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Lyapunov Exponents of Linear and
Stochastic
Functional Differential Equations Driven by Semimartingales, Part I:
The
Multiplicative Ergodic Theory (with M. Scheutzow), *Annals of
Institute
of Henri Poincare, Probabilites e'tet Statistiques* Vol.
32,
1, (1996), 69-105.

``Degenerate
Stochastic Differential Equations, Flows and Hypoellipticity," (with
Denis Bell) (Invited Paper), AMS Summer Research Institute on
Stochastic
Analysis, Ithaca, Cornell 1993,
*Proceedings of Symposia in Pure Mathematics,
American Mathematical Society* , Vol. 57, Stochastic Analysis,
American
Mathematical Society, Providence, Rhode Island (1995), 553-564.

(i)Existence of smooth densities for solutions of stochastic hereditary equations whose covariances degenerate polynomially (anywhere) on hypersurfaces in $R^d$.

(ii)Existence of smooth densities for diffusions with degeneracies of infinite order on a collection of hypersurfaces in $R^d$.

(iii)Extension and refinement of Hormander's hypoellipticity theorem
for a large class of highly degenerate second order parabolic
operators:
Hormander's Lie algebra condition is allowed to fail exponentially fast
on the degeneracy hypersurfaces, which are imbedded in submanifolds of
dimension less than $d$. The exponential decay rate near the
degeneracy
surface is found to be
*optimal.*

Our proofs are based on the Malliavin calculus and require new sharp
estimates for Ito processes in Euclidean space.

``An
Extension of Hormander's Theorem for Infinitely Degenerate Second-Order
Operators," (with Denis Bell), *Duke Mathematical Journal*,
Vol.78,
No. 3, (1995), 453-475.

``Smooth
Densities for Degenerate Stochastic Delay Equations with Hereditary
Drift"
(with Denis Bell),
*The Annals of Probability*, Vol. 23, No. 4, (1995),
1875-1894.

dx(t) = H(t,x) dt g(t,x(t - r)) dW(t).

In the above equation, $W$ is an $R^n$-dimensional
Wiener process, $r$
is a positive time delay, $H$ is a non-anticipating functional
defined
on the space of paths in $R^d$ and $g$ is an $R^{nxd}$-matrix-valued
function
defined on $[0,\infty)xR^d$ such that $gg^*$ has degeneracies of
polynomial
order on a hypersurface in $R^d$. In the course of proving
this result,
we establish a very general criterion for the hypoellipticity of a
class
of degenerate parabolic second-order time-dependent differential
operators
with space-independent principal part.

``Lyapunov
Exponents of Linear Stochastic Functional Differential Equations, Part
II: Examples and Case Studies" (With M. Scheutzow), (preprint
1995)
*The
Annals of Probability* , Vol. 25, No. 3, (1997), 1210-1240. (*Reprint
*available
upon request).

``Spatial Estimates for Stochastic Flows
in
Euclidean
Space" (With M. Scheutzow),
*The Annals of Probability*, Vol. 26,
No. 1, (1998), 56-77. (*Reprint
*available
upon request.)

We give an example of a one-dimensional s.d.e. with *sublinear *coefficients
but with the underlying stochastic flow growing *superlinearly *for
large $x$. In this example the stochastic flow has a.s.
unbounded
spatial
derivatives, even though the driving martingale has local
characteristics
with all derivatives globally bounded. It is interesting to note that
in
this example the driving noise is *infinite-dimensional*. However
the infinite-dimensionality of the driving noise is *not* the
crucial
factor. To illustrate this point we provide an example of a s.d.e.
driven
by *one-dimensional* Brownian motion, has coefficients with
globally
bounded derivatives, while its stochastic flow has a.s. unbounded
derivatives.
This result is surprising since it is in sharp contrast with well-known
behavior of deterministic flows driven by vector fields whose
derivatives
are globally bounded.

For one-dimensional s.d.e.'s, sufficient conditions on the coefficients are given in order for the stochastic flow to have sublinear growth and a.s. bounded derivatives.

