General Diffusion Processes Research Articles

Closed-form approximations for diffusion densities: a path integral approach

In this paper, we investigate the transition probabilities for diffusion processes. In a first part, we show how transition probabilities for rather general diffusion processes can always be expressed by means of a path integral. For several classical models, an exact calculation is possible, leading to analytical expressions for the transition probabilities and for the maximum probability paths. A second part consists of the derivation of an analytical approximation for the transition probability, which is useful in case the path integral is too complex to be calculated. The approximation we present, is based on a convex combination of a new analytical upper and lower bound for the transition probabilities. The fact that the approximation is analytical has some important advantages, e.g., for the investigation of Asian options. Finally, we demonstrate the accuracy of the approximation by means of some graphical illustrations.

Biased movement at a boundary and conditional occupancy times for diffusion processes

Motivated by edge behaviour reported for biological organisms, we show that random walks with a bias at a boundary lead to a discontinuous probability density across the boundary. We continue by studying more general diffusion processes with such a discontinuity across an interior boundary. We show how hitting probabilities, occupancy times and conditional occupancy times may be solved from problems that are adjoint to the original diffusion problem. We highlight our results with a biologically motivated example, where we analyze the movement behaviour of an individual in a network of habitat patches surrounded by dispersal habitat.

Biased movement at a boundary and conditional occupancy times for diffusion processes

Motivated by edge behaviour reported for biological organisms, we show that random walks with a bias at a boundary lead to a discontinuous probability density across the boundary. We continue by studying more general diffusion processes with such a discontinuity across an interior boundary. We show how hitting probabilities, occupancy times and conditional occupancy times may be solved from problems that are adjoint to the original diffusion problem. We highlight our results with a biologically motivated example, where we analyze the movement behaviour of an individual in a network of habitat patches surrounded by dispersal habitat.

On Multidimensional Generalized Diffusion Processes

We construct a multidimensional generalized diffusion process with the drift coefficient that is the (generalized) derivative of a vector-valued measure satisfying an analog of the Holder condition with respect to volume. We prove the existence and continuity of the density of transition probability of this process and obtain standard estimates for this density. We also prove that the trajectories of the process are solutions of a stochastic differential equation.

Probability distance inequalities on Riemannian manifolds and path spaces

We construct Otto–Villani's coupling for general reversible diffusion processes on a Riemannian manifold. As an application, some new estimates are obtained for Wasserstein distances by using a Sobolev–Poincaré type inequality introduced by Latała and Oleszkiewicz. The corresponding concentration estimates of the measure are presented. Finally, our main result is applied to obtain the transportation cost inequalities on the path space with respect to both of the L 2-distance and the intrinsic distance. In particular, Talagrand's inequality holds on the path space over a compact manifold.

International asset allocation: A new perspective

We consider an international economy where the purchasing power parity (PPP) is violated and financial asset returns and exchange rates follow, in real terms, general diffusion processes driven by K state variables. A country-specific representative individual trades on available assets to maximize the expected utility of her final consumption. Her optimal strategy is shown to contain, in addition to the usual speculative component, only two hedging components, however large is K. The first one is associated with domestic interest rate risk and the second one with the risk brought about by the co-movements of the interest rates and the market prices of risk. The implementation of the strategy thus is much easier than with the traditional Merton decomposition, as it involves estimating the characteristics of the yield curve and the market prices of risk only, rather than those of numerous (and a priori unknown) state variables. In view of the necessity for optimizing agents to account for the (partial) asset return predictability that derives from the investors’ hedging demands at equilibrium, our result significantly lessens the difficulty of achieving the optimal portfolio strategy. The second hedging term turns out to depend on interest rate differentials across countries and to encompass hedging against PPP deviations. Therefore, in contrast with previous models that obtained a (direct) currency risk hedging component in a rather ad hoc manner, our decomposition leads to optimal (indirect) currency risk hedging in a natural and general way. It also provides new insights as to the pricing of foreign exchange risk at equilibrium.

Optimal Strategies for Risk-Sensitive Portfolio Optimization Problems for General Factor Models

We consider constructing optimal strategies for risk-sensitive portfolio optimization problems on an infinite time horizon for general factor models, where the mean returns and the volatilities of individual securities or asset categories are explicitly affected by economic factors. The factors areassumed to be general diffusion processes. In studying the ergodic type Bellman equations of the risk-sensitive portfolio optimization problems, we introduce some auxiliary classical stochastic control problems with the same Bellman equations as the original ones. We show that the optimal diffusion processes of the problem are ergodic and that under some condition related to integrability by the invariant measures of the diffusion processes we can construct optimal strategies for the original problems by using the solution of the Bellman equations.

