^{1}

^{2}

^{1}

^{2}

Under the notable Issacs's condition on the Hamiltonian, the existence results of a saddle point are obtained for the stochastic recursive zero-sum differential game and mixed differential game problem, that is, the agents can also decide the optimal stopping time. The main tools are backward stochastic differential equations (BSDEs) and double-barrier reflected BSDEs. As the motivation and application background, when loan interest rate is higher than the deposit one, the American game option pricing problem can be formulated to stochastic recursive mixed zero-sum differential game problem. One example with explicit optimal solution of the saddle point is also given to illustrate the theoretical results.

The nonlinear backward stochastic differential equations (BSDEs in short) had been introduced by Pardoux and Peng [

BSDE plays an important role in the theory of stochastic differential game. Under the notable Isaacs's condition, Hamadène and Lepeltier [

Then, in Section

In fact, the recursive (mixed) zero-sum game problem has wide application background in practice. When the loan interest rate is higher than the deposit one. The American game option pricing problem can be formulated to the stochastic recursive mixed game problem in our Section

In this section, we will study the existence of the stochastic recursive zero-sum differential game problem using the result of BSDEs.

Let

for any stopping time

The

for any

for any

there exists a constants

Then, the equation

Now, we consider a compact metric space

for any

for any

there exists a constant

there exists a constant

For

Thanks to Girsanov's theorem, under the probability

Suppose that we have a system whose evolution is described by the process

In order to define the payoff, we introduce two functions

For

We suppose now that the Isaacs' condition is satisfied. By a selection theorem (see Benes [

Thanks to the assumption of

Now we give the main result of this section.

We consider the following BSDE:

For any

Now, we study the stochastic recursive mixed zero-sum differential game problem. First, let us briefly describe the problem.

Suppose now that we have a system, whose evolution also is described by

The problem is to find a saddle point strategy (one should say a fair strategy) for the controllers, that is, a strategy

Like in Section

From the result in [

One defines

Then

It is easy to know that the reflected BSDE (

We know that

Next, let

The payoff

In the same way, we can show that

Finally, let us show that the value of the game is

In this section, we present two examples to show the applications of Section

The first example is about the American game option pricing problem. We formulate it to be one stochastic recursive mixed game problem. This can be regarded as the application background of our stochastic game problem.

American game option when loan interest is higher than deposit interest is shown.

In El Karoui et al. [

We suppose that the investor is allowed to borrow money at time

An American game is a contract between a broker

We consider the problem of pricing an American game contingent claim at each time

We formulate the pricing problem of American game option to the stochastic recursive mixed zero-sum differential game problem which was studied in Section

In the following, we give another example, where we obtain the explicit saddle point strategy and optimal value of the stochastic recursive game. The purpose of this example is to illustrate the application of our theoretical results.

We let the dynamics of the system

First, we consider the following stochastic recursive zero-sum differential game:

Second, we consider the following stochastic recursive mixed zero-sum differential game:

We also can get the conclusion that the optimal game value

This work is supported by the National Natural Science Foundation of China (no. 10921101, 61174092), the National Science Fund for Distinguished Young Scholars of China (no. 11125102), and the Special Research Foundation for Young teachers of Ocean University of China (no. 201313006).