It is expected that the results would be of interest for the theory
of stochastic flows on non-compact manifolds as well as in the study of
non-linear filtering, stochastic functional and partial differential
equations.

+ACY-nbsp;

``Stochastic Differential Systems with
Memory:
Theory, Examples and Applications", pp. 91, for *Proceedings of the
Sixth
Oslo-Silivri Workshop, Geilo, Norway, July 29-August 4, 1996*
(Invited
presentation of six hourly lectures) (*preprint*
is now available upon request), *Stochastic Analysis and Related
Topics
VI. The Geilo Workshop, 1996 *, ed. L. Decreusefond, Jon Gjerde, B.
Oksendal, A.S. Ustunel, Progress in Probability, Birkhauser (1998),
1-77.

+ACY-nbsp;

``The
Stable Manifold Theorem for Stochastic Differential Equations"
(With
M. Scheutzow), (*preprint*,
available upon request) (
MSRI Preprint 1998-015 and list of errata,
.dvi file), *The Annals of Probabilit*y, Vol. 27, No.
2,
(1999),
615-652.

``The Dirichlet problem for
superdegenerate
differential
operators" (with Denis Bell), *C.R. Acad. Sci. Paris (French Academy
of Sciences), t. 327, serie I (1998), 81 - 86 *(*
preprint* available upon request).

``The
Stable Manifold Theorem for Nonlinear Stochastic Systems with
Memory
I: Existence of the Semiflow" (with M.K.R. Scheutzow), *Journal
of
Functional Analysis, *205, (2003), 271-305 (communicated by L.
Gross)
(*preprint*
available
upon request).

``The
Stable Manifold Theorem for Nonlinear
Stochastic
Systems with Memory II: The Local Stable Manifold Theorem" (with
M.K.R.
Scheutzow), *Journal of Functional Analysis*,
206, (2004), 253-306 (communicated
by L. Gross) (*preprint*
available upon request).

Preprint:

``Stochastic Functional Differential Equations on Manifolds " (with R. Leandre),

**Preprint:**

``Discrete-time Approximations of Stochastic Differential Systems with Memory " (with Y. Hu and F. Yan), (2001) pp. 71 (

The convergence orders of the Euler-Maruyama and Milstein schemes are $0.5$ and $1$ respectively. In order to establish the Milstein scheme, we prove an infinite-dimensional Ito formula for ``tame" functions acting on the segment process of the solution of an SDDE. It is interesting to note that the presence of the memory in the SDDE requires the use of the Malliavin calculus together with the anticipating stochastic analysis of Nualart and Pardoux.

**Preprint:**

``Discrete-time Approximations of Stochastic Delay Equations: The Milstein Scheme " (with Y. Hu and F. Yan),

**Preprint:**

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`` The Stable Manifold Theorem for
Semilinear
SPDEs" (with Tusheng Zhang and Huaizhong Zhao), (2003) pp. 6 (*preprint*
available upon request).

**Abstract:**

The main objective of this work is to characterize the pathwise
local
structure of solutions of semilinear stochastic partial differential
equations
(spde's) near stationary solutions. We first prove general
existence
theorems for smooth compacting semiflows of semilinear spde's
and
stochastic evolution equations (see's). We then establish *local
stable manifold theorems* for these infinite-dimensional stochastic
dynamical systems. In particular, these results give a random family of
Frechet smooth stable and unstable manifolds in a neighborhood of a
hyperbolic
stationary solution. The unstable and stable manifolds are stationary,
asymptotically invariant under the stochastic semiflow and
have fixed
finite dimension and codimension, respectively.

``Hartman-Grobman
theorems along hyperbolic stationary trajectories" (with E. A.
Coayla-Teran and P. R. C. Ruffino), Discrete
and
Continuous Dynamical Systems-Series, vol. 17, no. 2,
281-292 (2007).

Abstract:

Hartman-Grobman theorems are proved for continuous
stochastic
dynamical systems near hyperbolic stationary
trajectories. A topological conjugacy is
established
between traveling neighbourhoods of trajectories and and
neighbourhoods of the origin in the corresponding tangent bundle. The
results apply to stochastic flows generated by sode's.

