Non-stochastic portfolio theory
Vladimir Vovk
q-fin.PM 91G10 (Primary) 60G05, 60G44 (Secondary)

NNDrone : a toolkit for the mass application of machine learning in High Energy Physics
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    December 30, 2023
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  This paper studies a non-stochastic version of Fernholz's stochastic portfolio theory
  for a simple model of stock markets with continuous price paths.
  It establishes non-stochastic versions of the most basic results of stochastic portfolio theory
  and discusses connections with Stroock–Varadhan martingales.

    

    The version at
    http://probabilityandfinance.com/articles/index.html#51http://probabilityandfinance.com
    (Working Paper 51)
    is updated most often.




§ INTRODUCTION


Fernholz's stochastic portfolio theory <cit.>,
as its name suggests, depends on a stochastic model of stock prices.
This paper proposes a non-stochastic version of this theory based on the framework of <cit.>
(see the end of this section for a brief discussion of its relation to <cit.>).

A key finding
(see, e.g., <cit.>, <cit.>, <cit.>)
of stochastic portfolio theory is that, under certain simplifying assumptions,
there is a long-only portfolio that outperforms the capital-weighted market portfolio.
The principal aim of this paper is to give a simple non-stochastic formalization of this phenomenon.

Section <ref> defines our model of a stock market
and introduces non-stochastic notions of a portfolio's value and its excess growth component.
Section <ref> is devoted to a non-stochastic version of the “master equation” of stochastic portfolio theory,
and Section <ref> to its applications.
In particular, the latter covers the entropy-weighted portfolio
(as in <cit.> and <cit.>)
and diversity-weighted portfolios
(<cit.>, <cit.>, going back to at least <cit.>).
Section <ref> is devoted to detailed interpretations and discussions of the results of the previous sections.
Section <ref> discusses connections with Stroock–Varadhan martingales,
which make the master equation very intuitive.
Finally, Section <ref> lists some directions of further research.

Another paper treating stochastic portfolio theory in a pathwise manner is <cit.>,
and it considers a wider class of portfolios.
However, that paper relies on some assumptions that are not justified by economic considerations:


  * it postulates a suitable “refining sequence of partitions”;

  * it postulates the existence of a continuous covariation between each pair of price paths
  w.r. to this refining sequence of partitions (in Föllmer's <cit.> sense);

  * a possible extension to non-smooth portfolio generating functions
  (as in <cit.>) would require postulating the existence of local times
  (perhaps along the lines of <cit.>).




§ MARKET AND PORTFOLIOS



This paper uses the definitions and notation of <cit.> and <cit.>
(the latter, however, will always be repeated).
The notation ∫ X  Y is used for the process whose value at time t
is ∫_0^t X(s)  Y(s), both for Itô and Lebesgue–Stieltjes integration.
The brackets […] always signify quadratic variation and are never used in the role of parentheses.
The abbreviations “q.a.” and “ucqa” stand for “quasi always” and “uniformly on compacts quasi always”;
see <cit.> for definitions.

We consider a financial market in which J idealized securities, referred to as stocks, are traded;
their price paths S_j:[0,∞)→(0,∞), j=1,…,J, are assumed to be continuous functions,
and they never pay dividends.
We let C[0,∞) stand for the set of all continuous real-valued functions on [0,∞).
As in <cit.>, we fix a sufficiently rich language for defining sequences of partitions;
all notions of non-stochastic Itô calculus used in this paper
(such as Itô integral and Doléans exponential and logarithm)
are relative to this language.

For convenience, we identify S_j(t) with the total market capitalization of the jth stock at time t∈[0,∞).
The total capitalization of the market is defined as the process

    S(t) := ∑_j=1^J S_j(t),
        
      t∈[0,∞),

and the market weight of the jth stock is

    μ_j(t) := S_j(t)/S(t),
        
      j=1,…,J.

We take the total capitalization of the market as our numéraire,
which allows us to regard μ_1,…,μ_J,1 as the traded securities
(cf. <cit.>),
the first J of them being just like our original securities S_j but constrained by μ_1+⋯+μ_J=1.
(In fact, the original securities S_j will never be used explicitly in the rest of this paper
apart from an informal remark.)

