Girsanov Research Articles

A fractional credit model with long range dependent default rate

Motivated by empirical evidence of long range dependence in macroeconomic variables like interest rates we propose a fractional Brownian motion driven model to describe the dynamics of the short and the default rate in a bond market. Aiming at results analogous to those for affine models we start with a bivariate fractional Vasicek model for short and default rate, which allows for fairly explicit calculations. We calculate the prices of corresponding defaultable zero-coupon bonds by invoking Wick calculus. Applying a Girsanov theorem we derive today’s prices of European calls and compare our results to the classical Brownian model.

A Malliavin Calculus method to study densities of additive functionals of SDE’s with irregular drifts

On introduit une méthode générale qui permet l’utilisation du Calcul de Malliavin pour des fonctionnelles additives générées par des équations stochastiques avec une dérive irrégulière. Cette méthode utilise le théorème de Girsanov avec l’expansion d’Itô–Taylor pour obtenir la régularité de la densité. On applique cette méthodologie pour au cas de l’intégrale en temps d’une diffusion avec derive mesurable bornée.

A continuum-tree-valued Markov process

We present a construction of a L\'evy continuum random tree (CRT) associated with a super-critical continuous state branching process using the so-called exploration process and a Girsanov's theorem. We also extend the pruning procedure to this super-critical case. Let $\psi$ be a critical branching mechanism. We set $\psi_\theta(\cdot)=\psi(\cdot+\theta)-\psi(\theta)$. Let $\Theta=(\theta_\infty,+\infty)$ or $\Theta=[\theta_\infty,+\infty)$ be the set of values of $\theta$ for which $\psi_\theta$ is a branching mechanism. The pruning procedure allows to construct a decreasing L\'evy-CRT-valued Markov process $(\ct_\theta,\theta\in\Theta)$, such that $\mathcal{T}_\theta$ has branching mechanism $\psi_\theta$. It is sub-critical if $\theta>0$ and super-critical if $\theta<0$. We then consider the explosion time $A$ of the CRT: the smaller (negative) time $\theta$ for which $\mathcal{T}_\theta$ has finite mass. We describe the law of $A$ as well as the distribution of the CRT just after this explosion time. The CRT just after explosion can be seen as a CRT conditioned not to be extinct which is pruned with an independent intensity related to $A$. We also study the evolution of the CRT-valued process after the explosion time. This extends results from Aldous and Pitman on Galton-Watson trees. For the particular case of the quadratic branching mechanism, we show that after explosion the total mass of the CRT behaves like the inverse of a stable subordinator with index 1/2. This result is related to the size of the tagged fragment for the fragmentation of Aldous' CRT.

Girsanov identities for Poisson measures under quasi-nilpotent transformations

We prove a Girsanov identity on the Poisson space for anticipating transformations that satisfy a strong quasi-nilpotence condition. Applications are given to the Girsanov theorem and to the invariance of Poisson measures under random transformations. The proofs use combinatorial identities for the central moments of Poisson stochastic integrals.

Exploring behaviors of stochastic differential equation models of biological systems using change of measures

Stochastic Differential Equations (SDE) are often used to model the stochastic dynamics of biological systems. Unfortunately, rare but biologically interesting behaviors (e.g., oncogenesis) can be difficult to observe in stochastic models. Consequently, the analysis of behaviors of SDE models using numerical simulations can be challenging. We introduce a method for solving the following problem: given a SDE model and a high-level behavioral specification about the dynamics of the model, algorithmically decide whether the model satisfies the specification. While there are a number of techniques for addressing this problem for discrete-state stochastic models, the analysis of SDE and other continuous-state models has received less attention. Our proposed solution uses a combination of Bayesian sequential hypothesis testing, non-identically distributed samples, and Girsanov's theorem for change of measures to examine rare behaviors. We use our algorithm to analyze two SDE models of tumor dynamics. Our use of non-identically distributed samples sampling contributes to the state of the art in statistical verification and model checking of stochastic models by providing an effective means for exposing rare events in SDEs, while retaining the ability to compute bounds on the probability that those events occur.

