| Abstract: |
| Stochastic It\^o--McKean--Vlasov equation is considered in $R^d$,
\[
dX_t = B[t,X_t, \mu_t]dt + \Sigma[t,X_t, \mu_t]dW_t, \qquad
X_0=x_0,
\]
under the convention
\[
B[t,x,\mu]=\int b(t,x,y)\mu(dy), \;\; \Sigma[t,x,\mu]=\int \sigma(t,x,y)\mu(dy).
\]
Here $W$ is a standard $d_1$-dimensional Wiener process, $b$ and $\sigma$ are, respectively, vector and matrix Borel functions of corresponding dimensions $d$ and $d\times d_1$, $\mu_t$ is the distribution of the process $X$ at $t$. The initial data $x_0$ may be random, but independent of $W$.
New weak and strong existence and weak and strong uniqueness results for the equation are established under relaxed regularity conditions as well as a ``propagation of chaos`` (convergence of multi-particle approximations).
The talk is based on a joint work with Yulia Mishura and on discussions and a work in progress with David \v{S}iska and Lukasz Szpruch. |
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