Preprint:

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``A stochastic Calculus for
Systems with
Memory" (with F. Yan), Stochastic
Analysis and Applications, vol. 23, no. 3, 613-657 (2005) (*preprint*
available upon request).

Abstract:

For a given stochastic process *X*, its *segment** X*_{t}
at time *t*, represents the "slice" of each path of *X*
over a fixed time-interval [*t*-*r*, *t*],
where *r*
is the length of the "memory" of the process. Segment processes are
important in the study of stochastic systems with memory (stochastic
functional differential equations or sfde's). The main objective of
this
paper is to study nonlinear transforms of segment processes. Toward
this end, we construct a stochastic integral with respect to the
Brownian segment process. The difficulty in this construction is the
fact that the stochastic integrator is infinite dimensional and is not
a (semi)martingale. We overcome this difficulty by employing Malliavin
(anticipating) calculus techniques. The segment integral is interpreted
as a Skorohod integral via a stochastic Fubini theorem. We then prove
Ito's formula for the segment of a continuous Skorohod-type
process and
embed the segment calculus in the theory of anticipating calculus.
Applications of the Ito formula include the weak infinitesimal
generator for the solution segment of a stochastic system with memory,
the associated Feynman-Kac formula, and the Black-Scholes pde for stock
dynamics with memory.

Preprint:

``Absolute
Continuity of Stationary Measure-valued Processes Generated by
Stochastic Equations with Interaction" (with A. Pilipenko), Theory of Stochastic Process, 2005,
vol. 11(27), No.1-2, pp. 17 (*preprint*
available upon request).

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``The Weak
Euler
Scheme for Stochastic
Delay Equations" (with E. Buckwar, R. Kuske and T. Shardlow), London
Mathematical Society Journal of Computation and Mathematics,
2008, 11, (60-99) (*preprint*
available upon request).

**Abstract:**

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``Large Deviations
for
Stochastic
Systems with Memory" (with Tusheng Zhang), Discrete and
Continuous Dynamical Systems-Series B, 2006, vol. 6, No. 4,
881-893 (*preprint*
available upon request).

**Abstract:**

**Preprint:**

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``The Stable Manifold Theorem for Semilinear Stochastic Evolution Equations and Stochastic Partial Differential Equations.

Part 1: The Stochastic Semiflow ",

Part 2: Existence of stable and unstable manifolds"

(with Tusheng Zhang and Huaizhong Zhao), pp. 98 (

Abstract:

The main objective of this work is to characterize the pathwise
local
structure
of solutions of semilinear stochastic evolution equations (see's) and
stochastic
partial differential equations (spde's) near stationary
solutions.
Such characterization is realized through the long-term behavior of the
solution field near stationary points. The analysis falls in
two
parts 1, 2.

In Part 1, we establish a general
existence and compactness theorem for cocycles of
semilinear
see's and spde's. Our results cover a large class of semilinear see's
as
well as certain semilinear spde's with non-Lipschitz terms such as
stochastic
reaction diffusion equations and the stochastic Burgers equation with
additive
infinite-dimensional noise. We adopt a notion of stationarity
employed
in previous work of Mohammed with M. Scheutzow.

In Part 2 of this work, we give a characterization
of the pathwise local structure of solutions of semilinear
stochastic
evolution equations (see's) and stochastic partial differential
equations
(spde's) near stationary solutions. The characterization is
expressed
in terms of the almost sure long-time behavior of trajectories of the
equation
in relation to the stationary solution. More specifically, we establish
*local
stable manifold theorems* for semilinear see's and spde's.
These results give smooth stable and unstable manifolds in the
neighborhood
of
a hyperbolic stationary solution of the underlying stochastic
equation.
The stable and unstable manifolds are stationary, live in a stationary
tubular neighborhood of the stationary solution and are
asymptotically
invariant under the stochastic semiflow of the see/spde. The proof uses
infinite-dimensional multiplicative ergodic theory techniques and
interpolation
arguments.

