Solution Stochastic Process Research Articles

Random matrix difference models arising in long-term medical drug strategies

This paper deals with the construction of random discrete solutions of coupled linear difference equations, incorporating uncertainty into both the initial condition and the source term. First, sufficient conditions in order to guarantee mean square stability of the solution are provided, then the main statistical functions, such as mean and covariance of the discrete solution stochastic process, are given. Finally, illustrative examples of potential interest in long-time medical drug strategies are shown.

Random Airy type differential equations: Mean square exact and numerical solutions

This paper deals with the construction of power series solutions of random Airy type differential equations containing uncertainty through the coefficients as well as the initial conditions. Under appropriate hypotheses on the data, we establish that the constructed random power series solution is mean square convergent over the whole real line. In addition, the main statistical functions, such as the mean and the variance, of the approximate solution stochastic process generated by truncation of the exact power series solution are given. Finally, we apply the proposed technique to several illustrative examples which show substantial speed-up and improvement in accuracy compared to other approaches such as Monte Carlo simulations.

An explicitly solvable multi‐scale stochastic volatility model: Option pricing and calibration problems

AbstractWe introduce an explicitly solvable multiscale stochastic volatility model that generalizes the Heston model. The model describes the dynamics of an asset price and of its two stochastic variances using a system of three Ito stochastic differential equations. The two stochastic variances vary on two distinct time scales and can be regarded as auxiliary variables introduced to model the dynamics of the asset price. Under some assumptions, the transition probability density function of the stochastic process solution of the model is represented as a one‐dimensional integral of an explicitly known integrand. In this sense the model is explicitly solvable. We consider the risk‐neutral measure associated with the proposed multiscale stochastic volatility model and derive formulae to price European vanilla options (call and put) in the multiscale stochastic volatility model considered. We use the thus‐obtained option price formulae to study the calibration problem, that is to study the values of the model parameters, the correlation coefficients of the Wiener processes defining the model, and the initial stochastic variances implied by the “observed” option prices using both synthetic and real data. In the analysis of real data, we use the S&P 500 index and to the prices of the corresponding options in the year 2005. The web site http://www.econ.univpm.it/recchioni/finance/w7 contains some auxiliary material including some animations that helps the understanding of this article. A more general reference to the work of the authors and their coauthors in mathematical finance is the web site http://www.econ.univpm.it/recchioni/finance. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:862–893, 2009

Mean square power series solution of random linear differential equations

This paper deals with the construction of random power series solutions of linear differential equations containing uncertainty through the diffusion coefficient, the source term as well as the initial condition. Under appropriate hypotheses on the data, we establish that the constructed random power series solution is mean square convergent in a certain interval whose length depends on the mean square norm of the random variable coefficient. Also, the main statistical functions of the approximating stochastic process solution generated by truncation of the exact power series solution are given. Finally, we apply the proposed technique to several illustrative examples.

Weak order for the discretization of the stochastic heat equation

In this paper we study the approximation of the distribution of $X_t$ Hilbert--valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as $$ dX_t+AX_t \, dt = Q^{1/2} d W_t, \quad X_0=x \in H, \quad t\in[0,T], $$ driven by a Gaussian space time noise whose covariance operator $Q$ is given. We assume that $A^{-\alpha}$ is a finite trace operator for some $\alpha>0$ and that $Q$ is bounded from $H$ into $D(A^\beta)$ for some $\beta\geq 0$. It is not required to be nuclear or to commute with $A$. The discretization is achieved thanks to finite element methods in space (parameter $h>0$) and implicit Euler schemes in time (parameter $\Delta t=T/N$). We define a discrete solution $X^n_h$ and for suitable functions $\varphi$ defined on $H$, we show that $$ |\E \, \varphi(X^N_h) - \E \, \varphi(X_T) | = O(h^{2\gamma} + \Delta t^\gamma) $$ \noindent where $\gamma<1- \alpha + \beta$. Let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations.

Random analytic solution of coupled differential models with uncertain initial condition and source term

This paper deals with the construction of random power series solution of vector initial value problems containing uncertainty in both initial condition and source term. Under appropriate hypothesis on the data, we prove that the random series solution constructed by a random Fröbenius method is convergent in the mean square sense. Also, the main statistical functions of the approximating stochastic process solution generated by truncation of the exact series solution are given. Finally, we apply the proposed technique to several illustrative examples.

