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Let

  • $T>0$
  • $(\Omega,\mathcal A,\operatorname P)$ be a probability space
  • $(\mathcal F_t)_{t\in[0,\:T]}$ be a complete filtration on $(\Omega,\mathcal A)$
  • $(B_t)_{t\in[0,\:T]}$ be an $(\mathcal F_t)_{t\in[0,\:T]}$-Brownian motion on $(\Omega,\mathcal A,\operatorname P)$
  • $X_0$ be a real-valued $\mathcal F_0$-measurable random variable on $(\Omega,\mathcal A,\operatorname P)$
  • $\varphi:\Omega\times[0,T]\times\mathbb R\to\mathbb R$
  • $\Phi:\Omega\times[0,T]\times\mathbb R\to\mathbb R$

I want to consider the equation $$X_t=X_0+\int_0^t\varphi(s,X_s)\:{\rm d}s+\int_0^t\Phi(s,X_s)\:\circ{\rm d}B_s\;\;\;\text{for all }t\in[0,T]\tag 1$$ where the last integral has to be understood in the Stratonovich sense.

Question: Under which condition on $\Phi$ is $(1)$ well-defined?

If $(Y_t)_{t\in[0,\:T]}$ is a stochastic process on $(\Omega,\mathcal A,\operatorname P)$, then $$\int_0^tY_s\:\circ{\rm d}B_s:=\int_0^tY_s\:{\rm d}B_s+\frac12\langle Y,B\rangle_t\;\;\;\text{for }t\in[0,T]\tag 2$$ where the integral on the right-hand side is understood in the Itō sense and $(\langle Y,B\rangle_t)_{t\in[0,\:T]}$ denotes the covariation of $Y$ and $B$. Of course, in order for $(2)$ to be well-defined we need that

  1. $Y$ is integrable in the Itō sense, i.e. $Y$ is $(\mathcal F_t)_{t\in[0,\:T]}$-progressively measurable and $$\operatorname P\left[\int_0^TY_s^2\:{\rm d}s<\infty\right]=1\tag 3$$
  2. the covariation of $Y$ and $B$ exists, i.e. $Y$ is an $(\mathcal F_t)_{t\in[0,\:T]}$-semimartingale

So, $(1)$ is well-defined if and only if 1. and 2. are satisfied by $$Y_t:=\Phi(t,X_t)\;\;\;\text{for }t\in[0,T]\;.$$ Since $X$ is the unknown solution (whose existence is not even known and cannot be proved until $(1)$ is at least well-posed), we somehow need to ensure this by a suitable assumption for $\Phi$. The question is which assumption this is.

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