Wishart Processes

Chapter 1 Introduction

In this thesis we consider a matrix variate extension of the Cox-Ingersoll-Ross process (see Cox et al. ), i.e. a solution of a one-dimensional stochastic differential equation of the form

with positive numbers $a,b,\sigma$ and a one-dimensional Brownian motion $B$. Our extension is defined by a solution of the $p\times p$-dimensional stochastic differential equation of the form

where $Q$ and $K$ are real valued $p\times p$-matrices, ${\alpha}$ a non-negative number and $B$ a $p\times p$-dimensional Brownian motion (that is, a matrix of $p^{2}$ independent one-dimensional Brownian motions). While it is well-known that solutions of (1), called CIR processes, always exist, the situation for (2) is more difficult. As we will see in this thesis, it is crucial to choose the parameter ${\alpha}$ in the right way, to guarantee the (strong) existence of unique solutions of (2), that are then called Wishart processes. We will derive that it is sufficient to choose the parameter $\alpha$ larger or equal to $p+1$. That is similar to a result given by Bru .

The characteristic fact of (1) is that this process remains positive for a certain choice of $b$. This makes it suitable for modeling for example an interest rate, which should always be positive because of economic reasons. Hence, this is an approach for pricing bonds (see the section 5.1). If we want to consider some (corporate) bonds jointly, e.g. because they are correlated, the need for a multidimensional extension comes up. See section 5.3 of this thesis or Gourieroux [2007, 3.5.2.] for a discussion of this topic. For the Wishart processes, we have in the case ${\alpha}\geq p+1$ that the unique solution of (2) remains positive definite for all times. Another well-known fact is that the conditional(on $s_{0}$) distribution of the CIR process at a certain point in time is noncentral chi-square. We will see that the conditional distribution of the Wishart process $S_{t}$ at time $t$ is a matrix variate extension of the noncentral chi-square distribution, that is called noncentral Wishart distribution. An application for matrix variate stochastic processes can be found in Fonseca et al. , which model the dynamics of a $p$ risky assets $X$ by

In chapter 2, we introduce some notations and give the necessary background for understanding the following chapters. In chapter 3, a review of fundamental terms and results of matrix variate stochastics and the theory of stochastic differential equations is given, and in section 3.5 some results are derived concerning a matrix variate extension of the Ornstein-Uhlenbeck processes. The main work on the theory of Wishart processes will be done in chapter 4, where we give a general theorem about the existence and uniqueness of Wishart processes in section 4.2. In section 4.3, we show that some soutions of (2) can be expressed in terms of the matrix variate Ornstein-Uhlenbeck process from section 3.5 and in section 4.4 we give an algorithm to simulate Wishart processes. Finally, we give an outlook on the applications of Wishart processes in mathematical finance in chapter 5. For further readings about Wishart processes, one could have a look at the paper of Bru and for more financial applications, one could consider Gourieroux or Fonseca et al. , for example.

Chapter 2 Preliminaries

In this section we summarize the technical prerequisites which are necessary for matrix variate stochastics.

Let us review some characteristics of positive (semi)definite matrices.

If $A\in{\mathcal{S}}_{p}^{+}$ then $A^{-1}\in{\mathcal{S}}_{p}^{+}$

$A\in\overline{{\mathcal{S}}_{p}^{+}}$ if and only if A is orthogonally diagonalizable with non-negative eigenvalues

See Muirhead [2005, Appendix A8] or Fischer . ∎

On $\overline{{\mathcal{S}}_{p}^{+}}$ we are able to define a matrix valued square root function by

The square root $\sqrt{A}$ is well-defined and independent of $U$, as can be seen in Fischer , for example.

If $A\in{\mathcal{S}}_{p}^{+}$ then $\sqrt{A}\in{\mathcal{S}}_{p}^{+}$.

Now, we talk about the differentiation of matrix valued functions.

The following calculation rules are going to be helpful in the next chapters, so we state them here. In order not to lengthen this chapter, we only give outlines of the proofs.

If $A\in GL(p)$ or $B\in GL(p)$ then $\det(I_{p}+AB)=\det(I_{p}+BA)$

$\det(A)$ is the product of the eigenvalues of $A$.

$\frac{\partial^{2}}{\partial S_{ij}\,\partial S_{kl}}(\det(S))=\det(S)[(S^{-1})_{kl}(S^{-1})_{ij}-(S^{-1})_{ik}(S^{-1})_{lj}]$ where $(S^{-1})_{ij}$ denotes the $i,j$-th entry of $S^{-1}$

For (i) see Fischer , (ii) is a trivial consequence of (i). ∎

Before we continue, we introduce a new notation: For every linear operator ${\mathcal{A}}$ on a finite dimensional space we denote by $\sigma({\mathcal{A}})$ the spectrum of ${\mathcal{A}}$, that is the set of all eigenvalues of ${\mathcal{A}}$.

Then the inverse of ${\mathcal{A}}$ is given by

Can be shown with Lemma 2.11 (ii), for details see Stelzer [2007, p. 66] ∎

For symmetric matrices there also exists another operator which transfers a matrix into a vector.

Some Functions of Matrices

In this section we give a brief introduction to matrix variate functions that arise in the probability density function of the noncentral Wishart distribution.

(Borel-$\sigma$-algebra, cf. Jacod and Protter [2004, p. 48]) Let $(X,\mathcal{T})$ be a topological space. The Borel-$\sigma$-algebra on X is then given by the smallest $\sigma$-algebra that contains $\mathcal{T}$ and will be denoted by ${\mathcal{B}}(X)$.

