# Expected Solution of a Stochastic Differential Equation Expressed as Conditional Expectation

On all you geniusses out there: this is a tough one.

Preliminaries and Rigorous Technical Framework

• Let $$T \in (0, \infty)$$ be fixed.

• Let $$d \in \mathbb{N}_{\geq 1}$$ be fixed.

• Let $$(\Omega, \mathcal{G}, (\mathcal{G}_t)_{t \in [0,T]}, \mathbb{P})$$ be a complete probability space with a complete, right-continuous filtration $$(\mathcal{G}_t)_{t \in [0,T]}$$.

• Let $$B : [0,T] \times \Omega \rightarrow \mathbb{R}^d, \quad (t,\omega) \mapsto B_t(\omega)$$ be a standard $$d$$-dimensional $$(\mathcal{G}_t)_{t \in [0,T]}$$-adapted Brownian motion on $$\mathbb{R}^d$$ such that, for every pair $$(t,s) \in \mathbb{R}^2$$ with $$0 \leq t < s$$, the random variable $$B_s-B_t$$ is independent of $$\mathcal{G}_t$$.

• Let \begin{align} &\sigma: \mathbb{R}^d \rightarrow \mathbb{R}^{d \times d}, \ &\mu: \mathbb{R}^d \rightarrow \mathbb{R}^{d}, \end{align} be affine linear transformations, i.e. let there be matrices $$(A^{(\sigma)}_1,…,A^{(\sigma)}_d, \bar{A}^{(\sigma)}):= \theta_{\sigma} \in (\mathbb{R}^{d \times d})^{d+1}$$ such that, for all $$x \in \mathbb{R}^d$$, $$$$\sigma(x) = ( A^{(\sigma)}_1 x \mid … \mid A^{(\sigma)}_d x) + \bar{A}^{(\sigma)},$$$$ where $$A^{(\sigma)}_i x$$ describes the $$i$$-th column of the matrix $$\sigma(x) \in \mathbb{R}^{d \times d}$$, and let there be a matrix-vector pair $$(A^{(\mu)}, \bar{a}^{(\mu)}) := \theta_{\mu} \in \mathbb{R}^{d \times d} \times \mathbb{R}^d$$ such that, for all $$x \in \mathbb{R}^d$$, $$$$\mu (x) = A^{(\mu)}x + \bar{a}^{(\mu)}.$$$$

• Let $$$$\varphi : \mathbb{R}^d \rightarrow \mathbb{R}$$$$ be a fixed, continuous and at most polynomially growing function, i.e. let $$\varphi$$ be continuous and let there be a constant $$C \in [1, \infty)$$ such that, for all $$x \in \mathbb{R}^d$$ it holds that $$$$\lVert \varphi(x) \rVert \leq C (1+\lVert x \rVert )^C.$$$$

• Let $$x_0 \in \mathbb{R}^d$$ be fixed.

Question

Consider the following stochastic differential equation, given as an equivalent stochastic integral equation, where the multidimensional integrals are to be read componentwise:

$$$$S_t = x_0 + \int_{0}^{t} \mu(S_t) ds + \int_{0}^{t} \sigma (S_t) dB_s.$$$$

Under our assumptions, it is the case that an (up to indistinguishability) unique solution process

$$S^{(x_0, \theta_{\sigma}, \theta_{\mu})} :[0,T] \times \Omega \rightarrow \mathbb{R}^d, \quad (t, \omega) \mapsto S_t(\omega),$$

for this equation exists (to see this, consider for example Theorem 8.3. in Brownian Motion, Martingales and Stochastic Calculus from Le Gall).