Let Δ^J be the interior of the standard simplex in ^J,

    Δ^J
      :=
      {
        x = (x_1,…,x_J) ∈ (0,1)^J
        
        x_1 + ⋯ + x_J = 1
      }.

A basic portfolio is a continuous bounded function π:Δ^J→Δ^J
mapping Δ^J to its closure in ^J;
intuitively, it maps the current market weights μ=(μ_1,…,μ_J)
to the fractions π(μ)=(π_1(μ),…,π_J(μ)) of the current capital invested in the J stocks.
(In this paper we will only need these very primitive Markovian portfolios.)

The non-stochastic notions of Doléans exponential  and Doléans logarithm  used in this paper
are defined in <cit.>.
The most useful for us interpretation of Doléans logarithm
is that (Y) is the cumulative return of a price path Y∈ C[0,∞),
and Doléans exponential restores the price path from its cumulative return.
The value process of π is the Doléans exponential

    Z_π   :=
      (
        ∫π(μ)(μ)
      )
      
       :=
      (
        ∑_j=1^J
        ∫π_j(μ)(μ_j)
      )
      =
      (
        ∑_j=1^J
        ∫π_j(μ)/μ_jμ_j
      ),

where μ:[0,∞)→^J is defined by μ(t):=(μ_1(t),…,μ_J(t)),
π_j(μ):[0,∞)→ is defined by π_j(μ)(t):=π_j(μ(t)),
and π(μ):[0,∞)→^J is defined by π(μ)(t):=(π_1(μ)(t),…,π_J(μ)(t)).
The value process Z_π is defined and continuous quasi always.

The definition (<ref>) involves Doléans logarithm,
but stochastic portfolio theory emphasizes regular logarithm
(cf. the logarithmic model in <cit.>).
On the log scale the definition (<ref>) can be rewritten as

    ln
      Z_π   =
      ln(
        ∑_j=1^J
        ∫π_j(μ)(μ_j)
      )
       =
      ∑_j=1^J
      ∫π_j(μ)(μ_j)
      -
      1/2[
        ∑_j=1^J
        ∫π_j(μ)(μ_j)
      ]
       =
      ∑_j=1^J
      ∫π_j(μ)lnμ_j
      +
      1/2∑_j=1^J
      ∫π_j(μ)[lnμ_j]
           -
      1/2[
        ∑_j=1^J
        ∫π_j(μ)lnμ_j
      ]
        q.a..

The second equality in the chain (<ref>)–(<ref>) follows from the standard equality

    (X)
      =
      exp(X-[X]/2)
        q.a.

and the third equality in (<ref>)–(<ref>) follows from

    (Y)
      =
      ln Y_t + 1/2 [ln Y]
        q.a.

(showing that the first term in (<ref>) can be represented as (<ref>))
and a slight generalization of

    [(Y)]=[ln Y]
        q.a.

(showing that the second term in (<ref>) can be rewritten as (<ref>)).
See <cit.> for (<ref>)–(<ref>).

The part

    Γ^*_π   =
      1/2∑_j=1^J
      ∫π_j(μ)[lnμ_j]
      -
      1/2[
        ∑_j=1^J
        ∫π_j(μ)lnμ_j
      ]
      
       =
      1/2∑_j=1^J
      ∫π_j(μ)[lnμ_j]
      -
      1/2∑_i,j=1^J
      ∫π_i(μ)π_j(μ)
      [lnμ_i,lnμ_j]

of (<ref>)–(<ref>) consisting of the last two addends
will be called the excess growth term
(it corresponds to the cumulative excess growth rate in stochastic portfolio theory).
We can use it to summarize (<ref>)–(<ref>) as

    ln Z_π
      =
      ∑_j=1^J
      ∫π_j(μ)lnμ_j
      +
      Γ^*_π  q.a.

The addend ∑_j=1^J∫π_j(μ)lnμ_j is the naive expression for the cumulative log growth in the value of π,
and Γ^*_π is the adjustment required to obtain the true cumulative log growth.