The following definition is just for convenience and can also be found in Gupta and Nagar .

Gupta and Nagar [2000, Theorem 1.4.1.] show that, for $a>\frac{p-1}{2}$, the matrix variate gamma function can be expressed as a finite product of ordinary gamma functions. Thus, we do not need to worry about the existence of the matrix variate gamma function.

The Hypergeometric Function of matrix argument is defined by

Gupta and Nagar [2000, p.34] discuss conditions for the convergence and thus well-definedness of (4). A sufficient condition is $m<n+1$.

The following Lemma eases later on the calculation of expectations of functions of noncentral Wishart distributed random matrices.

This is a special case of Gupta and Nagar [2000, Theorem 1.6.2]∎

Chapter 3 Matrix Variate Stochastics

the filtration is right continuous, i.e. $\bigcap_{s>t}\mathcal{G}_{s}=\mathcal{G}_{t}$ for every $t\geq 0$, and

Now we summarize a few results and definitions from Gupta and Nagar .

A nonnegative measureable function $f_{X}$ such that

defines the probability density function(p.d.f.) of a $m\times n$-random matrix X.

The characteristic function of a $m\times n$-random matrix $X$ with p.d.f. $f_{X}$ is defined by

If the distribution of $X$ under $P$ is denoted by $P^{X}$, then (5) is the Fourier transform of the measure $P^{X}$ at point $Z$ and will be denoted by $\widehat{P^{X}}(Z)$.

From now on, with the term ”$X$ in ${\mathcal{S}}_{p}^{+}$ is a random matrix” we mean that $X$ is a random matrix with $X(\omega)\in{\mathcal{S}}_{p}^{+}$ for almost all $\omega\in\Omega$.

The Laplace transform of a $p\times p$-random matrix $X$ in ${\mathcal{S}}_{p}^{+}$ with p.d.f. $f_{X}$ is defined by

Basically, Levy’s Continuity Theorem says that weak convergence of probability measures is equivalent to the pointwise convergence of their respective Fourier transforms.

See Jacod and Protter [2004, Theorem 19.1] for a proof of a multivariate version that can be extended to the matrix variate case easily. ∎

With the term analytic function we mean a function that is locally given by a convergent power series.

Let $X$ be a $m\times n$-random matrix and Y be a $p\times q$-random matrix. The $mn\times pq$ covariance matrix is defined as

Let $X\thicksim{\mathcal{N}}_{p,n}(M,\Sigma\otimes\Psi)$. Then the characteristic function of $X$ is given by

Cf. Gupta and Nagar [2000, Theorem 2.3.2] ∎

If $\Sigma$ and $\Psi$ are not positive definite, but still positive semidefinite we will use (7) as a (generalized) definition for the matrix variate Normal distribution.

where $S\in{\mathcal{S}}_{p}^{+}$ and ${}_{0}F_{1}$ is the hypergeometric function. We write $X\thicksim\mathcal{W}_{p}(n,\Sigma,\Theta)$.

The requirement $n\geq p$ assures that the matrix variate gamma function is well-defined. For the case $p-1\leq n\leq p$ or resp. $n\in\{1,\ldots,p-1\}$ one could use the Laplace transform (9) or resp. Lemma 3.20 to define the Wishart distribution for this case. However, Olkin and Rubin [1961, Appendix] have shown for non-integer $n<p-1$ that (9) is not the Laplace transform of a probability distribution anymore.

If $\Sigma\in\overline{{\mathcal{S}}_{p}^{+}}\backslash{\mathcal{S}}_{p}^{+}$ the p.d.f. does not exist, but we can define the noncentral Wishart distribution using the characteristic function from Theorem 3.19

If $\Theta=0$, $X$ is said to have (central) Wishart distribution with parameters $p,n$ and $\Sigma\in{\mathcal{S}}_{p}^{+}$ and its p.d.f. is given by

where $S\in{\mathcal{S}}_{p}^{+}$ and $n\geq p$ (see Gupta and Nagar [2000, Definition 3.2.1]).

Let $S\thicksim\mathcal{W}_{p}(n,\Sigma,\Theta)$. Then the Laplace transform of S is given by

(Characteristic Function of the Noncentral Wishart Distribution, cf. Gupta and Nagar [2000, Theorem 3.5.3.]) Let $S\thicksim\mathcal{W}_{p}(n,\Sigma,\Theta)$. Then the characteristic function of S is given by

The next Lemma shows that the noncentral Wishart distribution is the square of a matrix variate normal distributed random matrix. Hence, it is the matrix variate extension of the noncentral chi-square distribution.

Consider $S\thicksim\mathcal{W}_{p}(n,\Sigma,\Theta)$. With the foregoing Lemma we can interpret the parameter $\Sigma$ as a scale and the parameter $\Theta$ as a location parameter for $S$. Especially, a central Wishart distributed matrix may be thought of a matrix square of normally distributed matrices with zero mean.

Processes and Basic Stochastic Analysis

$B_{t}\thicksim{\mathcal{N}}_{n,p}(0,tI_{np})$

Follows from Theorem 3.15 and the fact that $I_{np}=I_{n}\otimes I_{p}$ ∎

A matrix variate stochastic process $X$ is called a semimartingale if $X$ can be decomposed into $X=X_{0}+M+A$ where $M$ is a local martingale and $A$ an adapted process of finite variation.

