324 research outputs found
Mutual Fund Theorem for continuous time markets with random coefficients
We study the optimal investment problem for a continuous time incomplete
market model such that the risk-free rate, the appreciation rates and the
volatility of the stocks are all random; they are assumed to be independent
from the driving Brownian motion, and they are supposed to be currently
observable. It is shown that some weakened version of Mutual Fund Theorem holds
for this market for general class of utilities; more precisely, it is shown
that the supremum of expected utilities can be achieved on a sequence of
strategies with a certain distribution of risky assets that does not depend on
risk preferences described by different utilities.Comment: 17 page
On prescribed change of profile for solutions of parabolic equations
Parabolic equations with homogeneous Dirichlet conditions on the boundary are
studied in a setting where the solutions are required to have a prescribed
change of the profile in fixed time, instead of a Cauchy condition. It is shown
that this problem is well-posed in L_2-setting. Existence and regularity
results are established, as well as an analog of the maximum principle
Universal estimate of the gradient for parabolic equations
We suggest a modification of the estimate for weighted Sobolev norms of
solutions of parabolic equations such that the matrix of the higher order
coefficients is included into the weight for the gradient. More precisely, we
found the upper limit estimate that can be achieved by variations of the zero
order coefficient. As an example of applications, an asymptotic estimate was
obtained for the gradient at initial time. The constant in the estimates is the
same for all possible choices of the dimension, domain, time horizon, and the
coefficients of the parabolic equation. As an another example of application,
existence and regularity results are obtained for parabolic equations with time
delay for the gradient.Comment: 15 page
Predictability of band-limited, high-frequency, and mixed processes in the presence of ideal low-pass filters
Pathwise predictability of continuous time processes is studied in
deterministic setting. We discuss uniform prediction in some weak sense with
respect to certain classes of inputs. More precisely, we study possibility of
approximation of convolution integrals over future time by integrals over past
time. We found that all band-limited processes are predictable in this sense,
as well as high-frequency processes with zero energy at low frequencies. It
follows that a process of mixed type still can be predicted if an ideal
low-pass filter exists for this process.Comment: 10 page
Parabolic equations with the second order Cauchy conditions on the boundary
The paper studies some ill-posed boundary value problems on semi-plane for
parabolic equations with homogenuous Cauchy condition at initial time and with
the second order Cauchy condition on the boundary of the semi-plane. A class of
inputs that allows some regularity is suggested and described explicitly in
frequency domain. This class is everywhere dense in the space of square
integrable functions.Comment: 7 page
Regularity of a inverse problem for generic parabolic equations
The paper studies some inverse boundary value problem for simplest parabolic
equations such that the homogenuous Cauchy condition is ill posed at initial
time. Some regularity of the solution is established for a wide class of
boundary value inputs.Comment: 9 page
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