A particularly important special case is that of the market portfolio, π=μ.
To understand the intuition behind the excess growth term (<ref>) in this case,
we can rewrite 2Γ^*_μ as

    2Γ^*_μ(t)
         =
      ∑_j=1^J
      ∫_0^t
      μ_j(s)
      [lnμ_j](s)
      -
      [
        ∑_j=1^J
        ∫μ_j
        lnμ_j
      ](t)
      
       =
      ∑_j=1^J
      ∫_0^t
      μ_j(s)
      [lnμ_j](s)
      =
      ∑_j=1^J
      ∫_0^t
      [μ_j](s)/μ_j(s)
       ≥∑_j=1^J
      ∫_0^t
      [μ_j](s)
      =
      ∑_j=1^J
      [μ_j](t),

where we have used the fact that the subtrahend in (<ref>),
being the quadratic variation of a monotonic function
(remember that ∑_jμ_j=1),
is zero.
We can see that 2Γ^*_μ(t) is bounded below
by the total quadratic variation of the market weights.



§ MASTER EQUATION



Let  be a C^2 positive function defined on an open neighbourhood  of Δ^J in ^J.
For any C^2 function F (such as ln) defined on 
we let D_j stand for its jth partial derivative,

    D_j F(x)
      =
      ∂ F/∂ x_j(x),
        
      x=(x_1,…,x_J)∈,

and D_ij stand for its second partial derivative in x_i and x_j,

    D_ij F(x)
      =
      ∂^2 F/∂ x_i ∂ x_j(x).

The portfolio generated by  is defined by

    π_j(x)
      :=
      (
        D_j ln(x)
        +
        1
        -
        ∑_k=1^J
        x_k D_k ln(x)
      )
      x_j.

The main part of the expression in the parentheses is D_jln(x);
the rest is simply the normalizing constant c=c(x) making (D_jln(x)+c)x_j a portfolio
(it is a constant in the sense of not depending on j).

Now we can state a non-stochastic version of the “master equation” of stochastic portfolio theory
(see, e.g., <cit.>).


  The value process Z_π of the portfolio π generated by  satisfies
  
    ln Z_π(t)
        =
        ln(μ(t))/(μ(0))
        +
        Θ(t)
          q.a.,

  where
  
    Θ(t)
        :=
        ∫_0^t
        -1/2(μ(s))∑_i,j=1^J
        D_ij(μ(s))
        [μ_i,μ_j](s).




  The middle equality (<ref>) in the chain (<ref>)–(<ref>)
  gives for the left-hand side of (<ref>):
  
    ln Z_π(t)
           =
        ∑_j
        ∫_0^t
        π_j(μ(s))/μ_j(s)μ_j(s)
        -
        1/2∑_i,j∫_0^t
        π_i(μ(s))π_j(μ(s))/μ_i(s)μ_j(s)[μ_i,μ_j](s)
       =
        ∑_j
        ∫_0^t
        (
          D_j ln(μ(s))
          +
          1
          -
          ∑_k
          μ_k(s) D_k ln(μ(s))
        )
        μ_j(s)
         -
        1/2∑_i,j∫_0^t
        (
          D_i ln(μ(s))
          +
          1
          -
          ∑_k
          μ_k(s) D_k ln(μ(s))
        )
           ×(
          D_j ln(μ(s))
          +
          1
          -
          ∑_k
          μ_k(s) D_k ln(μ(s))
        )
        [μ_i,μ_j](s)
       =
        ∑_j
        ∫_0^t
        D_j(μ(s))/(μ(s))μ_j(s)
         -
        1/2∑_i,j∫_0^t
        D_i(μ(s))/(μ(s))D_j(μ(s))/(μ(s))[μ_i,μ_j](s)
          q.a.,

  where the last equality follows from ∑_kμ_k=1.
  Next we apply the Itô formula to the function
  ln
  on the right-hand side of (<ref>);
  the Itô formula still holds in our non-stochastic setting: cf. <cit.>.
  For the first addend on the right-hand side of (<ref>) it gives us the expression
  
    ln(μ(t))/(μ(0))   =
        ∑_j
        ∫_0^t
        D_j (μ(s))/(μ(s))μ_j(s)
        +
        1/2∑_i,j∫_0^t
        D_ij(μ(s))/(μ(s))[μ_i,μ_j](s)
         -
        1/2∑_i,j∫_0^t
        D_i(μ(s))D_j(μ(s))/(μ(s))^2[μ_i,μ_j](s)

  equal, q.a., to (<ref>)–(<ref>) minus Θ, as defined in (<ref>).