**Preprint:**

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``The
Substitution
Theorem for Semilinear Stochastic Partial Differential Equations"
(with Tusheng Zhang), Journal of
Functional Analysis, vol. 253, no. 1, (2007), 122-157.

Abstract:

In this article we establish a substitution theorem for semilinear
stochastic evolution equations (see's) depending on the initial
condition as an infinite-dimensional parameter. Due to the
infinite-dimensionality of the initial conditions and of the stochastic
dynamics, existing finite-dimensional results do not apply. The
substitution theorem is proved using Malliavin calculus techniques
together with new estimates on the underlying stochastic semiflow.
Applications of the theorem include dynamic characterizations of
solutions of stochastic partial differential equations (spde's) with
anticipating initial conditions and non-ergodic stationary solutions.
In particular, our result gives a new existence theorem for solutions
of semilinear Stratonovich spde's with anticipating initial
conditions.

**Preprint:**

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``A Delayed Black and Scholes Formula"
(with
M. Arriojas, Y. Hu and G. Pap), Journal
of Stochastic Analysis and Applications+AH0-, vol. 25, no. 2
(2007),
471 - 492 (preprint
available
upon request).

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``Stochastic
Dynamical Systems in Infinite Dimensions", Trends in Stochastic Analysis,
edited by Jochen Blath, Peter Morters and Michael Scheutzow,

London Mathematical Society Lecture Note Series, Cambridge University
Press,
(2009), pp. 30.

Abstract:

In this article, we summarize some results on the existence and qualitative behavior of stochastic dynamical systems in infinite dimensions. The three main examples covered are stochastic systems with finite memory (stochastic functional differential equations-sfde's), semilinear stochastic evolution equations (see's) and stochastic partial differential equations (spde's). Due to limitations of space, our summary is by no means intended to be exhaustive: The emphasis is mainly on the local behavior of infinite-dimensional stochastic dynamical systems near hyperbolic equilibria (or stationary solutions).**Preprint:**

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``Anticipating
stochastic differential systems with memory" (with T. S. Zhang),
Mittag-Leffler Institute Preprint Series -12, 2007/2008, fall,
pp.32.

Abstract:

This article establishes existence and uniqueness of solutions to two classes of stochastic systems with finite memory subject to anticipating initial conditions which are sufficiently smooth in the Malliavin sense. The two classes are semilinear stochastic functional differential equations (sfde's) and fully nonlinear sfde's with a sublinear drift term. For the semilinear case, we use Malliavin calculus techniques, existence of the stochastic semiflow and an infinite-dimensional substitution theorem. For the fully nonlinear case, we employ an anticipating version of the Ito-Ventzell formula due to Ocone and Pardoux. In both cases, the use of Malliavin calculus techniques is necessitated by the infinite-dimensionality of the initial condition.------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

``Anticipating
stochastic differential systems with memory" (with T. S. Zhang), Stochastic Processes and
their Applications (2009).

Abstract:

This article establishes existence and uniqueness of solutions to two classes of stochastic systems with finite memory subject to anticipating initial conditions which are sufficiently smooth in the Malliavin sense. The two classes are semilinear stochastic functional differential equations (sfde's) and fully nonlinear sfde's with a sublinear drift term. For the semilinear case, we use Malliavin calculus techniques, existence of the stochastic semiflow and an infinite-dimensional substitution theorem. For the fully nonlinear case, we employ an anticipating version of the Ito-Ventzell formula due to Ocone and Pardoux. In both cases, the use of Malliavin calculus techniques is necessitated by the infinite-dimensionality of the initial condition.**Preprint:**

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``Dynamics of Stochastic 2D Navier-Stokes Equations" (with T. S. Zhang), Journal of Functional Analysis, Volume 258, Issue 10, (2010), 3543-3591.

