Cfrgtkky

Suppose we are in the following setting

begin{gather}
dX_i(t)=X_i(t) big( b_i(t)dt+ sum limits_{nu=1}^{d} sigma_{i nu}(t) dW_{nu}(t) big) , qquad X_i(0)=x_i, i=1, ldots, n,
end{gather}

with $d geq n$, $d$-dimensionl standard Brownian motion $(W_1(cdot), ldots, W_d(cdot))^{intercal}$. Furthermore, the processes $b(cdot)=(b_1(cdot), ldots, b_n(cdot))^{intercal}$ and $sigma(cdot)=(sigma_{i nu}(cdot))_{1 leq i leq n, 1 leq nu leq d}$ are progressively measurable with respect to the filtration generated by the brownian motion and fulfill the usual intergability conditions

Question: Does there exist a closed form espression for the transition density function of the process $X$? Does anyone know of a closed form expression for the simpler case where $b$ and $sigma$ are constant? I'm interested in relatively straightforward derivations, i.e. not the one of the paper mentioned below (there is nothing wrong with it I just wonder if there is a simpler approach for the case of geometric brownian motion).

For example, in the one dimensional case, where $sigma$ and $b equiv 0$ are time independent we have the following transition density:

$$p(X(t),t;X(0),0)=frac{1}{X(t)sigma sqrt{2pi t}}exp{left(-frac{1}{2}left[frac{log(X_t)-log(X_0)-sigma^2 t/2}{sigma sqrt{t}}right]^2right)}$$

as seen here: Transition density of a Geometric Brownian-motion .

Explenation: I know how to derive the probability density function in the case where $n=1$, i.e., in the one dimensional case. The same approach does not work here because the components of $X$ are not independent and we can not simply write the density function as a product of the simpler density functions of the one dimensional case. There is some literature on the topic for general diffusion models, like here: https://www.princeton.edu/~yacine/multivarmle.pdf but the paper is very technical and several assumptions are made. This motivates my question about the existence of a simpler approach in the case of geometric brownian motion, or in the case where we make even the stronger restriction of considering time independent processes $b$ and $sigma$.

First, a simplification. Let $Y_i(t):=log(X_i(t))$ and $Y(t):=(Y_1(t),ldots,Y_n(t))^T$. It is enough to obtain a transition function for $Y$. By using Ito's formula for function $log x$, you can easily obtain the following.
$$
dY_i(t)=b'_i(t)dt + sum_j sigma_{ij}(t)dW_j(t),
$$
where $b'_i(t)=b_i(t)-frac 12 sum_j sigma_{ij}(t)^2$. Since the right hand side does not depend on $Y$, you can take integral of both sides and compute $Y(t)$ explicitly as an Ito integral.

Now, assume that $b$ and $sigma$ are constant. You can explicitly obtain
$$
Y_i(t)= tb'_i + sum_j sigma_{ij}W_j(t).
$$
Equivalently,
$$
Y(t)=Y(0)+tb'+sigma W(t).
$$
Hence, $Y$ is a multivariate normal random variable with mean $Y(0)+tb'$ and covariance matrix $sigmasigma^T$. In the case $sigma$ has rank $n$, this vector has a density which you can find here. If not, it does not have a density.

You can generalize this to the case where $b$ and $sigma$ are deterministic but may depend on $t$. In this case, $Y(t)$ is normal with mean $Y(0)+int_0^t b'(s)ds$ and covariance $int_0^t sigma(s)sigma(s)^T ds$.

If $b$ and $sigma$ are not deterministic, you cannot hope to find a general formula for transition probabilities. However, under some assumptions you can derive a differential equation for it, which is exactly what the Fokker-Planck equation does.