§ SPECIAL CASES



A positive C^2 function  defined on an open neighbourhood of Δ^J
is a measure of diversity if it is symmetric and concave.
In this section we will discuss three examples of measures of diversity.



 §.§ Fernholz's arbitrage opportunity


In <cit.>, Fernholz describes an arbitrage opportunity for his stochastic model of the market.
In the non-stochastic setting of this paper his portfolio ceases to be an arbitrage opportunity
but it is still interesting and suggests the possibility of beating the market
(as discussed in the next section).
Now we are interested in the measure of diversity

    (x)
      :=
      1
      -
      1/2∑_j=1^J
      x_j^2.

The components (<ref>) of the corresponding portfolio π are

    π_j(x)
      =
      (
        2-x_j/(x)
        -
        1
      )
      x_j.

Now Theorem <ref> gives the following non-stochastic version of <cit.>.


  The value process Z_π of the portfolio (<ref>) satisfies
  
    ln Z_π(t)
        =
        ln(μ(t))/(μ(0))
        +
        ∑_j=1^J
        ∫_0^t
        [μ_j](s)/2(μ(s))  q.a.




  Plugging D_ij(x)=-_i=j
  (where _E stands for the indicator function of E)
  into (<ref>),
  we indeed obtain
  
    Θ(t)
        =
        ∫_0^t
        1/2(μ(s))∑_j
        [μ_j](s).



A slightly cruder but simpler version of Corollary <ref> is:


  The value process Z_π of the portfolio (<ref>) satisfies
  
    ln Z_π(t)
        ≥
        -ln2
        +
        1/2∑_j=1^J
        [μ_j](t)
          q.a.




  It suffices to notice that ∈[1/2,1].




 §.§ Entropy-weighted portfolio


The archetypal measure of diversity <cit.> is the entropy function

    (x)
      :=
      -
      ∑_j=1^J
      x_j ln x_j.

Using (<ref>),
the components of the corresponding entropy-weighted portfolio can be computed as

    π_j(x)
      =
      -x_jln x_j/(x).

Calculating the drift term Θ in Theorem <ref>,
we obtain the following corollary
(a non-stochastic version of <cit.>).


  The value process Z_π of the entropy-weighted portfolio π satisfies
  
    ln Z_π(t)
        =
        ln(μ(t))/(μ(0))
        +
        ∫_0^t
        Γ^*_μ(s)/(μ(s))  q.a.




  Plugging D_ij(x)=-_i=j/x_j into (<ref>),
  we obtain
  
    Θ(t)
        =
        ∫_0^t
        1/2(μ(s))∑_j
        [μ_j](s)/μ_j(s).

  It remains to compare this expression with (<ref>).




 §.§ Diversity-weighted portfolios with parameter p


Fix p∈(0,1).
Define the measure of diversity with parameter p∈(0,1) <cit.> as

    _p(x)
      :=
      (
        ∑_j=1^J x_j^p
      )^1/p.

The p-diversity-weighted portfolio has components

    π_j(t)
      :=
      μ_j(t)^p/∑_i=1^J μ_i(t)^p.


The following corollary is a non-stochastic version of <cit.>.


  The value process Z_π of the diversity-weighted portfolio π with parameter p∈(0,1)
  satisfies
  
    ln Z_π(t)
        =
        ln_p(μ(t))/_p(μ(0))
        +
        (1-p)
        Γ^*_π(t)
          q.a.




  Now (<ref>) gives
  
    Θ(t)
           =
        ∫_0^t
        -1/2_p(μ(s))
         ×(
          ∑_i,j
          (1-p)
          (
            μ_i(s)
            μ_j(s)
          )^p-1(
            ∑_k μ_k(s)^p
          )^1/p-2[μ_i,μ_j](s)
         +
          ∑_j
          (p-1)
          (
            μ_j(s)
          )^p-2(
            ∑_k μ_k(s)^p
          )^1/p-1[μ_j](s)
        )
       =
        1-p/2∫_0^t
        (
          ∑_jπ_j(μ(s))
          [μ_j](s)/μ_j(s)^2
          -
          ∑_i,jπ_j(μ(s))
          [μ_i,μ_j](s)/μ_i(s)μ_j(s))
       =
        (1-p)
        Γ^*_π(t).