A conditional limit theorem for generalized diffusion processes

Let $\mathbf{X} = \{ X(t) :t \leq 0\}$ be a one-dimensional generalized diffusion process with initial state $X(0) > 0$, hitting time $\tau _{\mathbf{X}}(0)$ at the origin and speed measure m which is regularly varying at infinity with exponent $1/\alpha -1 > 0$. It is proved that, for a suitable function $u(c)$, the probability law of $\{ u(c)^{-1}X(ct) : 0 c \}$ converges as $c \to \infty$ to the conditioned $2(1-\alpha )$-dimensional Bessel excursion on natural scale and that the latter is equivalent to the $2(1-\alpha )$-dimensional Bessel meander up to a scale transformation. In particular, the distribution of $u(c)^{-1}X(c)$ converges to the Weibull distribution. From the conditional limit theorem we also derive a limit theorem for some of regenerative process associated with $\mathbf{X}$.

Discrete random walk models for space–time fractional diffusion

Abstract A physical–mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. By space–time fractional diffusion equation we mean an evolution equation obtained from the standard linear diffusion equation by replacing the second-order space derivative with a Riesz–Feller derivative of order α∈(0,2] and skewness θ (|θ|⩽min{α,2−α}), and the first-order time derivative with a Caputo derivative of order β∈(0,1]. Such evolution equation implies for the flux a fractional Fick’s law which accounts for spatial and temporal non-locality. The fundamental solution (for the Cauchy problem) of the fractional diffusion equation can be interpreted as a probability density evolving in time of a peculiar self-similar stochastic process that we view as a generalized diffusion process. By adopting appropriate finite-difference schemes of solution, we generate models of random walk discrete in space and time suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation.

Asymptotics of hitting probabilities for general one-dimensional pinned diffusions

We consider a general one-dimensional diffusion process and we study the probability of crossing a boundary for the associated pinned diffusion as the time at which the conditioning takes place goes to zero. We provide asymptotics for this probability as well as a first order development. We consider also the cases of two boundaries possibly depending on the time. We give applications to simulation.

Recovery of volatility coefficient by linearization

We study the problem of reconstruction of the asset price dependent local volatility from market prices of options with different strikes. For a general diffusion process we apply the linearization technique and we conclude that the option price can be obtained as the sum of the Black-Scholes formula and of an explicit functional which is linear in perturbation of volatility. We obtain an integral equation for this functional and we show that under some natural conditions it can be inverted for volatility. We demonstrate the stability of the linearized problem, and we propose a numerical algorithm which is accurate for volatility functions with different properties.

The Valuation of American Options for a Class of Diffusion Processes

We present an integral equation approach for the valuation of American-style derivatives when the underlying asset price follows a general diffusion process and the interest rate is stochastic. Our contribution is fourfold. First, we show that the exercise region is determined by a single exercise boundary under very general conditions on the interest rate and the dividend yield. Second, based on this result, we derive a recursive integral equation for the exercise boundary and provide a parametric representation of the American option price. Third, we apply the results to models with stochastic volatility or stochastic interest rate, and to American bond options in one-factor models. For the cases studied, explicit parametric valuation formulas are obtained. Finally, we extend results on American capped options to general diffusion prices. Numerical schemes based on approximations of the optimal stopping time (such as approximations based on a lower bound, or on a combination of lower and upper bounds) are shown to be valid in this context.

Optimal currency risk hedging

This paper investigates the optimal hedging strategy of a domestic expected utility maximizer endowed with a temporarily non-traded position in a foreign investment. The domestic and foreign yield curves, the exchange rate between the two involved currencies, and the foreign investment value are stochastic and obey fairly general diffusion processes. We compare the hedger's optimal strategies using either exchange rate forward contracts or futures contracts. For both strategies, the optimal number of contracts held can be broken down into several components having clear economic interpretations. However, while futures contracts are known in such a context to be more difficult to price than their forward counterparts, the optimal strategy using them is simpler. This is due to the fact that, when forward contracts are used, incurred profits or losses that accrue to the investor's wealth at each instant are locked-in in the forward position up to the contract maturity. Thus discounting these gains or losses back at the current date brings about an interest rate risk. Therefore the investor's hedging strategy itself generates an additional risk, which in turn induces the need for additional hedging. The magnitude of the difference between the two hedge ratios may be significant under a set of plausible assumptions. Since in general financial markets are not complete, the additional interest rate risk cannot be perfectly hedged. Hence the marking-to-market mechanism that characterizes futures contracts and allows for the complete elimination of this risk is valuable for risk averse agents.