Stochastic stability of viscoelastic system under non-Gaussian colored noise excitation

This article examines a viscoelastic plate that is driven parametrically by a non-Guassian colored noise, which is simplified to an Ornstein-Uhlenbeck process based on the approximation method. To examine the moment stability property of the viscoelastic system, we use the stochastic averaging method, Girsanov theorem and Feynmann-Kac formula to derive the approximate analytic expansion of the moment Lyapunov exponent. Furthermore, the Monte Carlo simulation results for the original system are given to check the accuracy of the approximate analytic results. At the end of this paper, results are presented to show some quantitative pictures of the effects of the system parameters, noise parameters and viscoelastic parameters on the stability of the viscoelastic plate.

From Dynamic Linear Evaluation Rule to Dynamic CAPM in a Fractional Brownian Motion Environment

In this paper, we present the fundamental framework of the evaluation problem under which the evaluation operator satisfying some axioms is linear. Based on the dynamic linear evaluation mechanism of contingent claims, studying this evaluation rule in the market driven by fractional Brownian motions has led to a dynamic capital asset pricing model. It is deduced here mainly with the fractional Girsanov theorem and the Clark-Haussmann-Ocone theorem.

On the Existence and Stability of Pareto Optimal Endogenous Matching with Fairness

In the current paper, we study the asymmetric normal-form game between two heterogeneous groups of populations by employing the stochastic replicator dynamics driven by Levy process. A new game equilibrium, i.e., the game equilibrium of a stochastic differential cooperative game on time, is derived by introducing optimal-stopping technique into evolutionary game theory, which combines with the Pareto optimal standard leads us to the existence of Pareto optimal endogenous matching. Moreover, stability of the Pareto optimal endogenous matching is confirmed by essentially using the well-known Girsanov Theorem.

On SDE associated with continuous-state branching processes conditioned to never be extinct

We study the pathwise description of a (sub-)critical continuous-state branching process (CSBP) conditioned to be never extinct, as the solution to a stochastic differential equation driven by Brownian motion and Poisson point measures. The interest of our approach, which relies on applying Girsanov theorem on the SDE that describes the unconditioned CSBP, is that it points out an explicit mechanism to build the immigration term appearing in the conditioned process, by randomly selecting jumps of the original one. These techniques should also be useful to represent more general $h$-transforms of diffusion-jump processes.

Local kinetic interpretation of entropy production through reversed diffusion

The time reversal of stochastic diffusion processes is revisited with emphasis on the physical meaning of the time-reversed drift and the noise prescription in the case of multiplicative noise. The local kinematics and mechanics of free diffusion are linked to the hydrodynamic description. These properties also provide an interpretation of the Pope-Ching formula for the steady-state probability density function along with a geometric interpretation of the fluctuation-dissipation relation. Finally, the statistics of the local entropy production rate of diffusion are discussed in the light of local diffusion properties, and a stochastic differential equation for entropy production is obtained using the Girsanov theorem for reversed diffusion. The results are illustrated for the Ornstein-Uhlenbeck process.

Semilinear backward doubly stochastic differential equations and SPDEs driven by fractional Brownian motion with Hurst parameter in [formula omitted

We study the existence of a unique solution to semilinear fractional backward doubly stochastic differential equation driven by a Brownian motion and a fractional Brownian motion with Hurst parameter less than 1/2. Here the stochastic integral with respect to the fractional Brownian motion is the extended divergence operator and the one with respect to Brownian motion is Itôʼs backward integral. For this we use the technique developed by R. Buckdahn (1994) [3] to analyze stochastic differential equations on the Wiener space, which is based on the Girsanov theorem and the Malliavin calculus, and we reduce the backward doubly stochastic differential equation to a backward stochastic differential equation driven by the Brownian motion. We also prove that the solution of semilinear fractional backward doubly stochastic differential equation defines the unique stochastic viscosity solution of a semilinear stochastic partial differential equation driven by a fractional Brownian motion.