Corollary <ref> immediately implies:

  The value process Z_π of the diversity-weighted portfolio π with parameter p∈(0,1)
  satisfies
  
    ln Z_π(t)
        ≥
        (1-p)
        Γ^*_π(t)
        -
        1-p/pln J
          q.a.




  Since _p∈[1,J^(1-p)/p], we have
  
    ln_p(μ(t))/_p(μ(0))≥
        -1-p/pln J

  (cf. <cit.>);
  plugging this into (<ref>) gives (<ref>).




§ BEATING THE MARKET



The results of the previous section have striking implications for our idealized financial market.
The easiest to discuss is Corollary <ref>.
It can be interpreted, very informally, as the following Fisherian disjunction:
either the variation of each stock in the market decays,
in that the total quadratic variation [μ_j](∞) of each of the J market weights over [0,∞) is finite,
or we can beat the market in the sense that lim_t→∞Z_π(t)=∞
(cf. <cit.>, p. 42).
Notice that the second alternative of the disjunction also takes care of the “q.a.” in (<ref>).

The portfolio (<ref>) is particularly tame
(or admissible, in Fernholz's <cit.> terminology):
it is long-only,
it never loses more than 50% of its value relative to the market portfolio
(by (<ref>)),
and it never invests more than 3 times more than the market portfolio
in any of the stocks.

A more specific possible interpretation of Corollary <ref> is based on the efficient market hypothesis
in the form that was so forcefully advocated in the bestseller <cit.> by Burton G. Malkiel;
for him,
“the strongest evidence suggesting that markets are generally quite efficient is that professional investors do not beat the market.”
Even if there are ways to beat the market,
it is often believed that they should involve something unusual
rather than merely simple portfolios
such as (<ref>), (<ref>), or (<ref>)
(widely known since at least 2002).
According to this interpretation, Corollary <ref> implies that in efficient markets
we expect market variation to die down eventually.

If we believe that the variation in our stock market will never die down,
we are forced to admit that Corollary <ref>
“opens the door to superior long-term investment returns
through disciplined active investment management” <cit.>.
This is the interpretation on which typical practical applications of stochastic portfolio theory are based
(see, e.g., <cit.>, which, however,
is based on the stochastic versions of Corollaries <ref> and <ref>
rather than Corollary <ref>).

Corollary <ref> is a cruder version of Corollary <ref>
that replaces the first addend on the right-hand side of (<ref>) by its lower bound
and the denominator in the second addend by its upper bound.
Corollary <ref> is more precise in that it decomposes the growth in the portfolio's value
into two components:
one related to the growth in the diversity (μ) of the market weights
and the other related to the accumulation of the variation of the market weights.

It is standard in stochastic portfolio theory to assume
both that the market does not become concentrated, or almost concentrated, in a single stock
and that there is a minimal level of stock volatility;
precise versions of these assumptions are referred to as diversity and non-degeneracy, respectively.
We will see that the results of the previous section can be interpreted as saying that we can beat the market
unless it loses its diversity or degenerates.
Corollary <ref> says that, in fact,
the condition of non-degeneracy alone is sufficient;
this follows from the representation [lnμ_j]=[μ_j]/μ_j^2.
(But remember that our exposition is in terms of market weights μ_j rather than prices S_j,
which are usually used in stochastic portfolio theory.)

Corollary <ref> relies on both assumptions, diversity and non-degeneracy.
If the market maintains its diversity, we expect the first addend on the right-hand side of (<ref>) to stay bounded below,
and if, in addition, the market does not degenerate, we expect the second addend to increase steadily.
As a result, the entropy-weighted portfolio outperforms the market.

To discuss Corollaries <ref> and <ref>,
it is convenient to extend our discussion of Γ^*_μ given in Section <ref>
to more general Γ^*_π.
Let us now rewrite twice the excess growth term (<ref>),
2Γ^*_π, as

    2Γ^*_π(t)
      =
      ∫_0^t
      ∑_j=1^J
      π_j(μ(s))
      [lnμ_j](s)
      -
      [
        ∑_j=1^J
        ∫π_j(μ)
        lnμ_j
      ](t).

