We establish a simple criterion for locating points where the transition density of a degenerate diffusion is strictly positive. Throughout, we assume that the diffusion satisfies a stochastic differential equation (SDE) on R d with additive noise and polynomial drift. In this setting, we will see that it is often that case that local information of the flow, e.g. the Lie algebra generated by t...

We consider Itô’s stochastic differential equation (SDE). First, we would review a standard theory under standard assumptions. Then we would see how such a standard theory should be modified under different and more general assumptions and how and what new notions need be introduced to discuss such modifications. Consider the following SDE on R dX (t) = σ k(X(t))dB (t) + b(X(t))dt, X(0) = x (1)...

1 What follows is my attempt to understand a set of ideas being developed by Boris Tsirelson. I do this by studying a specific, and I hope interesting, example. Tanaka's SDE is one of the easiest examples of a stochastic differential equation with no strong solution. Suppose X t ; t ≥ 0 is a real-valued Brownian motion starting from zero and we put B t = t 0 sgn(X s)dX s then B is also a Browni...

and Applied Analysis 3 For the sake of clarity in the presentation we recall that in a jump-diffusion model, the modified stochastic differential equation SDE for the underlying asset is dS S μdt σdz ( η − 1dq, 1.1 where S is the underlying stock price, μ is the drift rate, σ is the volatility, dz is the increment of Gauss-Wiener process, and dq is the Poisson process. The random variable repre...

The second order random walk (RW2) model is commonly used for smoothing data and for modelling response functions. It is computationally efficient due to the Markov properties of the joint (intrinsic) Gaussian density. For evenly spaced locations the RW2 model is well established, whereas for irregularly spaced locations there is no well established construction in the literature. By considerin...

The second order random walk (RW2) model is commonly used for smoothing data and for modelling response functions. It is computationally efficient due to the Markov properties of the joint (intrinsic) Gaussian density. For evenly spaced locations the RW2 model is well established, whereas for irregularly spaced locations there is no well established construction in the literature. By considerin...

We consider stochastic differential equations driven by some Volterra processes. Under time reversal, these equations are transformed into past dependent stochastic differential equations driven by a standard Brownian motion. We are then in position to derive existence and uniqueness of solutions of the Volterra driven SDE considered at the beginning.

An innovative theoretical framework for stochastic dynamics based on a decomposition of a stochastic differential equation (SDE) has been developed with an evident advantage in connecting deterministic and stochastic dynamics, as well as useful applications in physics, engineering, chemistry and biology. It introduces the A-type stochastic integration for SDE beyond traditional Ito’s or Straton...

We study an agent-based stock market model with heterogeneous agents and friction. Our model is based on that of [9]: the process of a stock price in a discrete-time framework is determined by temporary equilibria via agents’ excess demand functions, and the diffusion approximation approach is applied to characterize the continuous-time limit (as transaction intervals shorten) as a solution of ...

We consider Backward Stochastic Differential Equations (BSDEs) with generators that grow quadratically in the control variable. In a more abstract setting, we first allow both the terminal condition and the generator to depend on a vector parameter x. We give sufficient conditions for the solution pair of the BSDE to be differentiable in x. These results can be applied to systems of forward-bac...