# “Conditional distribution” of Brownian sample paths

I would like to consider the "conditional distribution" of the Brownian sample paths conditional on certain sample path functionals, in a similar way that one considers the Brownian bridge. For example, consider the functional $$\phi: C[0,1] \rightarrow \mathbb{R}$$ defined by, say

$$\phi( W(\cdot) ) = \left( \int_0^1 W dW \right)^2,$$ or $$\phi( W(\cdot) ) = \int_0^1 W^2_t dt,$$ etc., what is the process obtained by conditioning $$W$$ on $$\phi$$? Are there any results of this type?

• The first one is easy. $\int_0^1 WdW = W_1^2-1$ so of we condition on the square of that to be a fixed positive number, the resultant distribution is easily obtained in terms of the Brownian bridge. – Shalop May 12 '19 at 2:16

I don't know of any literature on the subject of conditioning Brownian motion in this way, however, I believe that when the functional $$\phi$$ is nice enough (in the sense of Malliavin calculus), it is possible to write the conditioned process as the solution of some SDE: $$dX_t = dB_t+G(t,X_{\cdot})dt$$. Here $$G$$ is some predictable function from $$\Bbb R_+ \times C_0[0,1] \to \Bbb R$$ which depends on $$\phi$$ and also on the value we condition on.

I do not know the precise/rigorous conditions on $$\phi$$ which make this possible and I do not intend to pursue that here. Instead, I will give some intuitionist, "physics-level" derivation of how to compute the drift $$G$$ from the functional $$\phi$$.

So, suppose we want to condition on $$\phi(W)=C$$. Assume that the law of $$\phi(W)$$ has a continuous density $$p_{\phi}(x)$$ with respect to Lebesgue measure which is strictly positive near $$C$$ (there are conditions to ensure this using Malliavin Calculus, e.g. $$C$$ should be a regular value of $$\phi$$, and the Malliavin matrix of $$\phi$$ should be invertible and satisfy some regularity conditions in some neighborhood of $$\phi^{-1}(\{C\})$$). Then define $$F(\phi,C) := \frac1{p_{\phi}(C)}\partial_t|_{t=0} \big(\partial_x|_{x=C}\Bbb E[W_t\cdot 1_{\{\phi(W) < x\}}]\big),\tag{1}$$ which (as may be verified using regular conditioning) will just be the drift felt by the conditional process at $$t=0$$. Furthermore, if $$t \ge 0$$ and $$f,g \in C[0,1]$$ then we define $$\psi(t,f,g) := \phi\big(f(t^{-1}\cdot)*g((1-t)^{-1}\cdot)\big)$$, where $$f(t^{-1}\cdot)\in C[0,t]$$ is the map $$u \mapsto f(t^{-1}u)$$ (and similarly for $$g$$); and moreover "$$*$$" denotes concatenation of paths: more specifically, if $$\gamma \in C_0[0,t]$$ and $$\mu \in C_0[0,1-t]$$ then $$\gamma * \mu\;(u) = \gamma(u)$$ for $$u \le t$$ and $$\gamma * \mu\; (u) = \gamma(t) + \mu(u-t)$$ for $$u \in [t,1]$$. Finally, we are ready to define the drift function $$G(t,\omega) := \frac{F\big(\;\psi(t,\omega(t\cdot),(1-t)^{1/2}\cdot)\;, C\big)}{1-t}.\tag{2}$$ The way to interpret this last expression is to note that for fixed $$\omega \in C_0[0,1]$$ and $$t \ge 0$$, the map $$g \mapsto \psi\bigg(t\;,u\mapsto \omega(tu)\;, v\mapsto (1-t)^{1/2}g(v)\bigg)$$ is a functional on $$C[0,1]$$, hence we can plug it into $$F(\cdot,C)$$.

Now, with the way we have defined $$G$$, we claim that the solution of $$dX_t = dB_t + G(t,X_{\cdot})dt$$ is essentially Brownian motion conditioned on $$\phi=C$$. The intuitive reason for this (which is easy to visualize) is that at each infinitesimal step, $$X_t$$ feels a drift in the direction which allows the remainder of the path to be distributed according to a Brownian motion conditional on the constraint that if we concatenate this Brownian path with the history of $$X$$ so far, then applying $$\phi$$ to the resultant path gives $$C$$. Then one may check (perhaps tediously) that mathematically formulating this whole procedure gives the above SDE. Unfortunately it remains to be shown that the solution $$X$$ actually exists and that its law is the same as that obtained by other conditioning procedures, like regular conditioning / disintegration. However I did some computations which suggest that this is actually true, and in fact the resulting law even appears to be weakly continuous in $$C$$ if $$\phi$$ is nice.