Define (using our fixed language) a sequence of partitions T^1,T^2,…
that is fine for all processes used in this paper and set,
for a given partition T^n=(T^n_k)_k=0^∞,

    μ_j,k   := μ_j(T^n_k∧ t),     k=0,1,…,
    Δlnμ_j,k   := lnμ_j,k - lnμ_j,k-1,     k=1,2,…,
    π_j,k   := π(μ_j,k),     k=0,1,…,

with the dependence on n suppressed.
We can then regard

    2Γ^*,n_π(t)
      :=
      ∑_k=1^∞∑_j=1^J
      π_j,k-1Δμ_j,k^2
      -
      ∑_k=1^∞(
        ∑_j=1^J
        π_j,k-1Δlnμ_j,k)^2

as the nth approximation to 2Γ^*_π(t);
it can be shown that

    2Γ^*,n_π(t)
      →
      2Γ^*_π(t)
        ucqa.

Rewriting (<ref>) as

    2Γ^*,n_π(t)
      =
      ∑_k=1^∞∑_j=1^J
      π_j,k-1(
        Δμ_j,k
        -
        ∑_i=1^J
        π_i,k-1Δlnμ_i,k)^2,

we can see that this expression is the cumulative variance of the logarithmic returns Δlnμ_j,k
over the time interval [T^n_k-1∧ t,T^n_k∧ t]
w.r. to the “portfolio probability measure” Q({j}):=π_j,k-1.
This makes the expression (<ref>) very intuitive:
the excess growth rate of the portfolio π over the naive expression
is determined by the volatility of the market weights w.r. to π.

As already mentioned,
the stochastic versions of Corollaries <ref> and <ref>
have been used for active portfolio management <cit.>.
The remarks made above about the relation between Corollaries <ref> and <ref>
are also applicable to Corollaries <ref> and <ref>;
the latter replaces the first addend on the right-hand side of (<ref>) by its lower bound.
Corollary <ref> decomposes the growth in the value of the diversity-weighted portfolio into two components,
one related to the growth in the diversity _p(μ) of the market weights
and the other related to the accumulation of the diversity-weighted variance of the market weights.
Corollary <ref> ignores the first component,
which does not make it vacuous since _p is bounded,
always being between 1 (corresponding to a market concentrated in one stock)
and J^(1-p)/p (corresponding to a market with equal capitalizations of all J stocks).

Several explanations have been suggested for the somewhat counterintuitive disjunction
stated at the beginning of this section:


  * If we include all stocks traded in a real-world market in our model,
  perhaps making J very large,
  the portfolio (<ref>)
  and its special cases (<ref>), (<ref>), and (<ref>)
  (particularly the last two) will not be efficient
  since they will be forced to invest into smaller and so less liquid stocks;
  it is known that portfolios generated by measures of diversity
  invest into smaller stocks more heavily than the market portfolio does
  <cit.>.

  * There is another explanation related to this common feature of the portfolios discussed in this paper that outperform the market
  (increased weights of smaller stocks as compared with the market).
  Over the last decades, such portfolios have been adversely affected by the tendency of larger companies
  to pay higher dividends
  (cf., e.g., <cit.>,
  describing the performance of an index that has been used in investment practice).
  The role of differential dividend rates in maintaining market diversity
  is emphasized in <cit.>.

  * If we restrict our attention only to J largest stocks traded in a real-world market,
  for a moderately large J (such as J=500 for S&P 500),
  the performance of portfolios such as (<ref>), (<ref>), or (<ref>)
  w.r. to this smaller “market” (which is now, in fact, a large cap market index)
  will be affected by the phenomenon of “leakage”
  <cit.>.




§ FERNHOLZ'S MASTER MARTINGALE AND STROOCK–VARADHAN MARTINGALES



One way to restate Theorem <ref> is to say that

    (μ(t))/(μ(0))exp(
        -1/2∑_i,j=1^J
        ∫_0^t
          D_ij(μ)/(μ)[μ_i,μ_j]
      )

is a value process q.a.;
in the terminology of <cit.>, it is a continuous martingale.
In this section we will see that Fernholz's master martingale (<ref>)
is in fact a very natural object,
and not just a product of formal manipulations with the Itô formula,
as might have appeared from its derivation in Section <ref>.
Connections with recent papers <cit.> and <cit.>
will be discussed later in the section.