Abstract:

In this paper, we study the dynamics of a two-dimensional stochastic Navier-Stokes equation on a smooth domain, driven by multiplicative white noise. We show that solutions of the 2D Navier-Stokes equation generate a smooth perfect and locally compacting $C^{1,1}$- cocycle. Using multiplicative ergodic theory techniques, we establish the existence of a discrete non-random Lyapunov spectrum for the cocycle. The Lyapunov spectrum characterizes the asymptotics of the cocycle near the zero equilibrium solution. We give sufficient conditions on the parameters of the Navier-Stokes equation and the geometry of the planar domain which guarantee hyperbolicity of the equilibrium, uniqueness of the stationary solution (viz. ergodicity), local almost sure asymptotic stability of the cocycle, and the existence of global invariant foliations of the energy space.Preprint:

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``Burgers
Equation with Affine Linear Noise: Dynamics and Stability" (with
T. S. Zhang) (2011).

Abstract:

The main objective of this article is to analyse the dynamics of Burgers equation on the unit interval, driven by multiplicative white noise.It is shown that the solution field of the stochastic Burgers equation generates a smooth perfect and locally compacting cocycle on the energy space. Using multiplicative ergodic theory techniques, we compute the discrete non-random Lyapunov spectrum of the linearized cocycle. The Lyapunov spectrum characterizes the asymptotics of the cocycle near the zero equilibrium solution. In particular, we construct a countable random family of local asymptotically flow-invariant manifolds, so that on each manifold the cocycle decays with fixed exponential speed. Each local manifold is smooth and has finite fixed codimension. On a global level, we show the existence of a flow-invariant random flag in the energy space. The global random flag is characterized by the Lyapunov spectrum of the linearized cocycle. In the presence of a linear drift, we also give sufficient conditions on the parameters of the stochastic Burgers equation which guarantee uniqueness of the stationary solution or its hyperbolicity. In the hyperbolic case, we establish a local stable manifold theorem near the zero equilibrium.

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``Invariant Manifolds for Stochastic Models in Fluid Dynamics

Abstract:

This paper is a survey of recent results on the dynamics of Stochastic Burgers equation (SBE) and two-dimensional Stochastic Navier–Stokes Equations (SNSE) driven by affine linear noise. Both classes of stochastic partial differential equations are commonly used in modeling fluid dynamics phenomena. For both the SBE and the SNSE, we establish the--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

``Burgers Equation with Affine Linear Noise: Dynamics and Stability" (with T. S. Zhang), Stochastic Processes and their Applications, Volume 122, Issue 4, April 2012, Pages 1887–1916.

Abstract:

The main objective of this article is to analyse the dynamics of Burgers equation on the unit interval, driven by affine linear (multiplicative) white noise. It is shown that the solution field of the stochastic Burgers equation generates a smooth perfect and locally compacting cocycle on the energy space. Using multiplicative ergodic theory techniques, we compute the discrete non-random Lyapunov spectrum of the linearized cocycle. The Lyapunov spectrum characterizes the asymptotics of the cocycle near the zero equilibrium solution. In particular, we construct a countable random family of local asymptotically flow-invariant manifolds, so that on each manifold the cocycle decays with fixed exponential speed. Each local manifold is smooth and has finite fixed codimension. On a global level, we show the existence of a flow-invariant random flag in the energy space. The global random flag is characterized by the Lyapunov spectrum of the linearized cocycle. In the presence of a linear drift, we also give sufficient conditions on the parameters of the stochastic Burgers equation which guarantee uniqueness of the stationary solution or its hyperbolicity. In the hyperbolic case, we establish a local stable manifold theorem near the zero equilibrium. To appear-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

``Stochastic
Burgers Equation with Random Initial Velocities: A Malliavin Calculus
Approach" (with
T. S. Zhang), SIAM Journal of
Mathematical Analysis, 45 (4), (2013), 2396-2420.

