Existence and Uniqueness of Solutions
As with deterministic ordinary and partial differential equations, it is important to know whether a given SDE has a solution, and whether or not it is unique. The following is a typical existence and uniqueness theorem for Itō SDEs taking values in n-dimensional Euclidean space Rn and driven by an m-dimensional Brownian motion B; the proof may be found in Øksendal (2003, §5.2).
Let T > 0, and let
be measurable functions for which there exist constants C and D such that
for all t ∈ and all x and y ∈ Rn, where
Let Z be a random variable that is independent of the σ-algebra generated by Bs, s ≥ 0, and with finite second moment:
Then the stochastic differential equation/initial value problem
has a Pr-almost surely unique t-continuous solution (t, ω) |→ Xt(ω) such that X is adapted to the filtration FtZ generated by Z and Bs, s ≤ t, and
Read more about this topic: Stochastic Differential Equation
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