We will only consider continuous semimartingales in this thesis.

with $D=(\frac{\partial}{\partial X_{ij}})_{i,j}$

Follows from Revuz and Yor [2001, Theorem 3.3 and Remark 2, Chapter IV] and Lemma 2.11 ∎

defines a sequence of positive stopping times. For every $n\geq N$, the stopped process $X^{T_{n}}:=X_{t\land T_{n}}$ is a continuous semimartingale with values in $U$, thus Theorem 3.27 can be applied to see that $f(X^{T_{n}})$ is a continuous semimartingale and (11) holds for $X^{T_{n}}$. Because $T_{n}$ converges to $T=\inf\{t\geq 0:\,X_{t}\in\partial U\}$, we can conclude $T>0$ and that $(f(X))_{t\in[0,T)}$ is a continuous semimartingale and (11) holds for $(X)_{t\in[0,T)}$. ∎

With the Definition of a ‘Matrix Quadratic Covariation’ we are able to state the matrix variate partial integration formula in a handy way.

where $A_{t-}$ denotes the limit from the left of $A_{t}$.

Stochastic Differential Equations

When we talk about (matrix variate) stochastic differential equations, we can classify between two different kind of solutions: Weak and strong solutions. Intuitively, a strong solution is constructed from a given Brownian motion and hence a ‘function’ of that Brownian motion.

A solution $(X,W)$ which is not strong will be termed a weak solution of (12).

(Uniqueness, Revuz and Yor [2001, Chapter IX, Definition 1.3]) Consider again the stochastic differential equation (12).

It is said that pathwise uniqueness holds for (12), if for every two solutions $(X,W)$ and $(X^{\prime},W^{\prime})$ defined on the same filtered probability space, $X_{0}=X_{0}^{\prime}$ and $W=W^{\prime}$ a.s. implies that $X$ and $X^{\prime}$ are indistinguishable, i.e. for $P$-almost all $\omega$ it holds that $X_{t}(\omega)=X_{t}^{\prime}(\omega)$ for every $t$, or equivalently

There is uniqueness in law for (12), if whenever $(X,W)$ and $(X^{\prime},W^{\prime})$ are two solutions with possibly different Brownian motions $W$ and $W^{\prime}$ (in particular if $(X,W)$ and $(X^{\prime},W^{\prime})$ are defined on two different filtered probability spaces) and $X_{0}\stackrel{{\scriptstyle\mathscr{D}}}{{=}}X_{0}^{\prime}$, then the laws $P^{X}$ and $P^{X^{\prime}}$ are equal. In other words, $X$ and $X^{\prime}$ are two versions of the same process, i.e. they have the same finite dimensional distributions (see Revuz and Yor [2001, Chapter I, Definition 1.6]).

As Yamada and Watanabe [1971a, Proposition 1] have shown, pathwise uniqueness implies uniqueness in law, which is not true conversely.

If pathwise uniqueness holds for (12), then every solution of (12) is strong (see Revuz and Yor [2001, Theorem 1.7]).

The definition of pathwise uniqueness implies that there exists at most one strong solution for (12) up to indistinguishability.

According to Skorohod [1965, p. 59f.], the stochastic differential equation (12) always has a weak (but not necessarily unique) solution if $b$ and $\sigma$ are continuous functions. If in this situation pathwise uniqueness holds for (12), then there exists a unique strong solution up to indistinguishability.

Similar to the theory of ordinary differential equations, the function on the right hand side of a stochastic differential equation being locally Lipschitz is sufficient in order to guarantee (strong) existence on a nonempty (stochastic) interval and (pathwise) uniqueness.

up to the time $T>0$ a.s., i.e. on the stochastic interval $[0,T)$. On $T<\infty$ we have that either $X$ hits the boundary $\partial U$ of $U$ at $T$, i.e. $X_{T}\in\partial U$ or explodes, i.e. $\limsup_{t\rightarrow T,t<T}||X_{t}||=\infty$. If $f$ satifies the linear growth condition

With the term unique it is meant that there holds pathwise uniqueness for (13). In other words, every two solutions $(X,Z)$ and $(X^{\prime},Z)$ of (13) defined on the same probability space and with the same continuous semimartingale $Z$ and the same initial value are indistinguishable.

If the process $Z$ from Theorem 3.34 is a continous Lévy process (Brownian motion with drift), it can be shown that the solution $X$ of (13) is a Markov process. Before we state this in a theorem, we give a general definition of the term ’Markov process’ and a necessary technical condition on our probability space.

then $Z$ is said to be a time homogeneous Markov process.

A time homogeneous Markov process is called a strong Markov process, if

At the beginning of this chapter we assumed $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},P)$ to be a filtered probability space where $(\mathcal{F}_{t})_{t\geq 0}$ is right continuous and all $\mathcal{F}_{t}$ are complete. Now, we need to enlarge our given probability space in order to get arbitrary initial values.

is a time homogeneous strong Markov process on $U$ under every probability measure of the family $(\overline{P}^{y})_{y\in U}$.

The Girsanov Theorem is a powerful tool that gives us the opportunity to construct a new probability measure $\widehat{Q}$ such that a drift changed $P$-Brownian motion (that is not a Brownian motion under the probability measure $P$ anymore) is a Brownian motion under the new measure $\widehat{Q}$. Before we state the Girsanov Theorem we make

Let $X$ be a stochastic process. The unique strong solution $Z=\mathcal{E}(X)$ of

From Theorem 3.34 we get immediately that (16) has a unique strong solution.

is a martingale, or, what is sufficient for (17), but not necessary, such that the Novikov condition is satisfied

is an equivalent probability measure, and

is a $\widehat{Q}$-Brownian motion on $[0,T)$.