Let f be a C^2 function defined on an open neighbourhood f of Δ^J in ^J.
The non-stochastic Itô formula <cit.> implies that

    f(μ(t))
      -
      f(μ(0))
      -
      1/2∑_i,j=1^J
      ∫_0^t
        D_ij f(μ)
      [μ_i,μ_j]
      =
      ∑_j=1^J
      ∫_0^t
        D_j f(μ)
      μ_j
        q.a.,

and so the left-hand side of (<ref>) is a continuous martingale,
which we will refer to as the Stroock–Varadhan martingale <cit.>;
it is a non-stochastic version of the classical martingales used by Stroock and Varadhan
in their study of diffusion processes.

Fernholz's master martingale (<ref>)
is the Doléans exponential of the Stroock–Varadhan martingale on the left-hand side of (<ref>)
for f:=ln.
Indeed, applying (<ref>) gives the Doléans exponential

    (μ(t))/(μ(0))exp(
        -
        1/2∑_i,j=1^J
        ∫_0^t
          D_ij f(μ)
        [μ_i,μ_j]
    
        -
        1/2∑_i,j=1^J
        ∫_0^t
          D_i f(μ)
          D_j f(μ)
        [μ_i,μ_j]
      )

of the left-hand side of (<ref>),
which is equal, by the identity

    D_ij f
      =
      D_ij/
      -
      D_i/D_j/
      =
      D_ij/
      -
      D_if
      D_jf,

to (<ref>).
If we regard the Stroock–Varadhan martingale to be an additive process,
Fernholz's master martingale (<ref>) becomes its multiplicative counterpart.
“Additive” and “multiplicative” is the terminology used in <cit.>
(more carefully than in this paper),
and the relation between the Stroock–Varadhan martingale on the left-hand side of (<ref>)
and Fernholz's master martingale (<ref>)
is somewhat analogous to the relation between the additive Bachelier formula <cit.>
and the multiplicative Black–Scholes formula <cit.> in option pricing.

The papers <cit.> and <cit.> study additive portfolio generation in depth.
In particular, these papers give numerous interesting examples
(including (<ref>) below).

We can rewrite (<ref>) in Corollary <ref> as

    Z_π(t)
      ≥1/2exp(
        1/2∑_j=1^J
        [μ_j](t)
      )
        q.a.,

which implies

    Z_π(τ_A)
      ≥1/2
      e^A/2  q.a.,

where A is a positive constant, τ_A is the stopping time

    τ_A
      :=
      min{
        t ∑_j [μ_j](t) = A
      },

and Z_π(∞):=∞.

Qualitatively, (<ref>) means that the market satisfies Fernholz's arbitrage-type property:
we can beat a non-degenerate market (interpreting non-degeneracy as τ_A<∞ for all A).
The Stroock–Varadhan martingale on the left-hand side of (<ref>) also gives a Fernholz-type arbitrage,
which is, however, polynomial (and even linear) in A, unlike (<ref>).
Setting

    f(x)
      :=
      -
      1/2∑_j=1^J
      x_j^2

(cf. (<ref>)),
we can rewrite the continuous martingale on the left-hand side of (<ref>) as

    Y(t)
      :=
      1/2∑_j=1^J
      μ_j(0)^2
      -
      1/2∑_j=1^J
      μ_j(t)^2
      +
      1/2∑_j=1^J
      [μ_j]
      ≥
      -1/2
      +
      1/2∑_j=1^J
      [μ_j];

therefore, X:=2Y+1 is a nonnegative continuous martingale satisfying X(0)=1 and

    X(τ_A)
      ≥
      A.

Therefore, this is an alternative method for achieving the same qualitative goal,
X(τ_A)→∞ as A→∞.
Quantitatively the result might appear weaker;
after all, we lose the exponential growth rate in A.
However, there is a range of A (roughly between 0.7 and 4.3)
where the Stroock–Varadhan martingale X performs better:
see Figure <ref>.





