Intrinsic stochastic differential equations as jets

Armstrong, John; Brigo, Damiano

doi:10.1098/rspa.2017.0559

Cited by 20 publications

(61 citation statements)

References 38 publications

(106 reference statements)

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“…However, Itô SDEs involve explicit second‐order effects, so that there is an actual difference in applying the tangent vector projection or the full metric projection, going beyond the linear term, in approximating a SDE on a submanifold. As we pointed out in , an Itô SDE can be interpreted as a 2‐jet. It is then not completely surprising that the second‐order terms of the metric projection play an important role in understanding the projection of SDEs.…”

Section: Projecting Stochastic Differential Equationsmentioning

confidence: 88%

“…We will say that an SDE on a manifold

M

driven by

m

‐dimensional Brownian motion

W_{t}

is defined by choosing at each point

x \in M

and time

t

a smooth map

γ_{x, t} : {double-struckR}^{m} \to M, γ false(0 false) = x .

For a given time interval

δ t

and starting point

X_{0}

, we may then define a process,

X^{δ t}

, by requiring that at times

(δ t, 2 δ t, 3 δ t, \dots)

we have

X_{t + δ t}^{δ t} = γ_{X_{t}^{δ t}, t} false(W_{t + δ t} - W_{t} false) .

We define

X_{t + ε}^{δ t} = X_{t}^{δ t}

0 ⩽ ε ⩽ δ t

. In the case of autonomous SDEs (where

γ_{x, t}

is independent of

t)

, it is shown in that

X^{δ t}

converges in mean square on compacts to a unique process

X

so long as the choice of

γ_{x}

at each point is made smoothly. Moreover, the limiting process depends only on the 2‐jet of

γ_{x}

at each point and coincides with the solution to the SDE given in a local chart by

…”

Section: The Jet Formulation Of Sdes On Manifoldsmentioning

confidence: 99%

“…Conversely, given an SDE of the form on

R^{d}

we define

γ_{x, t} false(V false) = b_{α} false(X, t false) V^{α} + \frac{1}{m} g_{α β}^{E} V^{α} V^{β},

where

V^{α}

are the components of the vector

V

and

g_{α β}^{E}

is the Euclidean metric on

R^{m}

(hence is equal as a matrix to the identity). The results of were given in the case of autonomous SDEs, but the generalization is straightforward.…”

Section: The Jet Formulation Of Sdes On Manifoldsmentioning

confidence: 99%

“…By this we mean that SDE defined on the manifold

M

transforms according to Itô's lemma as we change chart

ψ

. See for a more detailed discussion of the Itô formulation of SDEs on manifolds.…”

Section: Projecting Stochastic Differential Equationsmentioning

confidence: 99%

“…It is interesting to note that one can draw a diagram to show the Itô‐jet projection. In , it is discussed how the jet formulation of SDEs makes it possible to draw pictures of SDEs that transform according to Itô's lemma. For processes driven by one‐dimensional Brownian motion, one simply finds functions

γ_{x}

whose 2‐jet represents the SDE and then draws the image of an interval

[- ε, ε]

under the map

γ_{x}

at each point

x

.…”

Section: A Low‐dimensional Example: Cross Diffusion On a Unit Circlementioning

confidence: 99%

See 4 more Smart Citations

Optimal approximation of SDEs on submanifolds: the Itô‐vector and Itô‐jet projections

Armstrong

Brigo

Rossi

2018

Proc. London Math. Soc.

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We define two new notions of projection of a stochastic differential equation (SDE) onto a submanifold: the Itô‐vector and Itô‐jet projections. This allows one to systematically develop low‐dimensional approximations to high‐dimensional SDEs using differential geometric techniques. The approach generalizes the notion of projecting a vector field onto a submanifold in order to derive approximations to ordinary differential equations, and improves the previous Stratonovich projection method by adding optimality analysis and results. Indeed, just as in the case of ordinary projection, our definitions of projection are based on optimality arguments and give in a well‐defined sense ‘optimal’ approximations to the original SDE in the mean‐square sense over small times. We also explain how the Stratonovich projection satisfies an optimality criterion that is more ad hoc and less appealing than the criteria satisfied by the Itô projections we introduce. As an application, we consider approximating the solution of the non‐linear filtering problem with a Gaussian distribution. We show how the newly introduced Itô projections lead to optimal approximations in the Gaussian family and briefly discuss the optimal approximation for more general families of distributions. We perform a numerical comparison of our optimally approximated filter with the classical Extended Kalman Filter to demonstrate the efficacy of the approach.