With Kallenberg [1997, Corollary 16.25] the new measure $Q$ is then

McKean’s Argument

Lévy’s Theorem gives us an easy way to decide whether a continuous local martingale is a Brownian motion.

Let $B$ be a $p\times p$ dimensional continuous local martingale such that

for all $i,j,k\in\{1,\ldots,p\}$ and $B_{0}=0$. Then B is a $p\times p$ dimensional Brownian motion, $B\thicksim\mathcal{BM}_{p}$.

With Protter [2004, Theorem 40,Chapter 2] we can conclude that $X$ is a $p^{2}$-dimensional Brownian motion. ∎

Let $0<\tau_{0}<\infty$ be a stopping time and $M$ be a continuous local martingale on the interval $[0,\tau_{0}]$, which is not identically equal to zero. If we set

then $B_{t}:=M_{T_{t}}$ is a stopped $\mathcal{F}_{T_{t}}$-Brownian motion on the stochastic interval $[\,0,[M,M]_{\tau_{0}})$, $[M,M]_{\tau_{0}}>0$ a.s., i.e $B$ is a Brownian motion w.r.t. the $\sigma$-algebra

that is stopped at $[M,M]_{\tau_{0}}$. Furthermore, it holds that $M_{t}=B_{[M,M]_{t}}$, i.e. $M$ is a stopped, time-changed Brownian motion.

Hence $T_{t}<\infty$ as $\tau_{0}<\infty$. From Revuz and Yor [2001, Proposition 4.8 and 4.9, Chapter I] we know that the stopped process $M_{T_{t}}$ is $\mathcal{F}_{T_{t}}$-measurable. Furthermore,

as mentioned above. With Theorem 3.42 we can can conclude that $B$ is a stopped $\mathcal{F}_{T_{t}}$-Brownian motion on $[\,0,[M,M]_{\tau_{0}})$. To prove that $M$ is a time-changed Brownian motion, observe that $T_{[M,M]_{t}}>t$ if and only if $[M,M]$ is constant on $[t,T_{[M,M]_{t}}]$. As $M$ and $[M,M]$ are constant on the same intervals, we have $M_{T_{[M,M]_{t}}}=M_{t}$ and thus $B_{[M,M]_{t}}=M_{t}$. ∎

With the foregoing Theorem we can form an argument that will allow us in turn to proof some result on the existence of the Wishart process in the next chapter.

$h(r)$ is a continuous local martingale on the interval $[0,\tau_{0})$ for $\tau_{0}:=\inf\{s:\,r_{s}=0\}$

$\lim_{x\to 0,x>0}h(x)=\infty$ or resp. $\lim_{x\to 0,x>0}h(x)=-\infty$

If $h(r_{t})=\int_{0}^{t}f(s,r_{s})\,dW_{s}$ with a Brownian motion $W$, then for $h(r)$ being a continuous local martingale on $[0,\tau_{0})$ it is sufficient that $f$ is square integrable on $[0,\tau_{0})$.

Suppose $\tau_{0}<\infty$. Define $M:=h(r)$ and $T_{t}$ as in Theorem 3.43. $M$ is a continuous local martingale on $[0,\tau_{0}]$ and with Theorem 3.43 we can conclude that $B_{t}:=M_{T_{t}}$ is a stopped Brownian motion on $[\,0,[M,M]_{\tau_{0}})$. Observe, that $r_{0}>0$ and the continuity of $r$ impliy $\tau_{0}>0$. Consider the case $\lim_{x\to 0,x>0}h(x)=\infty$. On the interval $[0,\tau_{0})$, the process $h(r_{t})$ takes every value in $[h(r_{0}),\infty)$, especially $[h(r),h(r)]_{\tau_{0}}=[M,M]_{\tau_{0}}>0$. For $\tau_{0}<\infty$ we have

$[M,M]_{\tau_{0}}<\infty$, what is impossible because a Brownian motion can not go to infinity in finite time, see Revuz and Yor [2001, Law of the iterated logarithm, Corollary 1.12, Chapter II].

As both cases imply contradictions, we conclude $\tau_{0}=\infty$. The case for $\lim_{x\to 0,x>0}h(x)=-\infty$ is analogous. ∎

Ornstein-Uhlenbeck Processes

We give the following definition according to Bru such that we are later able to show that some solutions of the Wishart SDE can be constructed out of a matrix variate Ornstein-Uhlenbeck process.

is called $n\times p$-dimensional Ornstein-Uhlenbeck process. We write $X\thicksim\mathcal{OUP}_{n,p}(A,B,x_{0})$.

As $X\mapsto XB$ and $X\mapsto A$ are trivially globally Lipschitz and satisfy the linear growth condition we know from Theorem 3.34 that (21) has a unique strong solution on the entire interval $[0,\infty)$. Luckily, we are even able to give an explicit formula for the solution of (21).

For a Brownian motion $W\thicksim\mathcal{BM}_{n,p}$, the unique strong solution of (21) is given by

Furthermore, (22) is a strong solution by construction. ∎

The following Lemma will be useful for determining the conditional distribution of the Ornstein-Uhlenbeck process.

Now we are able to show that the (matrix variate) Ornstein-Uhlenbeck process is (matrix variate) normally distributed. Later we will see that some Wishart processes are therefore ‘square’ normally distributed, that is by Lemma 3.20 a noncentral Wishart distribution. To clarify terms, with distribution of a stochastic process $X$ we always mean from now on the distribution of $X_{t}$ at time $t$ conditional on the initial value $X_{0}$. Although it may be confusing, the terms distribution of a stochastic process and law of a stochastic process have a different meaning in this thesis.