Probability tree algorithm for general diffusion processes.

Motivated by path-integral numerical solutions of diffusion processes, PATHINT, we present a tree algorithm, PATHTREE, which permits extremely fast accurate computation of probability distributions of a large class of general nonlinear diffusion processes.

Mathematical formalism for isothermal linear irreversibility

Mathematical formalism for isothermal linear irreversibility

Understanding the shape of the hazard rate: a process point of view (With comments and a rejoinder by the authors)

Survival analysis as used in the medical context is focused on the concepts of survival function and hazard rate, the latter of these being the basis both for the Cox regression model and of the counting process approach. In spite of apparent simplicity, hazard rate is really an elusive concept, especially when one tries to interpret its shape considered as a function of time. It is then helpful to consider the hazard rate from a different point of view than what is common, and we will here consider survival times modeled as first passage times in stochastic processes. The concept of quasistationary distribution,which is a well-defined entity for various Markov processes, will turn out to be useful. We study these matters for a number of Markov processes, including the following: finite Markov chains; birth-death processes; Wiener processes with and without randomization of parameters; and general diffusion processes. An example of regression of survival data with a mixed inverse Gaussian distribution is presented. The idea of viewing survival times as first passage times has been much studied by Whitmore and others in the context of Wiener processes and inverse Gaussian distributions. These ideas have been in the background compared to more popular appoaches to survival data, at least within the field of biostatistics,but deserve more attention.

Probability Tree Algorithm for General Diffusion Processes

Motivated by path-integral numerical solutions of diffusion processes, PATHINT, we present a new tree algorithm, PATHTREE, which permits extremely fast accurate computation of probability distributions of a large class of general nonlinear diffusion processes.

The Stationary Arrival Process of Independent Diffusers from a Continuum to an Absorbing Boundary Is Poissonian

We consider the arrival process of infinitely many identical independent diffusion processes from an infinite bath to an absorbing boundary. Previous results on this problem were confined to independent Brownian particles arriving at an absorbing sphere. The present paper extends these results to general diffusion processes, without any symmetries and without resorting to explicit expressions for solutions to the relevant equations. It is shown that for general absorbing boundaries and force fields, the steady stream of arrivals is Poissonian with rate equal to the total flux on the absorbing boundary, as calculated from the continuum theory of diffusion with transport. The considered arrival problem arises in the theory of Langevin simulations of ions in electrolytic solutions. In a Langevin simulation ions enter and exit the simulation region, and it is necessary to compute the probability laws for their entrance times into the simulation. While the simulated ions inside the small simulation region interact with each other and with the far field of the surrounding bath and the applied voltage, the physical chemistry continuum description of the surrounding bath implies independent diffusion in a mean field. Under these conditions the result of this paper applies to the stream of new ions that arrive from the continuum bath into the discrete simulation region. The recirculation problem, of ions that have already visited and exited the simulation region, as well as the integration of these results into a simulation of interacting ions will be studied in separate papers.

Continuous-time monotone stochastic recursions and duality

A duality is presented for continuous-time, real-valued, monotone, stochastic recursions driven by processes with stationary increments. A given recursion defines the time evolution of a content process (such as a dam or queue), and it is shown that the existence of the content process implies the existence of a corresponding dual risk process that satisfies a dual recursion. The one-point probabilities for the content process are then shown to be related to the one-point probabilities of the risk process. In particular, it is shown that the steady-state probabilities for the content process are equivalent to the first passage time probabilities for the risk process. A number of applications are presented that flesh out the general theory. Examples include regulated processes with one or two barriers, storage models with general release rate, and jump and diffusion processes.

Continuous-time monotone stochastic recursions and duality

A duality is presented for continuous-time, real-valued, monotone, stochastic recursions driven by processes with stationary increments. A given recursion defines the time evolution of a content process (such as a dam or queue), and it is shown that the existence of the content process implies the existence of a corresponding dual risk process that satisfies a dual recursion. The one-point probabilities for the content process are then shown to be related to the one-point probabilities of the risk process. In particular, it is shown that the steady-state probabilities for the content process are equivalent to the first passage time probabilities for the risk process. A number of applications are presented that flesh out the general theory. Examples include regulated processes with one or two barriers, storage models with general release rate, and jump and diffusion processes.