(I did not read correctly before. Your equation is a non-linear pde.)

We have NIntegrate[ f, min, max ] so R[s]] * w[s] and R[s]^2 must each evaluate to being a list {?,min,max} as the return of the function (so to be input as the same in recursion): but i can say they will not evaluate to that (they should return as a single value if the integral evaluates). And NItegrate must evaluate (it is not symbolic it intends to return values).

Without knowing what your expecting to get I'm unsure how to comment on helping to set up a recursion other than the issues stated. A little comment on the problem type and method would be helpful, as others already said.

What does step parameter mean?

To calculate v, you need R and to compute R v is necessary.

Thank you. I have worked on my code. But it stil doesn't work. Here is the code

    pde = (-3/2) * 1* y[x, t] * D[y[x, t], t] + 
       1/(4*y[x, t]) * D[D[y[x, t], t], t] - D[y[x, t], x] == 0

sol2 = NDSolve[{pde, y[0, t] == Sqrt[(1.5 + Tanh[t - 100])^2]}, 
  y[x, t], {x, 0, 1}, {t, 0, 40}]

data = RandomFunction[WienerProcess[], {0, 10, 0.01}]

w[t_] = data["PathFunction"] [t]

f[x_, t_] = y[x, t] /. sol2[[1]]

h[x_, t_] = f[x^2, t] * x

Block[{$RecursionLimit = 1000}, 
 R[t_?NumberQ] := 
  NIntegrate[h[v[s], R[s]] * w[s], {s, 0, t}] + v[t] * w[t] - 
   NIntegrate[h[v[s], R[s]] ^2, {s, 0, t}]]; 

Block[{$RecursionLimit = 
   1000}, v[t_?NumberQ] := 
  NIntegrate[Exp[-f[v[s]^2, R[s]]], {s, 0, t}]];