Define $H_{t}=\int_{0}^{t}dW_{s}Ae^{-Bs}$. Lemma 3.48 shows us that

The equality $X_{t}=x_{0}e^{Bt}+H_{t}e^{Bt}$ and Theorem 3.15 gives us

Because ${\mathcal{A}}$ is a linear operator, ${\mathcal{A}}^{-1}$ is also linear and we can simplify

and, because ${\mathcal{A}}^{-1}$ is the integral from (23),

Now, we write the matrix $B$ in its Jordan normal form: There exists a matrix $T\in GL(p)$ such that

Without loss of generality, consider only the first Jordan block

Because $D$ and $N$ commute, $DN=ND$, we have

The matrix $N$ is nilpotent, as $N^{l}=0$. Hence, $\exp(Nt)$ is the finite sum

For $a<0$ and $t\rightarrow\infty$ we know that $e^{at}p_{t}$ converges to zero for every polynomial $p_{t}$ in $t$. Hence, $\lim_{t\rightarrow\infty}e^{at}\exp(Nt)=0$. As $e^{ibt}$ always has absolute value one, we also have that $\lim_{t\rightarrow\infty}\exp(Jt)=0$. With (26) it is now obvious that

The next step is to observe that ${\mathcal{S}}_{p}$ is a finite dimensional and normed space with the norm induced by the inner product trace. Hence, the linear operator ${\mathcal{A}}^{-1}:{\mathcal{S}}_{p}\rightarrow{\mathcal{S}}_{p}$ is continuous and with the above can conclude that $\Psi_{t}$ from (25) converges pointwise to

and the limit is independent of the initial value $x_{0}$. Thus, and by the fact that $X$ is a Markov process according to Theorem 3.39, the limit distribution in (28) is a stationary distribution. ∎

Chapter 4 Wishart Processes

In this chapter the main work on the theory of Wishart processes will be done. As an introduction we begin with the one dimensional case, called the square Bessel process. Afterwards, we focus on the general Wishart process and give a theorem about the existence and uniqueness of Wishart processes at the end of section 4.2. In section 4.3, we show that some Wishart processes can be expressed as matrix squares of the matrix variate Ornstein-Uhlenbeck process from section 3.5. Eventually, in section 4.4 we give an algorithm to simulate Wishart processes by using an Euler approximation.

The square Bessel process is the one-dimensional case of the Wishart process - in fact, even a special case of the one-dimensional Wishart process. We consider it separately, because it is much more elementary than for higher dimensions.

Let ${\alpha}\geq 0$, $\beta$ be a one-dimensional Brownian motion and $x_{0}\geq 0$. A strong solution of

Before we give a comprehensive theorem about existence, uniqueness and non-negativity of the square Bessel process, we state the following results for one-dimensional stochastic differential equations:

(Pathwise Uniqueness, cf. Yamada and Watanabe [1971a, Theorem 1]) Let

(Comparison Theorem, Revuz and Yor [2001, Chapter IX, Theorem 3.7]) Consider two stochastic differential equations

that both fulfill pathwise uniqueness and let $b^{1}$,$b^{2}$ be two bounded Borel functions such that $b^{1}\geq b^{2}$ everywhere and one of them is globally Lipschitz. If $(X^{1},\beta)$ is a solution of the first, and $(X^{2},\beta)$ a solution of the second stochastic differential equation ($X^{1}$,$X^{2}$ defined on the same probability space), w.r.t the same Brownian motion $\beta$ and if $X^{1}_{0}\geq X^{2}_{0}$ a.s., then

For ${\alpha}\geq 0$ and $x_{0}\geq 0$ there exists an unique strong and non-negative solution $X$ of

on the entire interval $[0,\infty)$. Moreover, if ${\alpha}\geq 2$ and $x_{0}>0$ the solution $X$ is positive a.s. on $[0,\infty)$.

We show with Theorem 4.3 that pathwise uniqueness holds for the stochastic differential equation

like in Revuz and Yor [2001, p. 439]. Since $|\sqrt{z}-\sqrt{z^{\prime}}|\leq\sqrt{|z-z^{\prime}|}$ for all $z,z^{\prime}\geq 0$, we have that

With Remark 3.33 we conclude that there exists a unique strong solution for (35). Next, we show that our solution never becomes negative: First, consider the case $x^{1}_{0}=0$ and ${\alpha}^{1}=0$. Obviously, $X^{1}\equiv 0$ is our unique strong solution in this case. Secondly, consider an arbitrary $x^{2}_{0}\geq 0$ and ${\alpha}^{2}\geq 0$. From the above, we have a unique strong solution $X^{2}$. The comparison theorem implies that

For ${\alpha}\geq 2$, one could also use McKean’s argument (Theorem 3.44) to show that the square Bessel process is positive, analogously to the Proof of Theorem 4.14.

General Wishart Processes: Definition and Existence Theorems

the Wishart SDE. A strong solution $S$ of (37) in $\overline{{\mathcal{S}}_{p}^{+}}$ is said to be a ($p\times p$-dimensional) Wishart process with parameters $Q,K,{\alpha},s_{0}$, written $S\thicksim\mathcal{WP}_{p}(Q,K,{\alpha},s_{0})$.