I am interested in the expectation of $$S^{(x_0, \theta_{\sigma}, \theta_{\mu})}$$ at time $$T$$ when passed through the function $$\varphi$$: $$\mathbb{E}[\varphi(S^{(x_0, \theta_{\sigma}, \theta_{\mu})}_T)].$$ More specifically, I want to express $$\mathbb{E}[\varphi(S^{(x_0, \theta_{\sigma}, \theta_{\mu})}_T)]$$ in the following way as a conditional expectation: $$\mathbb{E}[\varphi(S^{(x_0, \theta_{\sigma}, \theta_{\mu})}_T)] = \mathbb{E}[\varphi(S^{(X_0, \Theta_{\sigma}, \Theta_{\mu})}_T) \mid (X_0, \Theta_{\sigma}, \Theta_{\mu}) = (x_0, \theta_{\sigma}, \theta_{\mu})].$$

Here $$X_0 : \Omega \rightarrow \mathbb{R}^d,$$ $$\Theta_{\mu} : \Omega \rightarrow \mathbb{R}^{d \times d} \times \mathbb{R}^d,$$ $$\Theta_{\sigma} : \Omega \rightarrow (\mathbb{R}^{d \times d})^{d+1},$$ are $$\mathcal{G}_0$$-measurable random variables, which define the initial value $$x_0$$ of the process at $$t=0$$ as well as the entries of the affine-linear coefficient functions $$\mu$$ and $$\sigma$$. Moreover, $$\Sigma$$ is a random function.

The random variable

$$S^{(X_0, \Theta_{\sigma}, \Theta_{\mu})}_T : \Omega \rightarrow \mathbb{R}^d$$

is implicitly defined by the procedure of first “drawing” the random variables $$(X_0, \Theta_{\sigma}, \Theta_{\mu})$$ at time $$t = 0$$ in order to obtain fixed values $$(X_0, \Theta_{\sigma}, \Theta_{\mu}) = (\tilde{x}_0, \tilde{\theta}_{\sigma}, \tilde{\theta}_{\mu})$$ and then “afterwards” set $$S^{X_0, \Theta_{\sigma}, \Theta_{\mu})}_T := S^{(\tilde{x}_0, \tilde{\theta}_{\sigma}, \tilde{\theta}_{\mu})}_T,$$ where
$$S^{(\tilde{x}_0, \tilde{\theta}_{\sigma}, \tilde{\theta}_{\mu})} :[0,T] \times \Omega \rightarrow \mathbb{R}^d, \quad (t, \omega) \mapsto S^{(\tilde{x}_0, \tilde{\theta}_{\sigma}, \tilde{\theta}_{\mu})}_t(\omega)$$ is the (up to indistinguishability) unique solution process of the stochastic differential equation.

$$$$S_t = \tilde{x}_0 + \int_{0}^{t} \tilde{\mu}(S_t) ds + \int_{0}^{t} \tilde{\sigma} (S_t) dB_s.$$$$

Here, $$\tilde{\sigma}$$ and $$\tilde{\mu}$$ are the affine-linear maps associated with the parameter values $$\tilde{\theta}_{\sigma}$$ and $$\tilde{\theta}_{\mu}$$ as described above.

Now, my questions:

1. I know that there are technical problems with the way I “defined ” the random variable $$S^{(X_0, \Theta_{\sigma}, \Theta_{\mu})}$$, although I hope the idea is clear. How can I make the definition of $$S^{(X_0, \Theta_{\sigma}, \Theta_{\mu})}$$ rigorous in the above framework?
2. After having obtained a rigorous definition of $$S^{(X_0, \Theta_{\sigma}, \Theta_{\mu})}$$, how can I then show, that $$\mathbb{E}[\varphi(S^{(x_0, \theta_{\sigma}, \theta_{\mu})}_T)] = \mathbb{E}[\varphi(S^{(X_0, \Theta_{\sigma}, \Theta_{\mu})}_T) \mid (X_0, \Theta_{\sigma}, \Theta_{\mu}) = (x_0, \theta_{\sigma}, \theta_{\mu})] ?$$

If further regularity assumptions (for example on the random variables $$X_0, \Theta_{\sigma}, \Theta_{\mu}$$) are necessary in order to answer the above questions in a satisfactory way, then these can be made without second thoughts.

These questions are at the core of my current research. I am stuck and I would be extremely grateful for any advice!