Stochastic Differential Equations Involving Fractional Brownian Motion

Part of the Lecture Notes in Mathematics book series (LNM, volume 1929)

Existence and Uniqueness of Solutions: the Results of Nualart and Răşcanu

Consider the function \(\sigma = \sigma \left( {t,x} \right):\left[ {0,T} \right] \times {\rm R} \to {\rm R}\) satisfying the assumptions: σ is differentiable in x, there exist and for any \(M > 0,0 < \gamma ,\kappa \le 1\) R > 0 there exists M R > 0 such that
  1. (i)
    σ is Lipschitz continuous in x
    $$\left| {\sigma \left( {t,x} \right) - \sigma \left( {t,y} \right)} \right| \le M\left| {x - y} \right|,\forall t \in \left[ {0,T} \right],x,y \in {\rm R};$$
  2. (ii)
    x-derivative of σ is local Hölder continuous in x:
    $$\left| {\sigma _x \left( {t,x} \right) - \sigma _x \left( {t,y} \right)} \right| \le M_R \left| {x - y} \right|^\kappa ,\forall \left| x \right|,\left| y \right| \le {\rm R,}t \in \left[ {0,T} \right];$$
  3. (iii)
    σ is Hölder continuous in time:
    $$\left| {\sigma \left( {t,x} \right) - \sigma \left( {s,x} \right)} \right| + \left| {\sigma _x \left( {t,x} \right) - \sigma _x \left( {s,x} \right)} \right| \le M\left| {t - s} \right|^\gamma ,\forall x \in {\rm R,}t,s \in \left[ {0,T} \right].$$

Let \(0 < \beta < 1/2,f \in W_0^\beta \left[ {0,T} \right],g \in W_1^{1 - \beta } \left[ {0,T} \right].\) We need some preliminary estimates, in addition to Lemmas 2.1.9 and 2.1.10.

Consider on
$$W_0^\beta \left[ {0,T} \right]$$
the norm, equivalent to \(\left\| \cdot \right\|_{0,\beta } :\)
$$\left\| f \right\|_{0,\beta ,\lambda } : = \frac{{\sup }}{{t \in \left[ {0,T} \right]}}e^{ - \lambda t} \varphi _f^\beta \left( t \right).$$


Weak Solution Strong Solution Wiener Process Fractional Brownian Motion Orlicz Space 
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