To understand (37) intuitively, it may help to have a look at a approximation of the form

Here we can see that $Q$ controls the covariance of our normal distributed fluctuations, and that this fluctuations are proportional to the square root of our process:

From the theory of ordinary linear differential equations (see Timmann [2005, p. 83], for example), we know that the solution of (38) is given by

If the eigenvalues of $K$ only have negative real parts, $Re(\sigma(K))\subseteq(-\infty,0)$, then, because of $\sigma({\mathcal{C}})=\sigma(K)+\sigma(K)$, we also have $Re(\sigma({\mathcal{C}}))\subseteq(-\infty,0)$. From the proof of Theorem 3.50 we get that $\lim_{t\rightarrow\infty}e^{{\mathcal{C}}t}=0$ and thus from (40)

Hence, the deterministic solution of (38) converges to $-{\mathcal{C}}^{-1}M$. Thus, the stochastic solution of (37) will fluctuate around $-{\mathcal{C}}^{-1}M$.

for $\rho(U)=\sqrt{U}$. Hence the theorem can not be applied to our case. Furthermore, Yamada and Watanabe [1971b, Remark 2] have also shown that (41) is, for $p\geq 3$, nearly best possible in the sense that, if $\int_{U\in\overline{{\mathcal{S}}_{p}^{+}}:||U||\leq 1}\rho^{-2}(U)U\,du<\infty$ and $\rho$ is subadditive then, a stochastic differential equation can be constructed that has two solutions, and thus pathwise uniqueness cannot hold. But the fact the we cannot use Yamada and Watanabe [1971b, Theorem1] does not give any evidence whether pathwise uniqueness holds for the Wishart SDE or not. Before we can begin our mathematical analysis of (37), we need a few auxiliary results:

Let $S\thicksim\mathcal{WP}_{p}(Q,K,{\alpha},s_{0})$. Then

The first summand of $d[H_{ij},H_{kl}]_{t}$ is equal to

Because $S_{t}$ is symmetric this term can be simplified to

The other three summands can be evaluated in the same way. ∎

Observe that the denominator is zero if and only if the numerator is zero, so we make the convention $\frac{0}{0}:=1$. Then we have by definition

and therefore $\beta^{h}$ is a sum of continuous local martingales and thus a continuous local martingale itself. Furthermore

and with Lévy’s Theorem the proof is complete. ∎

The matrix variate square root function is locally Lipschitz on ${\mathcal{S}}_{p}^{+}$.

For every initial value $s_{0}\in{\mathcal{S}}_{p}^{+}$ there exists a unique strong solution $S$ of the Wishart SDE (37) in the cone ${\mathcal{S}}_{p}^{+}$ of all positive definite matrices up to the stopping time

is continuously differentiable, see Deuflhard and Hohmann [2002, Lemma 5.1]. Hence,

is a closed set; and also convex, as for all $M_{1},M_{2}\in U_{n},{\alpha}\in$:

Next, we define for $S\in{\mathcal{S}}_{p}^{+}$ the linear operator by

Because ${\mathcal{C}}$ is a bounded linear operator ($\dim{\mathcal{S}}_{p}<\infty$), we have

Let now be $Y\in{\mathcal{S}}_{p}^{+}$. As the matrix variate square root function is locally Lipschitz, there exists an open neighbourhood ${\mathcal{U}}(Y)$ of $Y$ and a constant $C(Y)$, such that for all $S,R\in{\mathcal{U}}(Y)$

Hence, we have for all $S,R\in{\mathcal{U}}(Y)$

i.e. ${\mathcal{Z}}$ is locally Lipschitz on ${\mathcal{S}}_{p}^{+}$. At all $f$ is locally Lipschitz because

Hence, we know that there exists a non-zero stopping time $T>0$ such that there exists a unique ${\mathcal{S}}_{p}^{+}$-valued strong solution $S$ of (37) for $t\in[0,T)$. If $T<\infty$, $S_{T}$ either hits the boundary of ${\mathcal{S}}_{p}^{+}$ or explodes. We show that the latter cannot happen. Fix $R\in{\mathcal{U}}(Y)$ and set $S=Y$. Then we from (43)

with $L=\max\{K^{2}||R||^{2},2K^{2}||R||,K^{2}\}$. Because of $||Y||\leq 1+||Y||^{2}$

Hence, $Y\mapsto f(Y)-f(R)$ satisfies the linear growth condition, and so does $f:Y\mapsto f(Y)$.

Thus, if $T<\infty$, we know that $S_{T}$ hits the boundary of ${\mathcal{S}}_{p}^{+}$ the first time, i.e. $S_{T}\in\overline{{\mathcal{S}}_{p}^{+}}$, but $S_{T}\notin{\mathcal{S}}_{p}^{+}$, and $S_{t}\in{\mathcal{S}}_{p}^{+}$ for all $t<T$. Hence

and hence all second derivatives are zero. With Itô’s formula (Theorem 3.27) we get

Let $S\thicksim\mathcal{WP}_{p}(Q,0,{\alpha},s_{0})$. Then, for $\xi\neq 0$

for $t\in[0,T)$ with $T=\inf\{s:\,\det(S_{s})=0\}$, where $\beta$ is a one-dimensional Brownian motion.

These results can also be found in Bru [1991, p. 747] without proof.

