# Questions tagged [stochastic-differential-equations]

Stochastic ordinary and partial differential equations generalize the concepts of ordinary and partial differential equations to the setting where the unknown is a stochastic process.

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### Malliavin differentiability of solutions to SDEs

In Bass's book on Diffusions and Elliptic Operators, the author gives a brief introduction into Malliavin Calculus. He calls a functional $F:C([0,1],\mathbb{R})\rightarrow \mathbb{R}$ $L^p-$smooth if ...
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### Malliavin derivative under change of measure

Let $\widetilde{B}$ be a Brownian Motion under the measure $\mathbb{P}$. Let $\theta$ be a stochastic process fulfilling the Novikov's condition and $Z_\theta$ the relative Radon–Nikodym derivative ...
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### When does the cumulative distribution function solve the Kolmogorov backward equation?

For a diffusion $X$ define the cumulative distribution function for $X_T$ started with $X_t=x$: $$u(t,x):=E^{t,x}(1_{X_T\ge y})$$ Under what conditions does $u$ solve $X$'s Kolmogorov backward ...
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### Cauchy Problem and stochastic representation for discontinuous initial data

Where can I read more about the Cauchy problem, i.e. solutions to $$\frac{\partial u}{\partial t}+Lu=0 \text{ and } u(0,x)=f(x)$$ for some elliptic differential operator $L$ where $f$ is not ...
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### Does $E^{x,t}(f(X_T))$ solve a PDE if $f$ is not continuous?

Many books [see below for references] explore the connections between partial differential equations and expectation values. Assume $X$ is a diffusion with generator $A$, then they conclude, that ...
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### Wong-Zakai smooth approximation in probabilty for stochastic differential equations

I'm looking for a result of the form: Let $B_\epsilon$ denote a "natural" smooth $\epsilon$-approximation to an $n$-dimensional Brownian motion $B$ (e.g. by mollification or simply piecewise linear) ...
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### Sobolev Bundle on Wiener Space

Right now I am learning about analysis of stochastic processes and the Malliavin calculus. It seems though, that most of the theory works for Brownian motion in $\mathbb{R}^n$, and it seems non-...
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### Analytic Solution to SDEs

Are there any example of SDEs with constant diffusion terms, other than the Ornstein Uhlenbeck process, which have exact solutions? I'm thinking of something of the form: dX_t = f(...
Let $M$ be a compact Riemannian manifold and let $X \in \mathfrak{X}(\mathbb{R}\times M)$ be a time-dependent vector field on $M$. I want to construct the Itô integral $$I(X) = \int_0^T \langle X(t, ... 1answer 738 views ### What is the idea behind interpolation spaces? I am working through a text on Numerics for SPDEs and there the concept an interpolation (Hilbert-)space associated to an operator is used. To be specific: Definition. Let H be an \mathbb{R}-... 1answer 131 views ### Finding a stochastic differential equation as limit of a discrete stochastic equation I'm dealing with the following problem: Choose Z_0 \in [0,1] and define a process governed by the following discrete stochastic equation: Z_{k+1}-Z_k=P_k(1-2Z_k) where P_k=0 with probability ... 1answer 353 views ### Question on viscosity solution through stochastic differential equations I have learned that for the equation \partial_tu+a(u)\partial_xu=0, the entropy solution could be obtained as the limit of the equation \partial_tu+a(u)\partial_xu=\epsilon u_{xx} with \epsilon&... 1answer 607 views ### Onsager-Machlup function and most probable path of a diffusion process Let X_{t} be a real, one-dimensional diffusion process satisfying the stochastic differential equation $$dX_{t} = f(X_{t})dt + dW_{t},$$ where f \in C_{b}^{2}(R) is a ... 1answer 378 views ### Does Brownian motion immediately visit both sides of a Jordan curve? Let C be a Jordan curve in \mathbb{R}^2. By the Jordan curve theorem, \mathbb{R}^2 \smallsetminus C is uniquely partitioned into two connected regions A and B (the interior and exterior). ... 0answers 84 views ### When the completed filtration of a process increases slowly If \mathcal{F}_t is the filtration of the evaluation process on C_T (continuous function on [0,T]). Can we find some law of continuous process \mathbb{P} so that for t\leq T$$\mathcal{F}^{\...
Let $M$ be a (compact) Riemannian manifold. Let $v$ be a smooth vector field on $M$ with flow $\Theta_t$. Let $L$ be an elliptic second order differential operator on $M$ that generates the Ito ...