A Williams decomposition for spatially dependent superprocesses

We present a genealogy for superprocesses with a non-homogeneous quadratic branching mechanism, relying on a weighted version of the superprocess introduced by Englander and Pinsky and a Girsanov theorem. We then decompose this genealogy with respect to the last individual alive (Williams' decomposition). Letting the extinction time tend to infinity, we get the Q-process by looking at the superprocess from the root, and define another process by looking from the top. Examples including the multitype Feller diffusion (investigated by Champagnat and Roelly) and the superdiffusion are provided.

A Girsanov Type Theorem Under G-Framework

This article establishes a Girsanov type theorem under the G-Framework of Peng [15]. Our result generalizes the classical Girsanov theorem for Brownian motion [10]. As an application, we price the European call option when the underlying asset's price follows the Geometric G-Brownian motion.

Applied Multidimensional Girsanov Theorem

The present article is meant as a bridge between theory and practice concerning Girsanov theorem. In the first part we give theoretical results leading to a straightforward three step process allowing to express an asset's dynamics in a new probability measure. In the following sections we apply this three step process. The first application consists in expressing a foreign asset's dynamics in domestic currency which leads to the well known quanto adjustment formula. The second application is the expression of the forwards' dynamics under Libor Market Model using a unique probability measure. The final application consists in expressing foreign short rate Hull & White dynamics under domestic measure.

Option Pricing and Hedging under Stochastic Verhulst-Gompertz Equation

Emphasis is placed on the valuation of plain vanilla option when the price process of underlying asset is described by the stochastic Verhulst-Gompertz Equation with network externality effects in a complete market. The method is based on the change of measure, Girsanov theorem and martingale valuation techinique. The application to an exchange option is made attempt and the valuation formula for this option like the Black-Scholes one is derived. A simple relationship similar to the put-call-parity between the exchange call option and the put option is provided. Our results will be useful to analyze and to hedge the price evolution at a sudden rise or crash of stock and commodity markets. 1 Introduction. The prices of some assets traded in the financial market have the ten- dency raised more when the prices begin to rise, it is because most traders expect that a price rises more and the purchase orders of other traders are induced as a result. Conversely, when the prices begin to fall, selling orders of other traders are induced. In the actual mar- ket, we often observe such a phenomenon called network externality price effects. These effects have been recognized as very significant concept in the field of marketing. Nerlove and Arrow(8) , Bass(2), Vidale and Wolfe(11) and Gould(4) made use of this deterministic version to study the dynamic behavior of new product and advertising policy. Their models were stimulated by a classical theory of logistic curve by Verhulst(10). Gompertz extended Verhulst's model to so-called a deterministic Verhulst-Gompertz model in order to investi- gate the growth of population. In the fields of finance, Schwartz(9) gave a stochastic version of Verhulst-Gompertz model and developed the random behavior of commodity price. His model was based on the biological problem studied by Goel and Richer-Dyn(3). The main purpose of this paper is devoted to describe the network externality effects by a generalized stochastic Verhulst-Gompertz equation and to derive pricing formulae for a plain vanilla and an exchange options when the price of underlying asset shows the network externality price effects. The organization of this paper is as follows. The next section briefly introduces the generalized stochastic Verhulst-Gompertz equation to express the network externality effects and in Section 3 we develope the martingale measure to evaluate the value of option. In Section 4 we derive a hedging strategy and a pricing formula of an exchange option that gives the holder the right which exchanges the underlying asset with the stochastic Verhulst-Gompertz equation for the asset with the geometric Brownian motion (Black-Scholes model) by means of martingale pricing method(7). If we regard the former as the spot price of energy commodity and the latter as the stock price of business firm that trades the energy commodity, such an exchang option will be very useful to hedge the sudden rise or heavy fall in price of energy and make a contribution to the stabilized price of energy. The paper concludes with a brief summary.