First, we prove Equation (44): According to Itô’s formula, we have

where we used Lemma 4.9 in the last equation. For the second order term, we get

where we used Lemma 4.8 and Lemma 2.6, again. At all, we get equation (44)

Let $K=0$, $s_{0}\in{\mathcal{S}}_{p}^{+}$ and ${\alpha}\geq p+1$. Then there exists a unique strong solution in ${\mathcal{S}}_{p}^{+}$ of the Wishart SDE (37) on $[0,\infty)$.

where we adopt the idea from Bru [1991, p.734] to use McKean’s argument. As a matrix norm we choose

First consider the case ${\alpha}=p+1$. Then we have from (46)

where $T_{n}<\infty$ because $T<\infty$, that converges to $T$ such that $\ln(\det(S_{\min\{t,T_{n}\}}))$ is a martingale:

for $t\in[0,T_{n})$. That means by definition that $\ln(\det(S_{t}))$ is a local martingale on $[0,T)$. Now we can apply McKean’s argument, Theorem 3.44, with $r_{t}=\det(S_{t})$ and $h\equiv ln$. By assumption we know $\det(S_{0})>0$ and we have shown above that $h(r_{t})=\ln(\det(S_{t}))$ is a local martingale on $[0,T)$. Obviously, $\ln(\det(S_{t}))$ converges to -$\infty$ for $t\rightarrow T$ and hence McKean’s argument implies $T=\infty$. That is a contradiction to our assumption. Logically consistent, we can conclude $T=\infty$.

In the case ${\alpha}>p+1$, we set $\xi=\frac{p+1-{\alpha}}{2}<0$ and

Hence, we can apply McKean’s argument for the local martingale $\det(S_{t})^{\xi}$ on $[0,T)$, because $\det(S_{t})^{\xi}$ converges to infinity for $t\rightarrow T$. The same reasoning as above implies the contradiction $T=\infty$. Finally, Theorem 4.11 proves the statement. Observe that, for ${\alpha}<p+1$, we have $\xi>0$ and thus $\det(S_{t})^{\xi}$ does converge to zero, so we cannot apply McKean’s argument in this case. ∎

Eventually, we are able to state the final theorem about the existence and uniqueness of the Wishart process, which is the main achievement of this section.

From Theorem 4.14 we know that there exists an unique strong solution $(\widehat{S},\widehat{B})$ of (51) on $[0,T^{\widehat{S}})$ with $T^{\widehat{S}}=\inf\{s:\det(\widehat{S_{s}})=0\}=\infty$. Define the process

In this proof, with solution we always mean an ${\mathcal{S}}_{p}^{+}$-valued solution.

With Girsanov’s Theorem (Theorem 3.41) we are able to conclude that

defines a Brownian motion w.r.t. the equivalentprobability measure $\widehat{Q}$ as defined in (19).

Hence $(\widehat{S},B)$ is a solution of (53) on $[0,\infty)$.

From Theorem 4.11 we know that there exists a unique strong solution $S$ of (55) on the interval $[0,T^{S})$ with $T^{S}=\inf\{s:\,\det(S_{s})=0\}>0$. The pathwise uniqueness implies uniqueness in law, thus $S$ and $\widehat{S}$ have the same distribution, and so have $T^{S}$ and $T^{\widehat{S}}$. Hence, $T^{S}=T^{\widehat{S}}=\infty$ and $(S,B)$ is an unique strong solution $S$ of (53) on the interval $[0,\infty)$. ∎

For the rest of this thesis, we will assume that (52) is a martingale.

Before we end this chapter, we want to compare our results to the one stated in Bru :

(Cf. Bru [1991, Theorem 2”]) If ${\alpha}\in\Delta_{p}:=\{1,\ldots,p-1\}\cup(p-1,+\infty)$, $A\in GL(p)$, $B\in{\mathcal{S}}_{p}^{-}$, $s_{0}\in{\mathcal{S}}_{p}^{+}$ and has all its Eigenvalues distinct, and $M$ is a $p\times p$-dimensional Brownian motion, then the stochastic differential equation

Compared to (56), our definition of the Wishart SDE (37) is more general, as we allow the drift matrix $K$ to be an arbitrary matrix, whereas in (56) the matrix $B$ has to be symmetric negative definite. In the case ${\alpha}\in[p+1,\infty)$, the result in Theorem 4.15 extends the one in Theorem 4.18 because we prove the existence of a strong solution, that is unique up to indistinguishability and takes almost surely values in the cone of all symmetric positive definite matrices. Furthermore, this solution has infinite lifetime independently of any collision of the eigenvalues of our solution.

Square Ornstein-Uhlenbeck Processes and their Distributions

$M_{t}$ is a local martingale. Furthermore, observe that

With Theorem 3.42 we con conclude that $M$ is a Brownian motion. Finally, using the partial integration formula shows us that our pair $(S,M)$ is a solution of the stochastic differential equation (57).

So far we have shown that $(S,M)$ is a weak solution of (57). From Theorem 4.15 we know that pathwise uniqueness holds for (57), and thus by Remark 3.33 (ii) we have that $(S,M)$ is also a strong solution of (57). ∎

From (58) and Theorem 3.19 we know for the solution $S$ of (57) that $S_{t}$ given $S_{0}$ has characteristic function

From the proof of Theorem 3.50 we know that

Simulation of Wishart Processes

Recall the stochastic differential equation of the Wishart process, that is

We now try to approximate the stochastic integral above to make the Wishart process suitable for numerical simulation. An easy way of doing this, is the Euler-Maruyama method (see Kloeden and Platen [1999, p. 340]):

where $Z_{ij}\thicksim{\mathcal{N}}_{1}(0,1)$ and $\stackrel{{\scriptstyle\mathscr{D}}}{{=}}$ denotes distributional equivalence. Written in matrix notation, we get for the increment of $B$

where $Z\thicksim{\mathcal{N}}_{p,p}(0,I_{2p})$. A sample implementation can be found in the appendix. Now we give a few examples.

