Stochastic Flows Of Diffeomorphisms Research Articles

Hedging options in a hidden Markov‐switching local‐volatility model via stochastic flows and a Monte‐Carlo method

AbstractThe hedging of European contingent claims in a continuous‐time hidden Markov‐regime‐switching diffusion model is discussed using stochastic flows of diffeomorphisms and Monte‐Carlo simulations. Specifically, the price dynamics of an underlying risky asset are governed by a continuous‐time hidden Markov‐modulated local‐volatility model. Filtering theory is used to estimate the unobservable drift given observable price information and to define a filtered market with complete observations. The delta–hedge ratio of a European option is derived using a martingale representation and stochastic flows of diffeomorphisms. The numerical computation of the delta–hedge ratio is estimated via Monte‐Carlo simulations. Numerical results for illustrating the proposed method and the (relative) importance of the impacts of the information risk and the local‐volatility parametrizations on the delta–hedge ratio are provided for the case of European call options.

HEDGING OPTIONS IN A DOUBLY MARKOV-MODULATED FINANCIAL MARKET VIA STOCHASTIC FLOWS

The hedging of a European-style contingent claim is studied in a continuous-time doubly Markov-modulated financial market, where the interest rate of a bond is modulated by an observable, continuous-time, finite-state, Markov chain and the appreciation rate of a risky share is modulated by a continuous-time, finite-state, hidden Markov chain. The first chain describes the evolution of credit ratings of the bond over time while the second chain models the evolution of the hidden state of an underlying economy over time. Stochastic flows of diffeomorphisms are used to derive some hedge quantities, or Greeks, for the claim. A mixed filter-based and regime-switching Black–Scholes partial differential equation is obtained governing the price of the claim. It will be shown that the delta hedge ratio process obtained from stochastic flows is a risk-minimizing, admissible mean-self-financing portfolio process. Both the first-order and second-order Greeks will be considered.

A Stochastic Flows Approach for Asset Allocation with Hidden Economic Environment

An optimal asset allocation problem for a quite general class of utility functions is discussed in a simple two-state Markovian regime-switching model, where the appreciation rate of a risky share changes over time according to the state of a hidden economy. As usual, standard filtering theory is used to transform a financial model with hidden information into one with complete information, where a martingale approach is applied to discuss the optimal asset allocation problem. Using a martingale representation coupled with stochastic flows of diffeomorphisms for the filtering equation, the integrand in the martingale representation is identified which gives rise to an optimal portfolio strategy under some differentiability conditions.

A Note on Differentiability in a Markov Chain Market Using Stochastic Flows

Using stochastic flows of diffeomorphisms relating to a Markov chain together with the Itô's differentiation rule, the differentiability of the price of a European-style contingent claim with respect to the underlying state variables is proved in a continuous-time Markov chain market. The differentiability results are also used to calculate the Greeks for hedging.

Markovian forward–backward stochastic differential equations and stochastic flows

Markovian forward–backward stochastic differential equations, (MFBSDEs), are discussed by exploiting techniques of stochastic flows. Using martingale representation, a differentiation rule, stochastic flows of diffeomorphisms and the unique decomposition of special semimartingales, we identify the solution of the backward system of the FBSDE. Applications of the result to convex risk measures are discussed.

A stochastic-Lagrangian particle system for the Navier–Stokes equations

This paper is based on a formulation of the Navier–Stokes equations developed by Constantin and the first author (Commun. Pure Appl. Math. at press, arXiv:math.PR/0511067), where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. In this paper, we take N copies of the above process (each based on independent Wiener processes), and replace the expected value with 1/N times the sum over these N copies. (We note that our formulation requires one to keep track of N stochastic flows of diffeomorphisms, and not just the motion of N particles.)We prove that in two dimensions, this system of interacting diffeomorphisms has (time) global solutions with initial data in the space C1,α which consists of differentiable functions whose first derivative is α Hölder continuous (see section 3 for the precise definition). Further, we show that as N → ∞ the system converges to the solution of Navier–Stokes equations on any finite interval [0, T]. However for fixed N, we prove that this system retains roughly O(1/N) times its original energy as t → ∞. Hence the limit N → ∞ and T → ∞ do not commute. For general flows, we only provide a lower bound to this effect. In the special case of shear flows, we compute the behaviour as t → ∞ explicitly.

Pesin's entropy formula for stochastic flows of diffeomorphisms

Pesin's entropy formula relating entropy and Lyapunov exponents within the context of random dynamical systems generated by (discrete or continuous) stochastic flows of diffeomorphisms (including solution flows of stochastic differential equations on manifolds) is proved.

Limit theorems for stochastic flows of diffeomorphisms of jump type

Limit theorems for stochastic flows of diffeomorphisms of jump type

On the convergence of solutions of stochastic ordinary differential equations as stochastic flows of diffeomorphisms

Soient F(τ, x, t, ω) et G(τ, x, t, ω) des champs vectoriels aleatoires sur R d a parametres de temps satisfaisant certaines proprietes statistiques. Soit ψ t e une solution de dx/dt=eF(e 2 t, x, t ω)+e 2 G(e 2 t, x, t, ω). On etudie la limite faible de Φ t e≡Ψ t/ e 2 e