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We develop and analyze a method, density tracking by quadrature (DTQ), to compute the probability density function of the solution of a stochastic differential equation. The derivation of the method begins with the discretization in time of the stochastic differential equation, resulting in a discrete-time Markov chain with continuous state space. At each time step, DTQ applies quadrature to solve the Chapman-Kolmogorov equation for this Markov chain. In this paper, we focus on a particular case of the DTQ method that arises from applying the Euler-Maruyama method in time and the trapezoidal quadrature rule in space. Our main result establishes that the density computed by DTQ converges in $L^1$ to both the exact density of the Markov chain (with exponential convergence rate), and to the exact density of the stochastic differential equation (with first-order convergence rate). We establish a Chernoff bound that implies convergence of a domain-truncated version of DTQ. We carry out numerical tests to show that the empirical performance of DTQ matches theoretical results, and also to demonstrate that DTQ can compute densities several times faster than a Fokker-Planck solver, for the same level of error.
We propose an unbiased Monte-Carlo estimator for $mathbb{E}[g(X_{t_1}, cdots, X_{t_n})]$, where $X$ is a diffusion process defined by a multi-dimensional stochastic differential equation (SDE). The main idea is to start instead from a well-chosen sim
We develop algorithms for computing expectations of the laws of models associated to stochastic differential equations (SDEs) driven by pure Levy processes. We consider filtering such processes and well as pricing of path dependent options. We propos
We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show well-posednes
We study convergence behaviors of degenerate and non-reversible stochastic differential equations. Our method follows a Lyapunov method in probability density space, in which the Lyapunov functional is chosen as a weighted relative Fisher information
We develop in this work a numerical method for stochastic differential equations (SDEs) with weak second order accuracy based on Gaussian mixture. Unlike the conventional higher order schemes for SDEs based on It^o-Taylor expansion and iterated It^o