tau$. Under minimal assumptions on the random time and on the driving Brownian motions, we study the behavior of the model in all the filtrations which naturally arise in this setting, establishing martingale representation results and characterizing the validity of the NA1 and NFLVR no-arbitrage conditions.     

A limit theorem for the time of ruin in a Gaussian ruin problem

For certain Gaussian processes X(t)$X\left(t\right)$ with trend −ct^β$-c{t}^{\beta }$ and variance V²(t)$V^2\left(t\right)$, the ruin time is analyzed where the ruin time is defined as the first time point t$t$ such that X(t)−ct^β≥u$X\left(t\right)-c{t}^{\beta }\ge u$. The ruin time is of interest in finance and actuarial subjects. But the ruin time is also of interest in other applications, e.g. in telecommunications where it indicates the first time of an overflow. We derive the asymptotic distribution of the ruin time as u→∞$u\to \infty$ showing that the limiting distribution depends on the parameters β$\beta$, V(t)$V\left(t\right)$ and the correlation function of X(t)$X\left(t\right)$.     

Tsirel'son's equation in discrete time

Summary Motivated by Tsirel'son's equation in continuous time, a similar stochastic equation indexed by discrete negative time is discussed in full generality, in terms of the law of a discrete time noise. When uniqueness in law holds, the unique solution (in law) is not strong; moreover, when there exists a strong solution, there are several strong solution. In general, for any time,n, the -field generated by the past of a solution up to timen is shown to be equal, up to negligible sets, to the -field generated by the 3 following components: the infinitely remote past of the solution, the past to the noise up to timen, together with an adequate independent complement.     

Sample path Large Deviations and optimal importance sampling for stochastic volatility models

Sample path Large Deviation Principles (LDP) of the Freidlin–Wentzell type are derived for a class of diffusions, which govern the price dynamics in common stochastic volatility models from Mathematical Finance. LDP are obtained by relaxing the non-degeneracy requirement on the diffusion matrix in the standard theory of Freidlin and Wentzell. As an application, a sample path LDP is proved for the price process in the Heston stochastic volatility model.     

Curve forecasting by functional autoregression

This paper deals with the prediction of curve-valued autoregression processes. It develops a novel technique, predictive factor decomposition, for the estimation of the autoregression operator. The technique is based on finding a reduced-rank approximation to the autoregression operator that minimizes the expected squared norm of the prediction error.Implementing this idea, we relate the operator approximation problem to the singular value decomposition of a combination of cross-covariance and covariance operators. We develop an estimation method based on regularization of the empirical counterpart of this singular value decomposition, prove its consistency and evaluate convergence rates.The method is illustrated by an example of the term structure of the Eurodollar futures rates. In the sample corresponding to the period of normal growth, the predictive factor technique outperforms the principal components method and performs on a par with custom-designed prediction methods.     

On free stochastic processes and their derivatives

We study a family of free stochastic processes whose covariance kernels K$K$ may be derived as a transform of a tempered measure σ$\sigma$. These processes arise, for example, in consideration of non-commutative analysis involving free probability. Hence our use of semi-circle distributions, as opposed to Gaussians. In this setting we find an orthonormal basis in the corresponding non-commutative L²$L^2$ of sample-space. We define a stochastic integral for our family of free processes.     

Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs

We consider the optimal financing and dividend control problem of the insurance company with fixed and proportional transaction costs. The management of the company controls the reinsurance rate, dividends payout as well as the equity issuance process to maximize the expected present value of the dividends payout minus the equity issuance until the time of bankruptcy. This is the first time that the financing process in an insurance model with two kinds of transaction costs, which come from real financial market has been considered. We solve the mixed classical-impulse control problem by constructing two categories of suboptimal models, one is the classical model without equity issuance, the other never goes bankrupt by equity issuance.     

Applications of the degree theorem to absolute continuity on Wiener space

Summary Let (,H, P) be an abstract Wiener space and define a shift on byT()=+F() whereF is anH-valued random variable. We study the absolute continuity of the measuresPºT ^–1and ( _F P)ºT ¹ with respect toP using the techniques of the degree theory of Wiener maps, where _F =det₂(1+F) × Exp{–F–1/2|F|²}.The work of the second author was supported by the fund for promotion of research at the Technion     

Stochastic optimal multi-modes switching with a viscosity solution approach

We consider the problem of optimal multi-modes switching in finite horizon, when the state of the system, including the switching cost functions are arbitrary (_gij(t,x)≥0$g_{ij}(t,x)\ge 0$). We show existence of the optimal strategy, via a verification theorem. Finally, when the state of the system is a Markov process, we show that the vector of value functions of the optimal problem is the unique viscosity solution to the system of m$m$ variational partial differential inequalities with inter-connected obstacles.     

Local times of ranked continuous semimartingales

Given a finite collection of continuous semimartingales, we derive a semimartingale decomposition of the corresponding ranked (order-statistics) processes. We apply the decomposition to extend the theory of equity portfolios generated by ranked market weights to the case where the stock values admit triple points.     

The Gapeev-Kühn stochastic game driven by a spectrally positive Lévy process

In Gapeev and Kühn (2005) [8], the Dynkin game corresponding to perpetual convertible bonds was considered, when driven by a Brownian motion and a compound Poisson process with exponential jumps. We consider the same stochastic game but driven by a spectrally positive Lévy process. We establish a complete solution to the game indicating four principle parameter regimes as well as characterizing the occurrence of continuous and smooth fit. In Gapeev and Kühn (2005) [8], the method of proof was mainly based on solving a free boundary value problem. In this paper, we instead use fluctuation theory and an auxiliary optimal stopping problem to find a solution to the game.     

Itô’s stochastic calculus: Its surprising power for applications

We trace Itô’s early work in the 1940s, concerning stochastic integrals, stochastic differential equations (SDEs) and Itô’s formula. Then we study its developments in the 1960s, combining it with martingale theory. Finally, we review a surprising application of Itô’s formula in mathematical finance in the 1970s. Throughout the paper, we treat Itô’s jump SDEs driven by Brownian motions and Poisson random measures, as well as the well-known continuous SDEs driven by Brownian motions.