White Noise Generalization of the Clark-Ocone Formula Under Change of Measure

We prove the white noise generalization of the Clark-Ocone formula under change of measure by using Gaussian white noise analysis and Malliavin calculus. Let W(t) be a Brownian motion on the filtered white noise probability space (Ω, ℬ, {ℱ t }0≤t≤T , P) and let be defined as , where u(t) is an ℱ t -measurable process satisfying certain conditions for all 0 ≤ t ≤ T. Let Q be the probability measure equivalent to P such that is a Brownian motion with respect to Q, in virtue of the Girsanov theorem. In this article, it is shown that for any square integrable ℱ T -measurable random variable, where 𝔼 Q is the expectation under Q and D · F(ω) is the (Hida) Malliavin derivative. The important point in this settlement is F does not have to be in stochastic Sobolev space 𝔻1, 2 ⊂ L 2(P). This makes the formula more useful in applications of finance. As an example, the replicating portfolio for a digital option with the payoff χ[K, ∞) W(T) ∉ 𝔻1, 2 is calculated by using this generalized Clark-Ocone formula under change of measure.

Stochastic financial models

Portfolio Choice Introduction Utility Mean-variance analysis The Binomial Model One-period model Multi-period model A General Discrete-Time Model One-period model Multi-period model Brownian Motion Introduction Hitting-time distributions Girsanov's theorem Brownian motion as a limit Stochastic calculus The Black-Scholes Model Introduction The Black-Scholes formula Hedging and the Black-Scholes equation Path-dependent claims Dividend-paying assets Interest-Rate Models Introduction Survey of interest-rate models Gaussian random-field model Appendix A: Mathematical Preliminaries Appendix B: Solutions to the Exercises Further Reading References Index Exercises appear at the end of each chapter.

Unconstrained recursive importance sampling

We propose an unconstrained stochastic approximation method of finding the optimal measure change (in an a priori parametric family) for Monte Carlo simulations. We consider different parametric families based on the Girsanov theorem and the Esscher transform (or exponential-tilting). In a multidimensional Gaussian framework, Arouna uses a projected Robbins-Monro procedure to select the parameter minimizing the variance. In our approach, the parameter (scalar or process) is selected by a classical Robbins-Monro procedure without projection or truncation. To obtain this unconstrained algorithm we intensively use the regularity of the density of the law without assume smoothness of the payoff. We prove the convergence for a large class of multidimensional distributions and diffusion processes. We illustrate the effectiveness of our algorithm via pricing a Basket payoff under a multidimensional NIG distribution, and pricing a barrier options in different markets.

Modal identification based on Gaussian continuous time autoregressive moving average model

A new time-domain modal identification method of the linear time-invariant system driven by the non-stationary Gaussian random force is presented in this paper. The proposed technique is based on the multivariate continuous time autoregressive moving average (CARMA) model. This method can identify physical parameters of a system from the response-only data. To do this, we first transform the structural dynamic equation into the CARMA model, and subsequently rewrite it in the state-space form. Second, we present the exact maximum likelihood estimators of parameters of the continuous time autoregressive (CAR) model by virtue of the Girsanov theorem, under the assumption that the uniformly modulated function is approximately equal to a constant matrix over a very short period of time. Then, based on the relation between the CAR model and the CARMA model, we present the exact maximum likelihood estimators of parameters of the CARMA model. Finally, the modal parameters are identified by the eigenvalue analysis method. Numerical results show that the method we introduced here not only has high precision and robustness, but also has very high computing efficiency. Therefore, it is suitable for real-time modal identification.

Modal identification of system driven by lévy random excitation based on continuous time AR model

Based on the continuous time AR model, this paper presents a new time-domain modal identification method of LTI system driven by the uniformly modulated levy random excitation. The structural dynamic equation is first transformed into the observation equation and the state equation (namely, stochastic differential equation). Based on the property of the strong solution of the stochastic differential equation, the uniformly modulated function is identified piecewise. Then by virtue of the Girsanov theorem, we present the exact maximum likelihood estimators of parameters. Finally, the modal parameters are identified by eigen analysis. Numerical results show that the method not only has high precision and robustness but also has very high computing efficiency.