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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.

5 votes
0 answers
136 views

Functional inverse problem based on a variational principle

I am trying to solve an inverse problem based on variational principle. I will first present a forward problem that is already solved, and then present the inverse problem that I am trying currently t …
can't stop me now's user avatar
5 votes
1 answer
504 views

Riemannian metric induced by a stochastic differential equation

Following this paper, a diffusion process in $\mathcal{R}^d$ $$dX_t = f(X_t) \, dt + \sigma(X_t) \, dW_t ,$$ with $\sigma(x) \in \mathbb{R}^{d \times m}$ and $m$ dimensional Brownian motion can be con …
can't stop me now's user avatar
2 votes
1 answer
200 views

Comparing diffusion processes in different metrics

I would like to know if it is possible to compare two diffusion processes defined on the same manifold $\mathcal{M}$ but with respect to different metrics say $g_1$ and $g_2$. Is there a way to apply …
can't stop me now's user avatar
2 votes
1 answer
418 views

Textbook definition for "path measure" or "probability measure over paths"

I need a formal definition for the path measure for stochastic differential equations. Which textbook or paper should I consult?
can't stop me now's user avatar
1 vote
1 answer
579 views

Expectation of stochastic integral

Let us consider a diffusion process defined as $dX_t = g(X_t,t) \, dt + \sigma \, dW_t$ which induces a path measure $Q$ in the time interval $[0,T]$. Is the following expectation $$ \left\langle \int …
can't stop me now's user avatar
1 vote
Accepted

Riemannian metric induced by a stochastic differential equation

What is missing above perspective is that by adding drift the most probable path for a diffusion is
can't stop me now's user avatar
1 vote
1 answer
547 views

Is there an inverse Lamperti transformation for diffusions?

The Lamperti transformation is commonly used to transform SDEs with state dependent coefficients into SDEs with constant diffusion. For multidimensional processes there are some conditions on the drif …
can't stop me now's user avatar