Stochastic exponential

Three realizations of a standard Wiener process (above) and its stochastic exponential (below)

A stochastic exponential is a stochastic process which , in the mathematical subfield of stochastic analysis, is an analogue to the exponential function of ordinary analysis . After the French mathematician Catherine Doléans-Dade , it is also known as the Doléans-Dade exponential or, for short, the Doléans exponential .

The exponential function can be characterized by the fact that it agrees with its derivative . If one wants to achieve an analogous behavior for the exponential function of a stochastic process, then, because of the Itō's lemma, its quadratic variation must be taken into account, if this does not vanish, for example in the Wiener process .

Stochastic exponentials play an important role in the explicit solution of stochastic differential equations and appear in Girsanow's theorem , which describes the behavior of stochastic processes when the measure changes . An important question in this context is the conditions under which a stochastic exponential is a martingale . Many models of financial mathematics contain processes that are stochastic exponentials, for example the geometric Brownian motion in the Black-Scholes model .

introduction

The exponential function is uniquely determined by the two conditions and . Somewhat more generally it follows from the chain rule that the uniquely determined solution is the linear ordinary differential equation with the initial condition . ${\displaystyle u(t)=\mathrm {e} ^{t}}$ ${\displaystyle u'(t)=u(t)}$ ${\displaystyle u(0)=1}$ ${\displaystyle u(t)=\mathrm {e} ^{x(t)-x(0)}}$ ${\displaystyle u'(t)=u(t)x'(t)}$ ${\displaystyle u(0)=1}$

These relationships no longer apply to stochastic differential equations in this simple form, since the chain rule has to be replaced by Itō's lemma, which takes into account the quadratic variation of the processes. If, for example, a standard Wiener process, then for the differential of the process results due to the Itō lemma ${\displaystyle (W_{t})}$ ${\displaystyle U_{t}=u(W_{t})=\mathrm {e} ^{W_{t}}}$ ${\displaystyle u=u'=u''}$

${\displaystyle \mathrm {d} U_{t}=\mathrm {e} ^{W_{t}}\,\mathrm {d} W_{t}+{\frac {1}{2}}\mathrm {e} ^{W_{t}}\,\mathrm {d} t=U_{t}\left(\mathrm {d} W_{t}+{\frac {1}{2}}\,\mathrm {d} t\right)}$.

The additional term in this stochastic differential equation can be avoided if the “corrected” approach is used instead of the exponential function : Then the result is analogous to the case of ordinary differential equations. In addition, the process, like the Wiener process, is now a martingale. ${\displaystyle U_{t}=\mathrm {e} ^{W_{t}-{\frac {1}{2}}t}}$ ${\displaystyle \mathrm {d} U_{t}=U_{t}\,\mathrm {d} W_{t}}$ ${\displaystyle U}$

definition

It is a semi-martingale . Then the (unambiguously determined) semimartingale is called the solution of the stochastic differential equation ${\displaystyle (X_{t})_{t\in \mathbb {R} _{+}}}$ ${\displaystyle U=(U_{t})_{t\in \mathbb {R} _{+}}}$

${\displaystyle \mathrm {d} U_{t}=U_{t-}\,\mathrm {d} X_{t}}$

with initial condition is the stochastic exponential of and is denoted by, i. H. . ${\displaystyle U_{0}=1}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {E}}(X)}$ ${\displaystyle {\mathcal {E}}(X)_{t}:=U_{t}}$

With this being left-side limit of the process at the site designated. If is continuous, then is also continuous; it then applies . ${\displaystyle U_{t-}}$ ${\displaystyle U}$ ${\displaystyle t}$ ${\displaystyle X}$ ${\displaystyle U}$ ${\displaystyle U_{t-}=U_{t}}$

The fact that the process is the solution of the aforementioned initial value problem explicitly means that it uses the Itō integral equation ${\displaystyle U}$

${\displaystyle U_{t}=1+\int _{0}^{t}U_{s-}\,\mathrm {d} X_{s}}$

Fulfills.

Explicit representation and calculation rules

If a continuous semimartal, then the stochastic exponential has the explicit representation ${\displaystyle X}$

${\displaystyle {\mathcal {E}}(X)_{t}=\mathrm {e} ^{X_{t}-X_{0}-{\frac {1}{2}}[X,X]_{t}}}$,

where denotes the quadratic variation of . ${\displaystyle [X,X]}$ ${\displaystyle X}$

In the general case, the jump points must also be taken into account. Here results ${\displaystyle X}$

${\displaystyle {\mathcal {E}}(X)_{t}=\mathrm {e} ^{X_{t}-X_{0}-{\frac {1}{2}}[X,X]_{t}}\prod _{s\leq t}(1+\Delta X_{s})\mathrm {e} ^{-\Delta X_{s}+{\frac {1}{2}}(\Delta X_{s})^{2}}}$

with the jump process . ${\displaystyle \Delta X_{s}=X_{s}-X_{s-}}$

Instead of the functional equation of the exponential function, the stochastic exponential of semimartingals and the calculation rule apply ${\displaystyle X}$ ${\displaystyle Y}$

${\displaystyle {\mathcal {E}}(X)_{t}{\mathcal {E}}(Y)_{t}={\mathcal {E}}(X+Y+[X,Y])_{t}}$.