It is interesting to note that if we apply these heuristics to $$\phi(\omega) = \omega(1)$$ then we recover the SDE of Brownian bridge: $$dX_t = dB_t + \frac{C-X_t}{1-t}dt$$. For another example, lets say that $$\phi(\omega) = p(\omega(1))$$ where $$p$$ is a quadratic polynomial with two real roots $$r,s$$. Then we set $$f(a,b) = \frac{ae^{-a^2/2}+be^{-b^2/2}}{e^{-a^2/2}+e^{-b^2/2}}$$. Then, one way to understand the conditional process (with $$\phi(W)=0$$) is in terms of a Brownian bridge (see my comment), but another way is as the solution of the SDE $$dX_t = dB_t +\frac{f(r-X_t,s-X_t)}{1-t}dt$$. With some modifiction, this SDE formulation seems to make sense even for problems such as conditioning $$W$$ to stay positive (e.g., take $$\phi = \inf$$ and let $$C=0$$).

Another remark is that it seems from this description that the conditioned process is necessarily absolutely continuous wrt Brownian motion, but actually this is not true because the drift term $$G$$ can have small blow-ups along the way which can make it true that $$\int_0^t G(s,X_{\cdot})^2ds = +\infty$$ (think about $$\phi(\omega) = \omega(1/2)$$, which gives a Brownian bridge up to time $$1/2$$, but then just evolves as Brownian motion). A final remark is that in practice it might be difficult to compute $$F$$ and $$G$$ (the quantities appearing in $$(1)$$ and $$(2)$$). To compute $$F$$, in principle one only needs to know how to compute $$p_{\phi}$$ and $$\Bbb E[W_t|\phi(W)]$$, and in certain cases (like Brownian bridge) this is quite explicit, however in other cases it might be very hard. Computing $$G$$ can be even harder, however in certain cases it can be computed easily from $$F$$, even if we cannot explicitly compute $$F$$ itself (this is the case for your second example where $$\phi(f) = \int f^2$$, because this functional satisfies a nice additivity property with respect to concatenation of paths).

• One more remark: there is a discrete-time version of this procedure. Specifically if $X$ is a markov chain on some discrete state space $S$, and if $f:S^N \to \Bbb R$ is some functional, then we can condition on $f(X_1,...,X_N)=C$ and the result is an (inhomogeneous) markov chain $(\bar X_1,...,\bar X_N)$ whose (time-dependent) transition densities can be easily derived from those of $X$ (it will be like a "time-varying Doob h-transform"). – Shalop May 14 '19 at 22:57
• Another remark on the "computations" mentioned above: Basically if $U \subset C_0[0,1]$ is open, then we can (similarly to above) define $F(U) = \partial_t|_{t=0} \Bbb E[W_t|W\in U]$, and analogously we can also define $G(U;t,\omega)$ using the same procedure as above. Then we claim that the solution of the SDE $dX_t = dB_t +G(U;t,X_{\cdot})dt$ has the same law as that of $W$ conditioned to stay in $U$. This may be proved by noting that if $Y$ denotes $W$ conditioned to stay in $U$,then $Y_t - \int_0^t G(U;s,Y_{\cdot})ds$ is a martingale whose quadratic variation... – Shalop May 15 '19 at 0:47
• ...equals $t$, and thus the latter is a standard BM (note by Girsanov that this also gives the seemingly nontrivial identity $\Bbb E[1_U|\mathcal F_t] = \Bbb P(W \in U) e^{\int_0^t G(U;s,W_{\cdot})ds -\frac12\int_0^t G(U;s,W_{\cdot})^2ds}$). Once you have this characterization of the conditioned process for open sets, you may apply it to the open sets $U_{\epsilon}:=\phi^{-1}(C-\epsilon, C+\epsilon)$; then as $\epsilon \to 0$ it is easily shown that $F(U_{\epsilon}) \to F(\phi,C)$ and similarly for $G$. Then one may argue that the respective laws of the SDE's also converge as $\epsilon \to 0$. – Shalop May 15 '19 at 19:49
• did you come up with this? If so, wouldn't it deserve a paper? – Tom Sep 8 '19 at 13:33
• @Tom Yes I came up with this, but I’m not sure if it’s publishable because this kind of thing seems classical in spirit. I imagine that there might already be existing results from ~30 years ago on this type of subject, though I’m not sure. – Shalop Sep 8 '19 at 16:52