Maximum likelihood estimation of the parameters of a system of stochastic differential equations that models the returns of the index of some classes of hedge funds

Abstract In this paper we study an inverse problem for a parabolic partial differential equation. The parabolic partial differential equation considered is the Fokker Planck equation associated to a system of stochastic differential equations and the inverse problem studied consists in finding from suitable data the values of the parameters that appear in the coefficients of this Fokker Planck equation. The data used in the reconstruction of the parameters are observations made at discrete times of the stochastic process solution of the system of stochastic differential equations. That is, the data of the inverse problem are a “sample” taken at discrete times of some of the components of the random vector solution of the stochastic differential equations and not, as usual, observations made on the solution of the parabolic equation. The choice of the system of stochastic differential equations and of the data used in the inverse problem are motivated by applications in mathematical finance. The stochastic differential equations presented can be used to model the dynamics of the log-returns of the index of some classes of hedge funds, such as, for example, the so called “long short equity” hedge funds and of some auxiliary variables. The solution of the inverse problem proposed is obtained through the solution of a filtering and of an estimation problem. The solution of these last two problems is based on the knowledge of the joint probability density function of the state variables of the model conditioned to the observations made and to the initial condition. This joint probability density function is solution of an initial value problem for the Kushner equation that in the circumstances considered here can be written as a sequence of initial value problems for the Fokker Planck equation associated to the system of stochastic differential equations with appropriate initial conditions. An integral representation formula for this probability density function is derived and used to develop a numerical procedure to solve the estimation problem using the maximum likelihood method. The Kushner equation provides the relation between the data and the Fokker Planck equation used to solve the inverse problem considered. The computational method proposed has been tested on synthetic data and the results obtained are presented. Some auxiliary material useful to understand this paper including some animations and some numerical experiments can be found in the website http://www.econ.univpm.it/recchioni/finance/w5. A more general reference to the work in mathematical finance of the authors and of their coauthors is the website http://www.econ.univpm.it/recchioni/finance.

Self-interacting diffusions. III. Symmetric interactions

Let M be a compact Riemannian manifold. A self-interacting diffusion on M is a stochastic process solution to $$dX_{t}=dW_{t}(X_{t})-\frac{1}{t}\biggl(\int_{0}^{t}\nabla V_{X_{s}}(X_{t})\,ds\biggr)\,dt,$$ where {Wt} is a Brownian vector field on M and Vx(y)=V(x,y) a smooth function. Let $\mu_{t}=\frac{1}{t}\int_{0}^{t}\delta_{X_{s}}\,ds$ denote the normalized occupation measure of Xt. We prove that, when V is symmetric, μt converges almost surely to the critical set of a certain nonlinear free energy functional J. Furthermore, J has generically finitely many critical points and μt converges almost surely toward a local minimum of J. Each local minimum has a positive probability to be selected.

Positivity of the density for the stochastic wave equation in two spatial dimensions

We consider the random vector , where t > 0, x1 ,...,xd are distinct points of and u denotes the stochastic process solution to a stochastic wave equation driven by a noise white in time and correlated in space. In a recent paper by Millet and Sanz–Sole [10], sufficient conditions are given ensuring existence and smoothness of density for . We study here the positivity of such density. Using techniques developped in [1] (see also [9]) based on Analysis on an abstract Wiener space, we characterize the set of points where the density is positive and we prove that, under suitable assumptions, this set is .

A variance reduction method for parametrized stochastic differential equations using the reduced basis paradigm

In this work, we develop a reduced-basis approach for the efficient computation of parametrized expected values, for a large number of parameter values, using the control variate method to reduce the variance. Two algorithms are proposed to compute online, through a cheap reduced-basis approximation, the control variates for the computation of a large number of expectations of a functional of a parametrized Ito stochastic process (solution to a parametrized stochastic differential equation). For each algorithm, a reduced basis of control variates is pre-computed offline, following a so-called greedy procedure, which minimizes the variance among a trial sample of the output parametrized expectations. Numerical results in situations relevant to practical applications (calibration of volatility in option pricing, and parameter-driven evolution of a vector field following a Langevin equation from kinetic theory) illustrate the efficiency of the method.