SDE of Mean-Field Type
We consider the system of mean-field type SDEs
begin{align}
dX_t &= sigma(t,X_t, P(X_t) ) dB_t, \
dY_t &= E[ phi(Y_t) X_t ] d B_t
end{align}
In a paper I am studying it is claimed that a unique solution exists if: $sigma$ is Lipschitz-continuous and $phi$ is Lipschitz continuous and bounded. From my point of view this is not enough, it should also be required that the function
$$
(x,y) mapsto phi(x)y
$$
is Lipschitz continuous. Maybe someone is familiar with these types of SDEs and could explain if I am missing something and if the assumptions in the paper are indeed enough to obtain existence and uniqueness.
sde
asked Nov 30 at 9:59
White
729
add a comment |
We consider the system of mean-field type SDEs
begin{align}
dX_t &= sigma(t,X_t, P(X_t) ) dB_t, \
dY_t &= E[ phi(Y_t) X_t ] d B_t
end{align}
In a paper I am studying it is claimed that a unique solution exists if: $sigma$ is Lipschitz-continuous and $phi$ is Lipschitz continuous and bounded. From my point of view this is not enough, it should also be required that the function
$$
(x,y) mapsto phi(x)y
$$
is Lipschitz continuous. Maybe someone is familiar with these types of SDEs and could explain if I am missing something and if the assumptions in the paper are indeed enough to obtain existence and uniqueness.
sde
asked Nov 30 at 9:59
White
729
add a comment |
We consider the system of mean-field type SDEs
begin{align}
dX_t &= sigma(t,X_t, P(X_t) ) dB_t, \
dY_t &= E[ phi(Y_t) X_t ] d B_t
end{align}
In a paper I am studying it is claimed that a unique solution exists if: $sigma$ is Lipschitz-continuous and $phi$ is Lipschitz continuous and bounded. From my point of view this is not enough, it should also be required that the function
$$
(x,y) mapsto phi(x)y
$$
is Lipschitz continuous. Maybe someone is familiar with these types of SDEs and could explain if I am missing something and if the assumptions in the paper are indeed enough to obtain existence and uniqueness.
sde
asked Nov 30 at 9:59
White
729
We consider the system of mean-field type SDEs
begin{align}
dX_t &= sigma(t,X_t, P(X_t) ) dB_t, \
dY_t &= E[ phi(Y_t) X_t ] d B_t
end{align}
In a paper I am studying it is claimed that a unique solution exists if: $sigma$ is Lipschitz-continuous and $phi$ is Lipschitz continuous and bounded. From my point of view this is not enough, it should also be required that the function
$$
(x,y) mapsto phi(x)y
$$
is Lipschitz continuous. Maybe someone is familiar with these types of SDEs and could explain if I am missing something and if the assumptions in the paper are indeed enough to obtain existence and uniqueness.
sde
sde
asked Nov 30 at 9:59
White
729
asked Nov 30 at 9:59
White
729
asked Nov 30 at 9:59
White
729
asked Nov 30 at 9:59
White
729
asked Nov 30 at 9:59
White
729
729
add a comment |
add a comment |
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