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Section: Projecting Stochastic Differential Equationsmentioning

confidence: 88%

“…We will say that an SDE on a manifold

M

driven by

m

‐dimensional Brownian motion

W_{t}

is defined by choosing at each point

x \in M

and time

t

a smooth map

γ_{x, t} : {double-struckR}^{m} \to M, γ false(0 false) = x .

For a given time interval

δ t

and starting point

X_{0}

, we may then define a process,

X^{δ t}

, by requiring that at times

(δ t, 2 δ t, 3 δ t, \dots)

we have

X_{t + δ t}^{δ t} = γ_{X_{t}^{δ t}, t} false(W_{t + δ t} - W_{t} false) .

We define

X_{t + ε}^{δ t} = X_{t}^{δ t}

0 ⩽ ε ⩽ δ t

. In the case of autonomous SDEs (where

γ_{x, t}

is independent of

t)

, it is shown in that

X^{δ t}

converges in mean square on compacts to a unique process

X

so long as the choice of

γ_{x}

at each point is made smoothly. Moreover, the limiting process depends only on the 2‐jet of

γ_{x}

at each point and coincides with the solution to the SDE given in a local chart by

…”

Section: The Jet Formulation Of Sdes On Manifoldsmentioning

confidence: 99%

“…Conversely, given an SDE of the form on

R^{d}

we define

γ_{x, t} false(V false) = b_{α} false(X, t false) V^{α} + \frac{1}{m} g_{α β}^{E} V^{α} V^{β},

where

V^{α}

are the components of the vector

V

and

g_{α β}^{E}

is the Euclidean metric on

R^{m}

(hence is equal as a matrix to the identity). The results of were given in the case of autonomous SDEs, but the generalization is straightforward.…”

Section: The Jet Formulation Of Sdes On Manifoldsmentioning

confidence: 99%

“…By this we mean that SDE defined on the manifold

M

transforms according to Itô's lemma as we change chart

ψ

. See for a more detailed discussion of the Itô formulation of SDEs on manifolds.…”

Section: Projecting Stochastic Differential Equationsmentioning

confidence: 99%

γ_{x}

whose 2‐jet represents the SDE and then draws the image of an interval

[- ε, ε]

under the map

γ_{x}

at each point

x

.…”

Section: A Low‐dimensional Example: Cross Diffusion On a Unit Circlementioning

confidence: 99%

See 3 more Smart Citations

Optimal approximation of SDEs on submanifolds: the Itô‐vector and Itô‐jet projections

Armstrong

Brigo

Rossi

2018

Proc. London Math. Soc.

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Itô Stochastic Differential Equations as 2-Jets

Armstrong

Brigo

2017

Lecture Notes in Computer Science

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We explain how Itô Stochastic Differential Equations on manifolds may be defined as 2-jets of curves and show how this relationship can be interpreted in terms of a convergent numerical scheme. We use jets as a natural language to express geometric properties of SDEs. We explain that the mainstream choice of Fisk-Stratonovich-McShane calculus for stochastic differential geometry is not necessary. We give a new geometric interpretation of the Itô-Stratonovich transformation in terms of the 2-jets of curves induced by consecutive vector flows. We discuss the forward Kolmogorov equation and the backward diffusion operator in geometric terms. In the one-dimensional case we consider percentiles of the solutions of the SDE and their properties. In particular the median of a SDE solution is associated to the drift of the SDE in Stratonovich form for small times.

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Stochastic Port-Hamiltonian Systems

2022

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In the present work we formally extend the theory of port-Hamiltonian systems to include random perturbations. In particular, suitably choosing the space of flow and effort variables we will show how several elements coming from possibly different physical domains can be interconnected in order to describe a dynamic system perturbed by general continuous semimartingale. Relevant enough, the noise does not enter into the system solely as an external random perturbation, since each port is itself intrinsically stochastic. Coherently to the classical deterministic setting, we will show how such an approach extends existing literature of stochastic Hamiltonian systems on pseudo-Poisson and pre-symplectic manifolds. Moreover, we will prove that a power-preserving interconnection of stochastic port-Hamiltonian systems is a stochastic port-Hamiltonian system as well.

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Intrinsic stochastic differential equations as jets

Cited by 20 publications

References 38 publications

Optimal approximation of SDEs on submanifolds: the Itô‐vector and Itô‐jet projections

Optimal approximation of SDEs on submanifolds: the Itô‐vector and Itô‐jet projections

Itô Stochastic Differential Equations as 2-Jets

Stochastic Port-Hamiltonian Systems

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