How can Wiener process be integrated w.r.t. time?

Question

Consider the RDE $$X_t = \int_0^t e^{cW_s}ds$$ where $c>0$ and $W_t$ is Wiener process. Can this integral be solved, i.e. can Wiener process be integrated w.r.t. time?

Now consider the integral $$X_t = \int_0^t sin(s)dB_s.$$ This integral cannot be further solved, is this correct?

Answer 1

• The correct way to write this equation is either $$ X_t=\int_0^te^{cW_\color{red}{s}}\,d\color{red}{s} $$ or -in differential form- $$ dX_t=e^{cW_t}\,dt\,. $$ It looks like you mixed up both.

• This integral is well defined even as a Riemann integral because for each $\omega$ the path $t\mapsto W_t(\omega)$ is continuous.

• It cannot be solved like we solve $$ \int_0^te^{cs}\,ds=\frac{e^{ct}-1}{c}\, $$ but this has not stopped generations of mathematicians to work with such integrals in the theory of stochastic differential equations or financial mathematics.

For example: Since Black & Scholes it is popular to model a stock price essentially by $$ S_t=S_0e^{cW_t-c^2t/2}\,. $$ Then $$ X_t=\frac{1}{t}\int_0^tS_s\,ds $$ describes the average stock price which is the underlying of an Asian option.

Answer 2

This is a Random differential equation, which differs from a Stochastic differential equations.

An RDE (or RODE) is an (path-wise) ODE containing a random process, in this case a Wiener process. Because of this, in a path-wise sense, the solution can be handled using standard (deterministic) calculus.

An SDE (driven by standard Brownian motion) is an equation of the form

$$dX_t = f(t,X_t)dt + g(t,X_t)dW_t,\,\, W_t : \text{ Wiener process}$$ Your notation is incorrect, you are mixing differential and integral forms.

$$\frac{dX_t}{dt} = e^{cW_t} \text{ or } X_t = \int_0^t e^{cW_s}ds$$

An aside: Interestingly, calling them stochastic differential equations is a bit of a misnomer, since sample paths are nowhere differentiable, so really they are stochastic integral equations.

Here is one numerical solution using Julia programming language with initial condition $X_0 = 1.0$ and $c = 2.0$ (cf. Julia DiffEq)

I recommend the following book Random Ordinary Differential Equations

EDIT: OP has added a stochastic integral question below original RDE question, for such integrals I recommend the following book: Stochastic Differential Equations