Stochastic differential equation

The concept of the stochastic differential equation (abbreviation SDGL or English SDE for stochastic differential equation ) is a generalization of the concept of the ordinary differential equation to stochastic processes in mathematics . Stochastic differential equations are used in numerous applications to model time-dependent processes which, in addition to deterministic influences, are also exposed to stochastic interference factors ( noise ).

The mathematical formulation of the problem posed great problems for mathematicians, and so the formal theory of stochastic differential equations was not formulated until the 1940s by the Japanese mathematician Itō Kiyoshi . Together with stochastic integration , the theory of stochastic differential equations establishes stochastic analysis .

From the differential to the integral equation

As with deterministic functions, one would like to formulate the relationship between the value of the function and its momentary change (its derivative ) in an equation for stochastic processes . What leads to an ordinary differential equation in one case is problematic in the other, since many stochastic processes, such as the Wiener process , are nowhere differentiable .

However, an ordinary differential equation can be used

${\displaystyle {\frac {{\rm {d}}y(t)}{{\rm {d}}t}}=f(t,y(t))}$

always equivalent as an integral equation

${\displaystyle y(t)=y(t_{0})+\int _{t_{0}}^{t}f(\tau ,y(\tau ))\,{\rm {d}}\tau }$

write that does not require an explicit mention of the derivative. With stochastic differential equations one goes the opposite way, ie one defines the term with the help of the associated integral equation.

The formulation

Let two functions and a Brownian movement be given. The associated stochastic integral equation ${\displaystyle a,b\colon \mathbb {R} \times \mathbb {R} _{+}\to \mathbb {R} }$ ${\displaystyle (W_{t})_{t\geq 0}}$

${\displaystyle X_{t}=X_{0}+\int _{0}^{t}a(X_{\tau },\tau )\,{\rm {d}}\tau +\int _{0}^{t}b(X_{\tau },\tau )\,{\rm {d}}W_{\tau }}$

becomes through the introduction of the differential notation

${\displaystyle {\rm {d}}X_{t}=a(X_{t},t)\,{\rm {d}}t+b(X_{t},t)\,{\rm {d}}W_{t}}$

to the stochastic differential equation. The first integral is to be read as the Lebesgue integral and the second as the Itō integral . For given functions and (also referred to as drift and diffusion coefficient ) and a Brownian motion , a process is sought that fulfills the above integral equation. This process is then a solution to the above SDGL. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle (W_{t})}$ ${\displaystyle (X_{t})}$

Existence and uniqueness

If there is any random variable defined on the same probability space as , then by adding the condition to the above SDGL, a stochastic initial value problem as a counterpart to the initial value problem for ordinary differential equations will almost certainly become . ${\displaystyle A}$ ${\displaystyle W}$ ${\displaystyle X_{0}=A}$

There is also a correspondence to the existence and uniqueness theorem of Picard and Lindelöf : if the following three properties are fulfilled:

• ${\displaystyle A\in L^{2}(P)}$, d. i.e., has finite variance . ${\displaystyle A}$
• Lipschitz condition : There is a constant such that for all and all true ${\displaystyle K\geq 0}$ ${\displaystyle x,y\in \mathbb {R} }$ ${\displaystyle t\geq 0}$

${\displaystyle |a(x,t)-a(y,t)|+|b(x,t)-b(y,t)|\leq K|x-y|}$.

• Linear limitations: There is a constant such that for all and all true ${\displaystyle C\geq 0}$ ${\displaystyle x\in \mathbb {R} }$ ${\displaystyle t\geq 0}$

${\displaystyle |a(x,t)|+|b(x,t)|\leq C(1+|x|)}$.

Then the initial value problem has a unique solution (apart from almost certain equality) , which also has finite variance at any point in time . ${\displaystyle X}$ ${\displaystyle t}$

Examples

• The SDGL for the geometric Brownian motion is . It is used, for example, in the Black-Scholes model to describe stock prices. ${\displaystyle {\rm {d}}S_{t}=rS_{t}{\rm {d}}t+\sigma S_{t}{\rm {d}}W_{t}}$
• The SDGL for an Ornstein-Uhlenbeck process is . Among other things, it is used in the Vasicek model for the mathematical modeling of interest rates using the current interest rate . ${\displaystyle \mathrm {d} X_{t}=\theta (\mu -X_{t})\mathrm {d} t+\sigma \mathrm {d} W_{t}}$
• The SDGL for the root diffusion process according to William Feller is ${\displaystyle {\rm {d}}X_{t}=\kappa (\theta -X_{t}){\rm {d}}t+\sigma {\sqrt {X_{t}}}{\rm {d}}W_{t}}$

Solving stochastic differential equations and simulating the solutions

As with deterministic equations, there is no general approach to finding the solution for stochastic differential equations. In some cases (as in the above-mentioned Black-Scholes SDGL, whose solution is a geometric Brownian motion) it is also possible here to “guess” the solution and to verify it by deriving it (the differentiation here using the lemma of Itō takes place).

In most of the cases that arise in practice, such as in the case of the root diffusion process, however, a closed form of the solution cannot be achieved. But mostly one is only interested in simulating the random paths of the corresponding solution. This can be achieved approximately using numerical discretization methods, for example the Euler-Maruyama scheme (which is based on the explicit Euler method for ordinary differential equations) or the Milstein method .

Stochastic delay differential equations

In a stochastic delay differential equation (SDDE, stochastic delay differential equation ), the future increase depends not only on the current state, but also on the states in a preceding limited time interval. Existence and uniqueness are given under similar conditions as in "normal" SDGLs. Be

${\displaystyle f\colon [0,\infty )\times C([-r,0],\mathbb {R} ^{d})\to \mathbb {R} ^{d}}$,

${\displaystyle g\colon [0,\infty )\times C([-r,0],\mathbb {R} ^{d})\to \mathbb {R} ^{d\times m}}$

continuous, and be an m-dimensional Brownian motion. Then a stochastic delay differential equation is an equation of form ${\displaystyle r>0}$ ${\displaystyle W}$

${\displaystyle X(t)=X(0)+\int _{0}^{t}f_{\tau }(X_{\tau })\,{\rm {d}}\tau +\int _{0}^{t}g_{\tau }(X_{\tau })\,{\rm {d}}W(\tau )}$

in which ${\displaystyle X_{t}(s):=X(t+s)\ \forall s\in [-r,0]}$

The corresponding differential notation is then

${\displaystyle {\rm {d}}X(t)=f_{t}(X_{t})\,{\rm {d}}t+g_{t}(X_{t})\,{\rm {d}}W(t)}$.

See also

• Langevin's equation

literature

• Bernt Øksendal: Stochastic Differential Equations. An Introduction with Applications. 6th edition. Springer, Berlin 2003, ISBN 3-540-04758-1 .
• Philip E. Protter: Stochastic Integration and Differential Equations. Springer, Berlin 2003, ISBN 3-540-00313-4 .