First we begin with the one-dimensional square Bessel process, i.e. a solution of

where $\beta$ denotes a one-dimensional Brownian motion. From Theorem 4.5 we know that this process never becomes negative for any choice of ${\alpha},x_{0}\geq 0$ and remains positive for $x_{0}>0$ and ${\alpha}\geq 2$. Here are two sample paths over the interval $$and initial step size$h=10^{-3}$: In the case$\alpha=0.5$, the algorithm had to reduce the initial step size to$1.5625\cdot 10^{-5}$at at least one point in order to guarantee non-negativity, whereas for$\alpha=2$this was not necessary. Observe that in the case$\alpha=0.5<2$, the square Bessel process can become zero, but gets reflected instantaneously.

Our next example is the 2-dimensional Wishart process

with a 2-dimensional Brownian motion $B$, ${\alpha}=3$, $Q=(\begin{smallmatrix}1&2\\ 0&-3\end{smallmatrix})$, $K=-4I_{2}$ and $s_{0}=I_{2}$. We show sample paths for the three different entries of $S$, that are $S_{11}$, $S_{22}$ and $S_{12}$, and for $\frac{S_{12}}{S_{11}^{2}+S_{22}^{2}}$, what would be the correlation in a stochastic volatility model. The algorithm had to reduce the initial step size of $h=10^{-3}$ by half at some points. We have for the eigenvalues of $S$.

The last example are two 5-dimensional solutions of the stochastic differential equation

For the first one, we have ${\alpha}_{1}=7.2$, and the smallest and the largest eigenvalue of $S$ look like In the second case, ${\alpha}_{2}=3.5$, we don’t know if there exists any solution of (65) for $i=2$. Thus, we do not know whether our simulated process corresponds to a solution of (65) for $i=2$, and if it does, it does not have to be a Wishart process by Definition 4.7 (because we demand the Wishart process to be a strong solution). It may be noted, that the algorithm had to reduce the initial step size of $h=10^{-3}$ to $7.8125\cdot 10^{-6}$ at some points. Again, we shall have a look at the eigenvalues of our simulated process:

Chapter 5 Financial Applications

We first focus on two well-known financial models that are based on the one-dimensional Wishart processes, that is in fact a generalized squared Bessel process. Later, we consider the actual matrix variate case.

has a unique, strong and positive solution in $(0,\infty)$. For $k<0$ and the new parametrization $\sigma=2q$, $a=-2k$ and $b=\frac{-{\alpha}q^{2}}{2k}$ the stochastic differential equation gets the form

that is the stochastic differential equation of the CIR process. It is mean reverting as $a>0$ with $b\geq\frac{-q^{2}}{k}$. As Glasserman [2004, p.122] has shown, $S_{t}|s_{0}$ is distributed as $\frac{\sigma^{2}(1-\exp(-at))}{4a}$ times a noncentral chi-square random variable with $\frac{4ab}{\sigma^{2}}$ degrees of freedom and noncentrality parameter $\frac{4a\exp(-at)}{\sigma^{2}(1-\exp(-at))}s_{0}$. From now on we follow Gourieroux [2007, p.184f.]. Let’s assume that the interest rate $r$ follows a stochastic differential equation of the form (66),

where $W$ is a Brownian motion under a risk-neutral probability measure $Q$. Then, the price of a zero-coupon bond at time $t$ with time to maturity $h$ is given by

where $E_{t}^{Q}$ denotes the conditional expectation $E^{Q}[\cdot|\sigma\{r_{s}:\,s\leq t\}]$ under the measure $Q$. Then Cox et al. have shown that

with $c:=\sqrt{a^{2}+2\sigma^{2}}$. Hence, we have a closed form solution for $B(t,t+h)$ that is exponential affine in $r_{t}$.

Heston Model

A process $S$ of the form (66) can also be used to model the volatility in a stochastic volatility Black-Scholes model according to Heston

where the stock price at time $t$ is denoted by $X_{t}$. We refer to Gourieroux [2007, Chapter 2.2.2.] for details.

Factor Model for Bonds

In contrast to section 12 where the risk-free rate $r$ followed a stochastic differential equation of the form (67), we now want to use a factor model to consider corporate bonds jointly with a long term government bond (e.g. T-bond). Again, we summarize the results of Gourieroux [2007, chapter 3.5.2.]. Denote by $\lambda_{i,t}$ the default intensity for firm $i$, $i=1,\ldots,K$. Let us assume that

where $c,d_{i}\geq 0$ are nonnegative, $C,D_{i}\in{\mathcal{S}}_{p}^{+}$ and $S\thicksim\mathcal{WP}_{p}(Q,K,{\alpha},s_{0})$, i.e.

With the convention $\lambda_{0,t}=0$ we get for the price $B_{0}$ of a zero-coupon bond and the prices of corporate bonds $B_{i}$ the formula

under a risk neutral measure $Q$. If we insert (69) and (70) into (72) we get

with the convention $d_{0},D_{0}\equiv 0$. Gourieroux shows that there exists a closed form expression for (73).

Matrix Variate Stochastic Volatility Models

An application for matrix variate stochastic processes can be found in Fonseca et al. , which model the dynamics of $p$ risky assets by

In contrast to a continous model of the form (3), another multivariate stochastic volatility model for the logarithmic stock price process can be given by

If one wants to model the volatility with a time continuous stochastic process, e.g. for economic reasons, one could also suggest the model

where the process $\Sigma$ was substituted by the Wishart process of (71).

Chapter 6 MATLAB Code for Simulating a Wishart Process

This is an implementation in MATLAB of the algorithm described in section 11.