If is continuous with , then applies ${\displaystyle X}$ ${\displaystyle X_{0}=0}$

${\displaystyle ({\mathcal {E}}(X)_{t})^{-1}={\mathcal {E}}(-X+[X,X])_{t}}$.

Martingale properties

In the following, a continuous semi-martingale applies without restriction , that is . By definition, the stochastic exponential is always a semimartingale. If there is a local martingale , the representation as an Itō integral shows that there is also a local martingale. However, even if there is a martingale, the stochastic exponential need not be a true martingale; as a non-negative local martingale, however, it is then a super martingale . ${\displaystyle X}$ ${\displaystyle X_{0}=0}$ ${\displaystyle {\mathcal {E}}(X)_{t}=\mathrm {e} ^{X_{t}-{\frac {1}{2}}[X,X]_{t}}}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {E}}(X)}$ ${\displaystyle X}$

For many applications it is important to have easily verifiable criteria that guarantee that the stochastic exponential of a local martingale is a (real) martingale. The best known sufficient condition is the Novikov condition (after the Russian mathematician Alexander Novikov ): Let be a continuous local martingale with . Applies to everyone , then a martingale is on . ${\displaystyle X}$ ${\displaystyle X_{0}=0}$ ${\displaystyle \operatorname {E} {\bigl (}\mathrm {e} ^{{\frac {1}{2}}[X,X]_{t}}{\bigr )}<\infty }$ ${\displaystyle t\leq T}$ ${\displaystyle {\mathcal {E}}(X)}$ ${\displaystyle [0,T]}$

Applications

Linear stochastic differential equations

With the help of the stochastic exponential, the solutions of linear stochastic differential equations can be given explicitly. A linear stochastic differential equation has the form

${\displaystyle \mathrm {d} X_{t}={\bigl (}\alpha _{t}+\beta _{t}X_{t}{\bigr )}\,\mathrm {d} t+{\bigl (}\gamma _{t}+\delta _{t}X_{t}{\bigr )}\,\mathrm {d} W_{t}}$

with continuous functions or continuously adapted stochastic processes . The corresponding homogeneous equation ${\displaystyle \alpha ,\beta ,\gamma ,\delta }$

${\displaystyle \mathrm {d} U_{t}=\beta _{t}U_{t}\,\mathrm {d} t+\delta _{t}U_{t}\,\mathrm {d} W_{t}=U_{t}(\beta _{t}\,\mathrm {d} t+\delta _{t}\,\mathrm {d} W_{t})}$

owns the solution with and without restriction . The general solution is therefore explicit ${\displaystyle U_{t}={\mathcal {E}}(Y)_{t}}$ ${\displaystyle \mathrm {d} Y_{t}=\beta _{t}\,\mathrm {d} t+\delta _{t}\,\mathrm {d} W_{t}}$ ${\displaystyle Y_{0}=0}$

${\displaystyle U_{t}=U_{0}\mathrm {e} ^{Y_{t}-{\frac {1}{2}}[Y,Y]_{t}}}$

With

${\displaystyle Y_{t}=\int _{0}^{t}\beta _{s}\,\mathrm {d} s+\int _{0}^{t}\delta _{s}\,\mathrm {d} W_{s}}$

and

${\displaystyle [Y,Y]_{t}=\int _{0}^{t}\delta _{s}^{2}\,\mathrm {d} s}$.

A particular solution of the inhomogeneous equation can be found from this by varying the constants , i.e. by the approach . ${\displaystyle X_{t}=Z_{t}U_{t}}$

Girsanow's theorem

→ Main article : Girsanow's theorem

Let there be a Wiener process on the interval with regard to the probability measure and a process with . If the stochastic exponential is a martingale, then the following applies and can be interpreted as the Radon-Nikodým density of a probability measure with regard to: ${\displaystyle (W_{t})}$ ${\displaystyle [0,T]}$ ${\displaystyle P}$ ${\displaystyle X}$ ${\displaystyle \mathrm {d} X_{t}=\theta _{t}\,\mathrm {d} W_{t}}$ ${\displaystyle L_{t}={\mathcal {E}}(X)_{t}}$ ${\displaystyle \operatorname {E} (L_{T})=\operatorname {E} (L_{0})=1}$ ${\displaystyle L_{T}}$ ${\displaystyle Q}$ ${\displaystyle P}$

${\displaystyle {\frac {\mathrm {d} Q}{\mathrm {d} P}}=L_{T}}$.

With regard to the dimension so defined is the drift process ${\displaystyle Q}$

${\displaystyle B_{t}=W_{t}-\int _{0}^{t}\theta _{s}\,\mathrm {d} s}$

a standard Wiener process.

literature

• Nicholas H. Bingham, Rüdiger Kiesel: Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives. 2nd edition, Springer, London / Berlin / Heidelberg 2004, ISBN 1-85233-458-4 , pp. 197, 215-217.
• Fima C. Klebaner: Introduction to Stochastic Calculus with Applications. 3rd edition, Imperial College Press, London 2012, ISBN 978-1-84816-831-2 .
• Philip E. Protter: Stochastic Integrals and Differential Equations. 2nd edition, version 2.1, Springer, Berlin 2005, ISBN 3-540-00313-4 .