§ CONCLUSION



Figure <ref> gives two functions g such that a final capital of g(A) is achievable at time τ_A.
It would be interesting to characterize the class of such functions g.
A related question is:
what is the best growth rate of g(A) as A→∞?
This question can be asked in both stochastic and non-stochastic settings.
These are some directions of further research for non-stochastic theory:


  * A natural direction is to try and strip other results of stochastic portfolio theory
  of their stochastic assumptions.
  First of all, it should be possible to extend Theorem <ref> to functions 
  that are not smooth (as in <cit.>);
  the existence of local time in a non-stochastic setting is shown in <cit.>
  and, in the case of continuous price paths, can be deduced from the main result of <cit.>.

  * Another direction is to extend this paper's results to general numéraires
  (this paper uses the value of the market portfolio as our numéraire).

  * Finally, it would be very interesting to extend some of the results to càdlàg price paths.




 §.§ Acknowledgements


The first draft of this paper was prompted by discussions with Martin Schweizer and Dániel Bálint in December 2017.
I am grateful to Ioannis Karatzas for information on the existing literature
(in particular, he drew my attention to connections between Section <ref> of this paper
and additive portfolio generation in <cit.> and <cit.>)
and for his advice on presentation, notation, and terminology.
Thanks to Peter Carr, Marcel Nutz, Philip Protter, and Glenn Shafer for useful comments.

10

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§ CONNECTIONS WITH THE FOUNDATIONS OF GAME-THEORETIC PROBABILITY


In this appendix we will see yet another method of achieving the qualitative goal
of lim_A→∞X(τ_A)=∞ for a nonnegative supermartingale X.
The result will be weaker than both functions in Figure <ref>,
but it will shed light on a seemingly paradoxical feature of continuous-time game-theoretic probability.

The method uses the non-stochastic Dubins–Schwarz theorem presented in <cit.>
and is based on the following apparent paradox,
which we first discuss informally.
As agreed in Section <ref>,
we regard μ_1,…,μ_J,1 as tradable securities.
According to the non-stochastic Dubins–Schwarz theorem and a standard property of Brownian motion,
with very high lower probability all J securities will eventually hit zero
if their volatility is appreciable.
When this happens, the normalized value of the market μ_1+⋯+μ_J will be 0 rather than 1,
which is impossible.
Therefore, we expect an event of a low upper game-theoretic probability to happen,
i.e., we expect to be able to outperform the market.
This is formalized in the following statement:


  For any constant A>0, there is a nonnegative supermartingale X such that X(0)=1 and
  
    X(τ_A)
        ≥
        1.25
        J^-3/2 A^1/2  q.a.,

  where τ_A is the stopping time (<ref>)
  and X(∞) is interpreted as ∞.



  For each j∈{1,…,J},
  we will construct a nonnegative continuous martingale X_j satisfying X_j(0)=1 and
  
    X_j(τ_j)
        ≥
        1.25
        (A/J)^1/2  q.a.,

  where
  
    τ_j
        :=
        min{
          t  [μ_j](t) = A/J
        }.

  (In this case we can set X to the average of all J of X_j stopped at time τ_j.)
  According to <cit.>,
  the probability that a Brownian motion started from 1
  (in fact μ_j is started from μ_j(0)≤ 1)
  does not hit zero over the time period A/J is
  
    1 - √(2/π)∫_(J/A)^1/2^∞ e^-x^2/2 x
        =
        √(2/π)∫_0^(J/A)^1/2 e^-x^2/2 x
        ≤√(2/π) (J/A)^1/2.

  In combination with the non-stochastic Dubins–Schwarz result <cit.>
  applied to μ_j,
  this gives (<ref>) with
  
    √(π/2)
        >
        1.25

  in place of 1.25.


The processes Z_π in (<ref>) and X in (<ref>) are nonnegative supermartingales
in the sense of <cit.>
(in fact, nonnegative continuous martingales).
On the other hand, the process X in (<ref>)
is a nonnegative supermartingale in the sense of the more cautious definitions in <cit.>.
This can be regarded as advantage of (<ref>) over (<ref>) and (<ref>).
However, a disadvantage of (<ref>)
is that quantitatively it is much weaker than both (<ref>) and (<ref>);
the right-hand side of (<ref>) is always smaller than the right-hand side of (<ref>),
and it is greater than the right-hand side of (<ref>) only for a small range of A
(approximately A∈(0,0.2) for J=2).