Abstract:

In this article we prove an existence theorem for solutions of the stochastic Burgers equation (SBE) on the unit interval with Dirichlet boundary conditions and anticipating initial velocities. The SBE is driven by affine (additive + linear) noise. In order to establish the existence theorem, we adopt a somewhat counter-intuitive perspective in which stochastic dynamical systems ideas lead to existence of solutions rather than vice versa. More specifically, our approach uses the Malliavin calculus and is based on the existence and regularity of a perfect cocycle on the energy space for the SBE. The proof of the existence theorem requires Malliavin regularity of the infinite-dimensional initial velocity field together with new spatial estimates on the cocycle, its Fr\'echet and Malliavin derivatives. The existence theorem provides a dynamic characterization of solutions of the {\it non-anticipating} SBE on its unstable invariant manifolds. Furthermore, as a corollary of the existence theorem, we show that random cocycle-invariant points on the energy space correspond to (possibly non-ergodic) stationary pathwise solutions for the SBE.------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

``Anticipating
Stochastic $2D$ Navier-Stokes Equations" (with
T. S. Zhang), Journal of Functional Analysis, vol. 264,
6, (2013), 1380-1408.

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`` Sobolev
Differentiable Stochastic Flows for SDE's with Singular Coefficients:
Applications to the Transport Equation" (with
T. Nilssen and F. Proske), Annals of Probability, Volume 43, Number 3
(2015), 1535-1576, (pp.
41)

In this paper, we establish
the existence of a stochastic flow of Sobolev
diffeomorphisms
for an SDE driven by a bounded measurable drift and additive
Brownian motion. The result is counter-intuitive, since the
dominant
`culture' in stochastic (and deterministic) dynamical
systems is that the
flow `inherits' its spatial regularity from the driving vector
fields.
The spatial regularity of the stochastic flow yields existence
and
uniqueness of a Sobolev differentiable weak solution of the
(Stratonovich) stochastic transport equation.
It is well-known that
the
deterministic transport equation does not in
general have a solution.

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`` Sobolev Differentiable Stochastic Flows for SDE's with Singular Coefficients: Applications to the Transport Equation" (with T. Nilssen and F. Proske) (2014) (pp. 37) (preprint)

Abstract:

In this paper, we establish the existence of a stochastic flow of Sobolev diffeomorphisms for an SDE driven by a bounded measurable drift and additive Brownian motion. The result is striking, since the dominant `culture' in stochastic (and deterministic) dynamical systems is that the flow `inherits' its spatial regularity from the driving vector fields. The spatial regularity of the stochastic flow yields existence and uniqueness of a Sobolev differentiable weak solution of the (Stratonovich) stochastic transport equation. It is well-known that the deterministic transport equation does not in general have a solution. Using stochastic perturbations and our analysis of the above SDE, we establish a deterministic flow of Sobolev diffeomorphisms for classical one-dimensional (deterministic) ODE's driven by discontinuous vector fields. Furthermore, and as a corollary of the latter result, we construct a Sobolev stochastic flow of diffeomorphisms for one-dimensional SDE's driven by discontinuous diffusion coefficients.

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``Optimal control of systems with noisy memory and BSDEs with Malliavin derivatives " (with B. Oksendal, E. E. Rose and K. R. Dahl ) Journal of Functional Analysis, 271, (2016), 289-329 (pp. 41).

Abstract:

In this article we consider
a stochastic optimal control problem where the dynamics of the state
process, $X(t)$, is a controlled stochastic differential equation with
jumps, delay and noisy memory.
The term noisy memory is, to
the best of our knowledge, new. By this we
mean that the dynamics of the state $X(t)$ depends on
$\int_{t-\delta}^t X(s) dB(s)$ (where $B(t)$ is a Brownian motion).
Hence, the dependence is noisy because of the Brownian motion, and it
involves memory due to the influence from the previous values of the
state process. We derive necessary and sufficient maximum principles
for this stochastic control problem in two different ways, resulting in
two sets of maximum principles. The first set of maximum principles is
derived using Malliavin calculus techniques, while the second set comes
from reduction to a discrete delay optimal control problem, and
application of previously known results by Oksendal, Sulem and Zhang.
The maximum principles also apply to the case where the controller only
has partial information, in the sense that the admissible controls are
adapted to a sub-$\sigma$-algebra of the natural filtration.

**Preprint:**

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May 19, 2016